HOMEOMORPHISMS OF ONE–DIMENSIONAL
HYPERBOLIC ATTRACTORS
by
James Leo Jacklitch
A dissertation submitted in partial fulfillment
of the requirements for the degree
of
Doctor of Philosophy
in
Mathematical Sciences
MONTANA STATE UNIVERSITY
Bozeman, Montana
April 2001
ii
APPROVAL
of a dissertation submitted by
James Leo Jacklitch
This dissertation has been read by each member of the dissertation committee and
has been found to be satisfactory regarding content, English usage, format, citations,
bibliographic style, and consistency, and is ready for submission to the College of
Graduate Studies.
Marcy Barge
(Signature)
Date
Approved for the Department of Mathematical Sciences
John Lund
(Signature)
Date
Approved for the College of Graduate Studies
Bruce McLeod
(Signature)
Date
iii
STATEMENT OF PERMISSION TO USE
In presenting this dissertation in partial fulfillment of the requirements for a
doctoral degree at Montana State University, I agree that the Library shall make it
available to borrowers under rules of the Library. I further agree that copying of
this dissertation is allowable only for scholarly purposes, consistent with “fair use” as
prescribed in the U. S. Copyright Law. Requests for extensive copying or reproduction
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Signature
Date
iv
ACKNOWLEDGEMENTS
I would like to thank my committee members for their time, suggestions, and
comments. I would also like to thank my friends and family for their support. Lastly,
I would like to especially thank my advisor, Dr. Marcy Barge, for his guidance and
patience, without which, this manuscript would not have been possible.
v
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Finite Topological Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inverse Limit Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Matchbox Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
11
14
3. MAIN RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Orientable Bouquets of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Orientable Finite Connected Graphs without Endpoints . . . . . . . . . . . . . . . . . . . . .
Non-Orientable Finite Connected Graphs without Endpoints . . . . . . . . . . . . . . .
Finite Connected Graphs without Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Two Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
One-Dimensional Hyperbolic Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
36
43
53
54
55
55
56
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
vi
LIST OF FIGURES
Figure
1. A star U with arc structure A(U ) = {I1 , I2 , I3 , I4 } . . . . . . . . . . . . . . . . . . . . . . .
2. A non-orientable graph G with corresponding orientable covering
e ertw wert wert wert wret wert wert wert wert wert wert
space G
wert wert tre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page
6
44
vii
ABSTRACT
Williams has shown that one-dimensional connected hyperbolic attractors can be
realized as the inverse limits of bouquets of circles with single bonding maps that fix
branch points. Suppose f : X → X and g : Y → Y are diffeomorphisms on manifolds
with one-dimensional connected hyperbolic attractors Λ and Γ, respectively. Let A
and B denote the transition matrices for the bonding maps on the bouquets of circles
whose inverse limits correspond to X and Y with spectral radii µA and µB . We show
that if Λ and Γ are orientable and homeomorphic then A and B are weakly equivalent.
We also show that if Λ and Γ are non-orientable and homeomorphic then there exist
Perron numbers α and β and positive integers m and n such that αµA = µm
B and
βµB = µnA . These two results together imply, the weaker result, if Λ is homeomorphic
to Γ and the topological entropies htop (f |Λ ) = log(µA ) and htop (g|Γ ) = log(µB ) then
the algebraic extension fields Q(µA ) and Q(µB ) are identical.
1
CHAPTER 1
INTRODUCTION
One important area in smooth dynamical systems is the study of structurally
stable diffeomorphisms. In fact, one of the main problems introduced by Smale in
his paper Differentiable Dynamical Systems [22] was the classification of structurally
stable diffeomorphisms.
J. Palis and S. Smale made two fundamental conjectures in this area which were
subsequently shown to be valid. The first was the Structural Stability Conjecture
which states that a C 1 diffeomorphism is structurally stable if and only if it satisfies
Axiom A and strong transversality. Robinson [20] showed the sufficiency and Mañé
[14] showed the necessity of this conjecture. necessity of this conjecture. The second
was the Ω-Stability Conjecture which states that a C 1 diffeomorphism is Ω-stable if
and only if it satisfies Axiom A and has no cycles. Smale [22] showed the sufficiency
and Palis [19] showed the necessity of this conjecture.
One theorem, due to Smale [21], concerning the non-wandering sets of C 1 Axiom
A diffeomorphisms is the Spectral Decomposition Theorem which states that if f
is a C 1 Axiom A diffeomorphism then there is a unique way of writing the nonwandering set Ω(f ) as the finite union of disjoint, closed, invariant, indecomposable
subsets Ω(f ) = Ω1 ∪ . . . ∪ Ωk in such a way that on each subset f is topologically
2
transitive. The simplest subsets that appear in the spectral decomposition of an
Ω-stable diffeomorphism, after isolated periodic sinks, are expanding attractors.
R. F. Williams realized an important tool for the study of expanding attractors
was inverse limits. Williams [24] showed that if a diffeomorphism F has a onedimensional hyperbolic attractor Λ then F |Λ is topologically conjugate to the shift
homeomorphism fˆ on an inverse limit of a branched one-manifold G with a single
bonding map f satisfying certain properties. In the same paper, he proved a converse
for diffeomorphisms on S 4 . Williams [26] showed similar results for all diffeomorphisms with expanding attractors.
This relationship between expanding attractors and inverse limits sparked interest
in the investigation of inverse limits of finite graphs in hopes that by studying the
dynamical properties of the bonding maps that topological properties of the inverse
limit spaces could be obtained, and vice versa.
One such result, relating the dynamics of a map on a finite graph and the topological properties of the associated inverse limit space, is the following due to Barge
and Diamond [3]. Suppose f : G → G is a piecewise monotone map on a finite graph
G. The following are equivalent: the topological entropy of f is positive; f has a
horseshoe; there are positive integers r and M such that f has a periodic point of
prime period rm for all m ≥ M ; and the inverse limit space lim{G, f } contains an
←−
indecomposable subcontinuum.
3
Barge and Diamond [4] introduced the relation of weak equivalence for nonnegative square integral matrices to study the inverse limits of finite graphs. Barge,
Jacklitch, and Vago [7] then introduced the stronger relation of weak equivalence for
substitutions to further study the inverse limits of finite graphs.
The two main results of this dissertation are the following:
Theorem 1. Let G and G0 be orientable bouquets of circles with branch points
b and b0 , respectively, let f : G → G and f 0 : G0 → G0 be collapsing surjective
immersions such that f (b) = b and f 0 (b0 ) = b0 , let Mf be the transition matrix for
f relative to an ordering of the components of G\{b}, and let Mf 0 be the transition
matrix for f 0 relative to an ordering of the components of G0 \{b0 }. If lim{G, f } is
←−
homeomorphic to lim{G0 , f 0 } then Mf is weakly equivalent to Mf 0 .
←−
Theorem 2. Let G and G0 be finite connected graphs with branch points but
without end points and let f : G → G and f 0 : G0 → G0 be aperiodic collapsing
surjective immersions such that the branch sets of the graphs are invariant under the
immersions. If lim{G, f } is homeomorphic to lim{G0 , f 0 } and the topological entropies
←−
←−
htop (f ) = log(λf ) and htop (f 0 ) = log(λf 0 ) then there exist Perron numbers α and β
n
and positive integers m and n such that αλf = λm
f 0 and βλf 0 = λf .
The second theorem includes collapsing surjective immersions on bouquets of
circles which fix branch points since they are automatically aperiodic. Both theorems
give us corresponding results for one-dimensional connected hyperbolic attractors by
results of Williams [24], [25].
4
CHAPTER 2
PRELIMINARIES
Finite Topological Graphs
A compactum is a nonempty compact metrizable space and a continuum is a
connected compactum.
Suppose x is a point in a one-dimensional compactum K. If x is an end point of
every arc in K containing x then x is called an end point of K. If K contains a simple
n-od (n ≥ 3) with vertex x then x is called a branch point of K. The set of all end
points of K will be denoted by E(K) and the set of all branch points of K will be
denoted by B(K).
A finite graph is a compactum which can be written as the union of finitely many
arcs each pair of which intersect in at most a common end point. We will call a
finite connected graph without endpoints, having only one branch point, a bouquet of
circles. By definition, a finite graph is locally connected and locally path connected.
Suppose M is a metric space with metric d. The metric d is called Menger-convex
if for every pair of distinct points x, z ∈ M there exists a point y ∈ M \{x, z} such
that d(x, y) + d(y, z) = d(x, z). The following result is due to Bing [8] and Moise [17],
independently.
5
Theorem 2.1. Every locally connected continuum admits a compatible Mengerconvex metric.
Since every finite connected graph is a locally connected continuum, it admits a
compatible Menger-convex metric, by the above theorem.
If U is an arc or simple n-od in a compactum K such that U \E(U ) is open in K
then we will call U a star in K. If U is a star in a finite graph without end points
then Int(U ) = U \E(U ).
Proposition 2.2. If V is an open neighborhood of x in a finite graph graph G
without end points then there exists a star neighborhood U of x such that U ⊂ V .
The following theorem is due to Menger [15].
Theorem 2.3. Given any two distinct points in a compact Menger-convex metric
space, there exists an arc between the points such that the length of the arc in the
metric is the same as the distance between the points.
Suppose (M, d) is a compact Menger-convex metric space. By the above theorem,
the distance between any two distinct points is the same as the length of the shortest
arc between the points. So Bd (x, ²) is arc connected for each x ∈ M and ² > 0.
Suppose U is a star in a finite graph G and A(U ) is a finite collection of distinct
arcs in G whose union is U . If the end points of U can be partitioned into two subsets
such that:
i) given an arc in A(U ) the end points of that arc are not contained in the same
element of the partition,
6
ii) given a point from each element of the partition, the arc between the points
that is contained in U is contained in A(U )
then A(U ) is called an arc structure on U. If U has n end points then there are
2n−1 − 1 different arc structures that can be placed on U . An arc structure on U
naturally induces an arc structure on any star contained in U .
`1
`1
r1
r1
r1
`1
`2
PSfrag replacements
U
I1
`2
I3
r2
I2
r2
`2
I4
r2
Figure 1. A star U with arc structure A(U ) = {I1 , I2 , I3 , I4 }.
Let G be a finite graph without end points. Since the collection of all stars in G
with end points removed forms a basis for the topology on G, we only need to work
with these kind of neighborhoods when studying G.
Suppose U and V are arc structured stars in G. Since a nondegenerate component
of the intersection of two stars in G is a star, the arc structures on U and V induce
arc structures on each nondegenerate component of U ∩ V . If the arc structures on
each nondegenerate path component of U ∩ V , induced by the arc structure on U
and the arc structure on V , are the same then U and V are said to have coherent arc
structures. A choice of coherent arc structures on each star contained in G is called
an arc structure system for G.
7
Suppose U is a arc structured star in G with arc structure A(U ). If ρ : U → R
is a map, by which we always mean a continuous function, such that ρ|I : I → R is a
topological embedding for each arc I ∈ A(U ) then we will call ρ a directing map on U.
Two directing maps, ρ1 and ρ2 , on U are called equivalent if for each arc I ∈ A(U ) the
homeomorphism (ρ2 |I ) ◦ (ρ1 |I )−1 : ρ1 (I) → ρ2 (I) is increasing. This is an equivalence
relation that partitions the collection of all directing maps on U into two equivalence
classes, each of which we will call a direction for U. We will call an arc structured
star together with a direction for it a directed star.
If U is a directed star then the direction for U naturally induces a direction for
any star contained in U .
Suppose U and V are directed stars. If the directions for each nondegenerate
path component of U ∩ V , induced by the direction for U and the direction for V ,
are the same then we say U and V are coherently directed.
Suppose ρ1 is a directing map on a star U1 in G and ρ2 is a directing map on a
star U2 in G. If x ∈ Int(U1 ) ∩ Int(U2 ) and there exists a star neighborhood V of x
contained in U1 ∩ U2 such that ρ1 |V and ρ2 |V are equivalent then ρ1 and ρ2 are said
to be equivalent near x. This is an equivalence relation that partitions the collection
of all directing maps on all star neighborhoods of x into two equivalence classes, each
of which is called a direction at x. We will let Dx denote the collection of these two
directions at x. The direction for a directed star naturally induces a direction at each
point in the interior of that star.
8
Suppose (U, DU ) and (V, DV ) are directed stars in G. If x ∈ Int(U ) ∩ Int(V )
and there exists a star neighborhood W of x contained in U ∩ V such that the
directions for W induced by DU and DV are equal then we say (U, DU ) and (V, DV )
are coherently directed near x. If (U, DU ) and (V, DV ) are coherently directed near x
for each x ∈ Int(U ) ∩ Int(V ) then (U, DU ) and (V, DV ) are coherently directed.
Suppose G is endowed with an arc structure system. We say G is orientable if
there exists a collection of coherently directed stars whose interiors cover G. If G is
orientable then a choice of coherently directed stars whose interiors cover G induces a
direction at each point of G. This continuous choice of directions at each point of G
is called an orientation of G. If G is connected and orientable then there exist exactly
two distinct orientations of G.
Suppose G0 is finite graph without end points endowed with an arc structure
system. We call a map f : G → G0 an immersion if for each point x in G and each
star neighborhood V of f (x) in G0 there exists a star neighborhood U of x in G such
that the restriction of f to each arc in A(U ) is a topological embedding into an arc
in A(V ). We call an immersion f : G → G0 collapsing if for each point x in G and
each star neighborhood V of f (x) in G0 there exists a star neighborhood U of x in G
such that the image of U under f is contained in an arc in A(V ).
If f : G → G is an immersion such that f (B(G)) = B(G) then B(G) ⊂ f −1 (B(G))
and f is one-to-one on each component of G\f −1 (B(G)). Thus f is a Markov map
and we can associate a matrix to f relative to an ordering of the components of
9
G\B(G) in the following manner. The transition matrix for f relative to the ordered
components, {J1 , . . . , Jm }, of G\B(G) is the m × m matrix whose ij-th entry is the
number of times the component Jj covers the component Ji under f , that is, the
number of components of G\f −1 (B(G)) in Jj that map onto Ji . If A and B are
transition matrices for f relative to different orderings of the components of G\B(G)
then A is similar to B by a permutation matrix.
We define λf to be the spectral radius of the transition matrix for f relative to
an ordering of the components of G\B(G). This definition is independent of which
ordering is chosen by the similarity remark made above.
It is a well known result, due to Block, Guckenheimer, Misiurewicz, and Young
[9], that the topological entropy of a Markov map on an interval or circle is equal to
the logarithm of the spectral radius of an associated transition matrix. We can easily
extend this result to Markov maps on finite graphs using similar techniques. Thus
htop (f ) = log(λf ).
The following result about topological entropies is due to Bowen [10].
Theorem 2.4. Let f : X → X and g : Y → Y be maps where X and Y
are compacta. If k : X → Y is a uniformly finite-to-one surjective map such that
k ◦ f = g ◦ k then htop (f ) = htop (g).
Two maps f : X → X and g : Y → Y are called shift equivalent provided there
are maps r : X → Y and s : Y → X such that r ◦ f = g ◦ r, f ◦ s = s ◦ g, s ◦ r = f m ,
and r ◦ s = g m for some positive integer m.
10
Lemma 2.5. If G and G0 are finite graphs and f : G → G and f 0 : G0 → G0 are
surjective immersions such that f is shift equivalent to f 0 then htop (f ) = htop (f 0 ).
Proof. Since f is shift equivalent to f 0 , there exist maps r : G → G0 and
s : G0 → G such that r ◦ f = f 0 ◦ r, f ◦ s = s ◦ f 0 , s ◦ r = f m , and r ◦ s = (f 0 )m for
some positive integer m. Since G and G0 are compact and f m and (f 0 )m are surjective
immersions, f m = s ◦ r and (f 0 )m = r ◦ s are uniformly finite-to-one surjective maps.
Then r and s are uniformly finite-to-one surjective maps. Thus htop (f ) = htop (f 0 ) by
Theorem 2.4.
A square nonnegative integer matrix A is called aperiodic if there exists a positive
integer m such that Am is a positive integer matrix. We will call f aperiodic if
the transition matrix for f relative to an ordering of the components of G\B(G) is
aperiodic. This definition is again independent of which ordering is chosen by the
earlier similarity remark.
A real number λ ≥ 1 is called a Perron number if it is an algebraic integer that
strictly dominates all of its other algebraic conjugates. The spectral radius of an
aperiodic matrix is a Perron number. Thus, if f is aperiodic then λf is a Perron
number.
We will say two square nonnegative integer matrices, A and B, are weakly equiv∞
alent if there exist sequences of positive integers, {ni }∞
i=1 and {mi }i=1 , and sequences
∞
ni
of nonnegative integer matrices, {Si }∞
and
i=1 and {Ti }i=1 , such that Si Ti = A
11
Ti Si+1 = B mi for all i ∈ N. If Aj and B k are weakly equivalent for positive integers j and k then A and B are weakly equivalent by definition. The following
theorem is due to Barge and Diamond [4].
Theorem 2.6. Let A and B be aperiodic nonnegative integer matrices with spectral radii λA and λB , respectively. If A is weakly equivalent to B then there are
Perron numbers, α and β, and positive integers, m and n, such that αλA = λm
B and
βλB = λnA .
It should be noted that this Perron relationship between spectral radii is weaker
than weak equivalence of the matrices. We provide an example of this in the next
chapter.
Inverse Limit Spaces
An inverse sequence of spaces is a double sequence {Xi , fi }∞
i=1 of spaces Xi , called
coordinate spaces, and maps fi : Xi+1 → Xi , called bonding maps. If {Xi , fi }∞
i=1
is an inverse sequence of spaces then the inverse limit of {Xi , fi }∞
i=1 , denoted by
lim{Xi , fi }∞
i=1 , is the subspace of the product space
←−
Q∞
{(xi )∞
i=1 ∈
i=1 Xi | fi (xi+1 ) = xi for each i ∈ N}.
Q∞
i=1
Xi defined by lim{Xi , fi }∞
i=1 =
←−
Theorem 2.7. If {Xi , fi }∞
i=1 is an inverse sequence of compacta (continua) then
lim{Xi , fi }∞
i=1 is a compactum (continuum).
←−
Proof. See [18].
12
Let X be a space and f : X → X be a map. If πj :
Q∞
i=1
X → X is the projection
map of the product space into the j-th coordinate space, defined by πj ((xi )∞
i=1 ) = xj ,
then πj |lim{X,f } : lim{X, f } → X is a map. It can be shown that if f is a surjective
←−
←
−
map then πj |lim{X,f } is a surjective map. Although we should write πj |lim{X,f } when
←
−
←
−
referring to the j-th projection map acting on the inverse limit space lim{X, f }, we
←−
will write πj without this restriction for brevity. It can be shown that the collection
{πi−1 (U ) | U is an open set in X and i ∈ N} is a basis for the subspace topology on
lim{X, f }.
←−
Let Y be a space, let g : Y → Y be a map, and let {pi }∞
i=1 be a collection of maps
{X, f } then
pi : X → Y such that pi ◦ f = g ◦ pi+1 for each i ∈ N. If (xi )∞
i=1 ∈ lim
←−
Q
∞
∞
{Y, g} since g(pi+1 (xi+1 )) = pi (f (xi+1 )) = pi (xi )
( ∞
i=1 pi )((xi )i=1 ) = (pi (xi ))i=1 ∈ lim
←−
Q
for each i ∈ N. So ( ∞
{X, f } into lim{Y, g}. We will call
i=1 pi )|lim{X,f } maps lim
←−
←−
←
−
Q
p∞ ≡ ( ∞
{X, f } → lim{Y, g} the ladder map induced by {pi }∞
i=1 .
i=1 pi )|lim{X,f } : lim
←−
←−
←
−
If p∞ : lim{X, f } → lim{Y, g} is the ladder map induced by {pi }∞
i=1 and πi and
←−
←−
σi are the i-th projection maps acting on lim{X, f } and lim{Y, g}, respectively, then
←−
←−
pi ◦ πi = σi ◦ p∞ for each i ∈ N. Note that p∞ is the only map from lim{X, f } into
←−
lim{Y, g} such that pi ◦ πi = σi ◦ p∞ for each i ∈ N.
←−
Theorem 2.8. Let X and Y be compacta, let f : X → X and g : Y → Y be
maps, and let {pi }∞
i=1 be a collection of maps pi : X → Y such that pi ◦ f = g ◦ pi+1
is
for each i ∈ N. If each pi is an open k-to-one surjective map such that f |p−1
i ({y})
13
a one-to-one map for each y ∈ pi (X) then the ladder map induced by {pi }∞
i=1 is an
open k-to-one surjective map.
Proof. Assume p∞ : lim{X, f } → lim{Y, g} is the ladder map induced by
←−
←−
is a
{pi }∞
i=1 where each pi is an open k-to-one surjective map such that f |p−1
i ({y})
one-to-one map for each y ∈ pi (X).
p∞ is surjective: Let (yi )∞
{Y, g}. Since pi is a surjective map, {yi } is
i=1 ∈ lim
←−
closed in Y , and X is a compactum, the subspace pi−1 ({yi }) of X is a compactum.
Then lim{p−1
({yi }), fi |p−1
}∞
i=1 is a compactum, by Theorem 2.7, which maps
i+1 ({yi+1 })
←− i
onto (yi )∞
i=1 .
p∞ is k-to-one: Let (yi )∞
{X, f }) = lim{Y, g}. Since yi ∈ pi (X) = Y
i=1 ∈ p∞ (lim
←−
←−
−1
k
and pi is k-to-one, p−1
i ({yi }) has k elements. Suppose {x1,j }j=1 = p1 ({y1 }). Since
−1
−1
pi ◦ f = g ◦ pi+1 , f (p−1
i+1 ({yi+1 })) ⊂ pi ({yi }). Since pi ({yi }) has k elements for each
−1
i ∈ N and f |p−1
is a one-to-one map, f (p−1
i+1 ({yi+1 })) = pi ({yi }). For each
i+1 ({yi+1 })
−1
xi,j ∈ p−1
i ({yi }), let xi+1,j be the unique element of pi+1 ({yi+1 }) such that f (xi+1,j ) =
∞
∞ k
xi,j . Then p−1
∞ ({(yi )i=1 }) = {(xi,j )i=1 }j=1 .
p∞ is open: For i ∈ N, let πi and σi be the i-th projection maps acting on
lim{X, f } and lim{Y, g}, respectively. Let πk−1 (U ) ∈ {πi−1 (U )|U is an open set in X
←−
←−
and i ∈ N}, which is a basis for lim{X, f }. Since pk is open and U is an open
←−
set in X, pk (U ) is an open set in Y . So σk−1 (pk (U )) is open in lim{Y, g}. Let
←−
−1
−1
(yi )∞
i=1 ∈ σk (pk (U )). Let xk ∈ U ∩ pk ({yk }); the latter set is nonempty since
−1
i −1
yk = σk ((yi )∞
i=1 ) ∈ pk (U ). Since f (pk+i ({yk+i })) = pk ({yk }) for i ∈ N, let xk+i ∈
14
i
k−1
p−1
(xk ), . . . , f (xk ), xk , xk+1 , . . .) ∈
k+i ({yk+i }) such that f (xk+i ) = xk . Then x = (f
−1
−1
−1
∞
πk−1 (U ) and p∞ (x) = (yi )∞
i=1 . So (yi )i=1 ∈ p∞ (πk (U )) and σk (pk (U )) ⊂ p∞ (πk (U )).
Since σk ◦ p∞ = pk ◦ πk , p∞ (πk−1 (U )) ⊂ σk−1 (pk (U )). So p∞ (πk−1 (U )) = σk−1 (pk (U )) is
an open set in lim{Y, g}.
←−
If p : X → Y is a map such that p ◦ f = g ◦ p then we will call (
Q∞
i=1
p)|lim{X,f } :
←
−
lim{X, f } → lim{Y, g} the ladder map induced by p. Let fˆ : lim{X, f } → lim{X, f }
←−
←−
←−
←−
be the ladder map induced by f . By definition, πk = πk+1 ◦ fˆ for each k ∈ N. The
map fˆ is a homeomorphism called the shift homeomorphism induced by f .
∞
If X is a compactum with metric d then d((xi )∞
i=1 , (zi )i=1 ) =
∞
P
d(xi , zi )/2i is
i=1
compatible with the subspace topology on lim{X, f }.
←−
Matchbox Continua
Let X be a continuum. Following closely notation introduced by Aarts and
Martens [1], we make the following definitions. If C is a zero-dimensional compactum
and h : C × [−1, 1] → X is a topological embedding such that h(C × [−1, 1]) is
closed in X and h(C × (−1, 1)) is open in X then the set M ≡ h(C × [−1, 1]) is
called a matchbox in X. For each c ∈ C, the set h({c} × [−1, 1]) is called a match in
M. If every point in X has an open neighborhood homeomorphic to the product of a
zero-dimensional compactum and an open arc then X is called a matchbox continuum.
Let G be a finite connected graph with branch points but without end points,
let d be a Menger-convex metric compatible with G, and let f : G → G be a
15
collapsing surjective immersion such that f (bi ) = bi for each bi ∈ {b1 , . . . , br } ≡
B(G). Let {J1 , . . . , Jm } be the path components of G\B(G) and {b1 , . . . , br , . . . , bs } ≡
f −1 (B(G)).
Proposition 2.9. For each k ∈ N and Ji ∈ {J1 , . . . , Jm }, πk−1 (Ji ) ⊂ lim{G, f }
←−
is homeomorphic to the product of a zero-dimensional compactum and an open arc.
Proof. Let {I1 , . . . , In } be the path components of G\f −1 (B(G)) and σ be
the discrete metric on the set {1, . . . , n}. Let Ci ≡ {(sj )∞
j=1 ∈
Q∞
j=1 {1, . . . , n}
|
Ji = f (Is1 ) and Isj ⊂ f (Isj+1 ) for each j ∈ N} which is a closed subset of the zerodimensional compactum
∞
P
j=1
Q∞
j=1 {1, . . . , n}
∞
with metric σ defined by σ((sj )∞
j=1 , (tj )j=1 ) =
σ(sj , tj )/2j . Let hi,k : Ci × Ji → πk−1 (Ji ) be defined by hi,k ((sj )∞
j=1 , x) =
(f k−1 (x), . . . , x, (f |Is1 )−1 (x), (f |Is2 )−1 ◦ (f |Is1 )−1 (x), . . .).
hi,k is surjective: Let x ∈ πk−1 (Ji ). Then πk (x) ∈ Ji and f ◦ π`+1 (x) = π` (x) for
each ` ∈ N. Since πk (x) ∈
/ B(G), πk+j (x) ∈
/ f −1 (B(G)) for each j ∈ N. For each j ∈ N,
let sj be the element of {1, . . . , n} such that πk+j (x) ∈ Isj . Since πk+1 (x) ∈ Is1 and f ◦
πk+1 (x) = πk (x) ∈ Ji , Ji = f (Is1 ) and (f |Is1 )−1 ◦ πk (x) = πk+1 (x). Since πk+j+1 (x) ∈
Isj+1 and f ◦ πk+j+1 (x) = πk+j (x) ∈ Isj , Isj ⊂ f (Isj+1 ) and (f |Isj+1 )−1 ◦ πk+j (x) =
−1
πk+j+1 (x). So (sj )∞
◦ . . . ◦ (f |Is1 )−1 ◦ πk (x) = πk+j (x). Thus
j=1 ∈ Ci and (f |Isj )
∞
k−1
((sj )∞
◦ πk (x), . . . , πk (x), (f |Is1 )−1 ◦
j=1 , πk (x)) ∈ Ci × Ji and hi,k ((sj )j=1 , πk (x)) = (f
πk (x), (f |Is2 )−1 ◦ (f |Is1 )−1 ◦ πk (x), . . .) = (π1 (x), . . . , πk (x), πk+1 (x), . . .) = x.
∞
hi,k is injective: Suppose hi,k ((sj )∞
j=1 , x) = hi,k ((tj )j=1 , z). Then x = πk ◦
∞
hi,k ((sj )∞
j=1 , x) = πk ◦ hi,k ((tj )j=1 , z) = z. Also for j ∈ N, sj = tj since πk+j ◦
16
∞
∞
hi,k ((sj )∞
j=1 , x) ∈ Isj , πk+j ◦ hi,k ((tj )j=1 , z) ∈ Itj , and πk+j ◦ hi,k ((sj )j=1 , x) = πk+j ◦
∞
∞
hi,k ((tj )∞
j=1 , z). Thus ((sj )j=1 , x) = ((tj )j=1 , z).
hi,k is continuous: Let ² > 0 and ((sj )∞
j=1 , x) ∈ Ci × Ji . Choose a positive integer K such that diam(G)/2K < ²/3 and a positive integer ` such that k + ` ≥ K.
Since f k−j is continuous for each j ∈ {1, . . . , k}, there exists a δj > 0 such that
if d(x, z) < δj then d(f k−j (x), f k−j (z)) < ²/3. Since (f |Isj−k )−1 ◦ . . . ◦ (f |Is1 )−1
is continuous on Ji for j ∈ {k + 1, . . . , k + `}, there exists a δj > 0 such that
if z ∈ Ji and d(x, z) < δj then d((f |Isj−k )−1 ◦ . . . ◦ (f |Is1 )−1 (x), (f |Isj−k )−1 ◦ . . . ◦
(f |Is1 )−1 (z)) < ²/3. Let δ = min{1/2k+` , δ1 , . . . , δk+` } and ((tj )∞
j=1 , z) ∈ Ci × Ji
∞
∞
∞
such that ρ(((sj )∞
j=1 , x), ((tj )j=1 , z)) < δ. Then σ((sj )j=1 , (tj )j=1 ) < δ and d(x, z) <
k+`
∞
, that is σ(sj , tj ) = 0, for j ∈
δ. So σ(sj , tj )/2j < σ((sj )∞
j=1 , (tj )j=1 ) < 1/2
∞
{1, . . . , k + `}. Thus d(hi,k ((sj )∞
j=1 , x), hi,k ((tj )j=1 , z)) =
j
hi,k ((tj )∞
j=1 , z))/2 =
k
P
d(f k−j (x), f k−j (z))/2j +
j=1
(f |Itj−k )−1 ◦. . .◦(f |It1 )−1 (z))/2j +
²/3 + ²/3 + diam(G)/2
k+`
∞
P
j=k+`+1
k+`
P
j=k+1
∞
P
j=1
d(πj ◦ hi,k ((sj )∞
j=1 , x), πj ◦
d((f |Isj−k )−1 ◦. . .◦(f |Is1 )−1 (x),
∞
j
d(πj ◦hi,k ((sj )∞
j=1 , x), πj ◦hi,k ((tj )j=1 , z))/2 <
< ².
−1
h−1
i,k is continuous: Let ² > 0 and x ∈ πk (Ji ). Then there exists a posi∞
tive integer ` such that 1/2` < ² and an (sj )∞
j=1 ∈ Ci such that ((sj )j=1 , πk (x)) =
h−1
i,k (x). For j ∈ {1, . . . , `}, since Isj is open in G, there exists a δj > 0 such that
Bd (πk+j (x), δj ) ⊂ Isj . Let δ = min{²/2k , δ1 /2k+1 , . . . , δ` /2k+` } and z ∈ πk−1 (Ji )
such that d(x, z) < δ. Then d(πj (x), πj (z))/2j < δ for each j ∈ N and there
−1
∞
exists a (tj )∞
j=1 ∈ Ci such that ((tj )j=1 , πk (z)) = hi,k (z). So d(πk (x), πk (z)) < ²
17
and d(πk+j (x), πk+j (z)) < δj for each j ∈ {1, . . . , n}. Then πk+j (z) ∈ Isj for each
∞
j ∈ {1, . . . , n} and σ((sj )∞
j=1 , (tj )j=1 ) =
∞
P
σ(sj , tj )/2j =
j=1
∞
P
j=`+1
σ(sj , tj )/2j ≤ 1/2` < ².
−1
∞
∞
∞
∞
Thus ρ(h−1
i,k (x), hi,k (z)) = ρ(((sj )j=1 , πk (x)), ((tj )j=1 , πk (z))) = max{σ((sj )j=1 , (tj )j=1 ),
d(πk (x), πk (z))} < ².
Corollary 2.10. For each k ∈ N and x ∈ G\B(G), πk−1 (U ) is a matchbox in
lim{G, f } for every star neighborhood U of x contained in G\B(G).
←−
Proof. Let Ji be the component of G\B(G) containing U and let Ci be the onedimensional compactum and hi,k : Ci × Ji → πk−1 (U ) the homeomorphism defined in
Proposition 2.9. Since U is an arc, let ρ : U → [−1, 1] be a homeomorphism. Then
(hi,k |Ci ×U ) ◦ (idCi × ρ−1 ) : Ci × [−1, 1] → πk−1 (U ) is a homeomorphism.
Proposition 2.11. For each k ∈ N and xi ∈ B(G), there exists a δi,k > 0 such
that πk−1 (U ) is a matchbox in lim{G, f } for every star neighborhood U of xi contained
←−
in Bd (xi , δi,k ).
Proof. For each i ∈ {1, . . . , r}, let Wi be a star neighborhood of xi such that the
only element of f −1 (B(G)) contained in Wi is xi and let ρi : Wi → [−1, 1] be a map
such that ρi |Ij : Ij → [−1, 1] is a homeomorphism for each Ij ∈ A(Wi ). We can choose
each Wi so {W1 , . . . , Wr } is a pairwise disjoint collection. For i ∈ {1, . . . , r}, let Vi be a
star neighborhood of xi contained in f −1 (Int(Wi ))∩Int(Wi ). For i ∈ {r+1, . . . , s}, let
Vi be an arc neighborhood of xi contained in (G\
r
S
j=1
Wj ) ∩ f −1 (
r
S
Int(Wj )). We can
j=1
choose each Vi so {Vr+1 , . . . , Vs } is a pairwise disjoint collection. For i ∈ {1, . . . , r},
let Ai = {I` ∈
s
S
j=1
A(Vj ) | I` ∩ f −1 ({xi }) 6= ∅}. Then
T
I` ∈Ai
ρ−1
i ◦ ρi ◦ f (I` ) is a star
18
neighborhood of xi contained in Vi . For i ∈ {1, . . . , s}, let Ui be the path component
of f −1 (
r
S
T
j=1 I` ∈Ai
ρ−1
i ◦ ρi ◦ f (I` ) contained in Vi . For each i ∈ {1, . . . , r}, we choose
a homeomorphism hi : Wi → Wi such that the end points and branch point of Wi
are fixed and the end points of
T
I` ∈Ai
ρ−1
i ◦ ρi ◦ f (I` ) are mapped to the end points
of Ui . Let h : G → G be the homeomorphism defined by h(x) = hi (x) for x ∈ Wi
and h(x) = x otherwise. By definition, h(Ji ) = Ji and h preserves the canonical
orientation of Ji . Let f 0 : G → G be the collapsing surjective immersion defined
by f 0 = h ◦ f . We now construct a homeomorphism p∞ : lim{G, f } → lim{G, fˆ}
←−
←−
in the following manner. Let p1 : G → G be the identity map. Suppose that for
j = 1, . . . , N , we have defined homeomorphisms pj : G → G such that pj (xi ) = xi
for each xi ∈ f −1 (B(G)), pj (Ji ) = Ji , pj preserves the canonical orientation of Ji for
each Ji ∈ {J1 , . . . , Jm }, and the following diagram commutes.
G ¾
f
p1
G ¾
f
G ¾
?
f0
f
...
p2
?
¾
G ¾
pN
?
¾
f0
G
f0
G
Let pN +1 |Ji,` : Ji,` → Ji,` be the homeomorphism defined by pN +1 |Ji,` = (f 0 |Ji,` )−1 ◦
pN ◦ f |Ji,` . Then pN +1 extends from
ri
m S
S
Ji,` to a homeomorphism on G such that
i=1 `=1
pN ◦ f = f 0 ◦ pN +1 . Let p∞ : lim{G, f } → lim{G, fˆ} be the ladder map induced by
←−
←−
19
{pi }∞
i=1 . Since pi is a homeomorphism for each i ∈ N, p∞ is a homeomorphism by
Theorem 2.8. Let πk0 be the k-th projection map acting on lim{G, f 0 }.
←−
Claim. For each k ∈ N and Ui ∈ {U1 , . . . , Ur }, (πk0 )−1 (U ) is a matchbox in
lim{G, f 0 } for every star neighborhood U of xi contained in Int(Ui ).
←−
Proof of Claim. Let {Us+1 , . . . , Ut } be the components of G\
Ss
i=1
Int(Ui )
and let {I1 , . . . , In } be the collection of open arcs such that {Cl(I1 ), . . . , Cl(In )} =
t
S
i=1
A(Ui ). For i ∈ {1, . . . , r}, we assume that Cl(Ii ) is the arc in A(Ui ) such that
f 0 (Ij ) = Ii for every Ij with Cl(Ii ) ∈ A(Ui ). Let Ci = {(sj )∞
j=0 ∈
Q∞
j=0 {1, . . . , n}
|
Cl(Is0 ) ∈ A(Ui ) and Isj ⊂ f 0 (Isj+1 ) for j = 0, 1, . . .}, which is a closed subset of
the zero-dimensional compactum
Q∞
j=0 {1, . . . , n}.
Let U be a star neighborhood of xi
contained in Int(Ui ) and ρ : U → [−1, 1] be a map such that ρ|U ∩Ij : U ∩Ij → [−1, 1] is
a homeomorphism for each Ij with Cl(Ij ) ∈ A(Ui ). Let hi,k : Ci ×[−1, 1] → (πk0 )−1 (U )
0 k−1
be defined by hi,k ((sj )∞
◦ (ρ|U ∩Is0 )−1 (t), . . . , (ρ|U ∩Is0 )−1 (t), (f 0 |Is1 )−1 ◦
j=1 , t) = ((f )
(ρ|U ∩Is0 )−1 (t), . . .).
hi,k is surjective: Let x ∈ (πk0 )−1 (U ). Then πk0 (x) ∈ U ⊂ Int(Ui ), ρ ◦ πk0 (x) ∈
0
[−1, 1], and f 0 ◦ πj+1
(x) = πj0 (x) for j ∈ N.
(f 0 )−1 (
Sr
i=1
0
Since πk0 (x) ∈
/ E(Ui ), πk+j
(x) ∈
/
E(Ui )) for j ∈ N.
0
0
If πk+j
(x) ∈ Int(Ui ) for each j ∈ {0, 1, . . .} then πk+j
(x) ∈ Ii for each j ∈
{0, 1, . . .}. So, let sj = i for each j ∈ {0, 1, . . .}. Then Cl(Is0 ) = Cl(Ii ) ∈ A(Ui ) and
0
Isj = Ii = f 0 (Ii ) = f 0 (Isj+1 ) for j ∈ {0, 1, . . .}. So (sj )∞
j=1 ∈ Ci . Since f is one-to-one
20
0
0
0
on Ii and πk+j
(x) ∈ Ii for each j ∈ {0, 1, . . .}, (f 0 |Isj )−1 (πk+j
(x) for each
(x)) = πk+j+1
j ∈ {0, 1, . . .}.
0
If πk+j
(x) ∈
/ Int(Ui ) for some j ∈ {0, 1, . . .} then there exists an ` ∈ {0, 1, . . .}
0
0
0
such that πk+`
(x) ∈ Int(Ui ) but πk+`+1
(x) ∈
/ Int(Ui ). So πk+j
(x) ∈
/
Sr
i=1
Int(Ui ) and
0
thus only one element of {I1 , . . . , In } contains πk+j
(x). For j ≥ ` + 1, let sj be the
0
unique element of {1, . . . , n} such that πk+j
(x) ∈ Isj . Let s` ∈ {1, . . . , n} such that
f 0 (Is`+1 ) = Is` . For 0 ≤ j < `, let sj = i. Then Cl(Is0 ) ∈ A(Ui ) and Isj ⊂ f 0 (Isj+1 )
0
0
for j ∈ {0, 1, . . .}. So (sj )∞
j=1 ∈ Ci . Since f is one-to-one on Ij and πk+j (x) ∈ Ij for
0
0
each j ∈ {0, 1, . . .}, (f 0 |Isj )−1 (πk+j
(x) for each j ∈ {0, 1, . . .}.
(x)) = πk+j+1
0
∞
0
0 k−1
Then ((sj )∞
◦
j=1 , ρ ◦ πk (x)) ∈ Ci × [−1, 1] and hi,k ((sj )j=1 , ρ ◦ πk (x)) = ((f )
(ρ|U ∩Is0 )−1 (ρ◦πk0 (x)), . . . , (ρ|U ∩Is0 )−1 (ρ◦πk0 (x)), (f 0 |Is1 )−1 ◦(ρ|U ∩Is0 )−1 (ρ◦πk0 (x)), . . .) =
0
(x), . . .) = x.
((f 0 )k−1 ◦πk0 (x), . . . , πk0 (x), (f 0 |Is1 )−1 ◦πk0 (x), . . .) = (π10 (x), . . . , πk0 (x), πk+1
0 ∞
0
hi,k is injective: Suppose that hi,k ((sj )∞
j=1 , t) = hi,k ((sj )j=1 , t ). Then t = ρ ◦
(ρ|U ∩Is0 )−1 (t) = ρ ◦ (ρ|U ∩Is0 )−1 (t0 ) = t0 .
For purposes of contradiction, assume
0
0 ∞
0
(sj )∞
j=1 6= (sj )j=1 . Let ` be the smallest index such that s` 6= s` . Without loss
of generality, assume s0` 6= i. Then Is0`+1 ∩ Int(Ui ) = ∅. So Is0`+1 ∩ Is`+1 = ∅ since
s`+1 6= s0`+1 . Then (f 0 |Is0 )−1 ◦ . . . ◦ (f 0 |Is0 1 )−1 ◦ (ρ|U ∩Is0 )−1 (t0 ) 6= (f 0 |Is`+1 )−1 ◦ . . . ◦
`+1
0
(f 0 |Is1 )−1 ◦ (ρ|U ∩Is0 )−1 (t0 ). But this contradicts the supposition that hi,k ((sj )∞
j=1 , t) =
0
∞
0 ∞
0
hi,k ((s0j )∞
j=1 , t ). Thus ((sj )j=1 , t) = ((sj )j=1 , t ).
hi,k is continuous: Similar to continuity argument in Proposition 2.9.
21
Since Ci ×[−1, 1] is compact and (πk0 )−1 (U ) is Hausdorff, hi,k is a homeomorphism
and (πk0 )−1 (U ) is a matchbox in lim{G, f 0 }, which proves the claim.
←−
To finish the proof of the proposition, let xi ∈ B(G). Since pk is a homeomorphism
such that pk (xi ) = xi , there exists a δi,k > 0 such that pk (Bd (xi , δi,k )) ⊂ Int(Ui ).
Let U be a star neighborhood of xi contained in Bd (xi , δi,k ). Then pk (U ) is a star
neighborhood of xi contained in Int(Ui ). So (πk0 )−1 (pk (U )) is a matchbox in lim{G, f 0 }
←−
by the above claim. Since p∞ is a homeomorphism and p∞ (πk−1 (U )) = (πk0 )−1 (pk (U )),
πk−1 (U ) is a matchbox in lim{G, f }
←−
Proposition 2.12. lim{G, f } is a matchbox continuum.
←−
Proof. By Theorem 2.7, lim{G, f } is a continuum. Let x ∈ lim{G, f }. If
←−
←−
πk (x) ∈
/ B(G) for some k ∈ N then πk (x) ∈ Ji for some Ji ∈ {J1 , . . . , Jm }. Let I be an
arc neighborhood of πk (x) contained in Ji . Then πk−1 (I) is a matchbox neighborhood
of x by Corollary 2.10. If πk (x) ∈ B(G) then there exists a δ > 0 such that πk−1 (U )
is a matchbox in lim{G, f } for every star neighborhood U of πk (x) contained in
←−
Bd (πk (x), δ) by Proposition 2.11. Let U be a star neighborhood of πk (x) contained
in Bd (πk (x), δ). Then πk−1 (U ) is a matchbox neighborhood of x.
Suppose U is a matchbox in a continuum. If ρ : U → R is a map such that
ρ|I : I → R is a topological embedding for each match I in U then we will call ρ a
directing map on U. Two directing maps, ρ1 and ρ2 , on U are called equivalent if for
each match I in U the homeomorphism (ρ2 |I ) ◦ (ρ1 |I )−1 : ρ1 (I) → ρ2 (I) is increasing.
This is an equivalence relation that partitions the collection of all directing maps on
22
U into equivalence classes, each of which we will call a direction for U. A matchbox
together with a direction is called a directed matchbox.
Proposition 2.13. Let M be a matchbox in a continuum X and let ρ1 and ρ2 be
directing maps on M . If M+ is the union of all matches I in M such that (ρ2 |I ) ◦
(ρ1 |I )−1 : ρ1 (I) → ρ2 (I) is increasing and M− is the union of all matches I in M such
that (ρ2 |I ) ◦ (ρ1 |I )−1 : ρ1 (I) → ρ2 (I) is decreasing then M+ and M− are matchboxes
or empty.
Proof. Let C be a zero-dimensional compactum and h : C × [−1, 1] → X
be topological embedding such that h(C × [−1, 1]) = M and h(C × (−1, 1)) =
Int(M ). Assume σ is the metric on C and d is the metric on C × [−1, 1] defined by
d((s1 , t1 ), (s2 , t2 )) = max{σ(s1 , s2 ), |t1 − t2 |}. Let C+ = pr1 (M+ ) and C− = pr1 (M− ).
For each s ∈ C, let ²s = min{|ρ1 ◦ h(s, −1) − ρ1 ◦ h(s, 1)|/2, |ρ2 ◦ h(s, −1) − ρ2 ◦
h(s, 1)|/2}. Since ρ1 ◦ h and ρ2 ◦ h are uniformly continuous, let δs > 0 be such that if
d((s1 , t1 ), (s2 , t2 )) < δs then |ρi ◦h(s1 , t1 )−ρi ◦h(s2 , t2 )| < ²s . Let Cs be a clopen neighborhood of s such that Cs ⊂ Bσ (s, δs ). For i ∈ {1, 2}, if ρi ◦ h(s, −1) < ρi ◦ h(s, 1) and
s0 ∈ C −s then d((s, t), (s0 , t)) < δs and |ρi ◦h(s, t)−ρi ◦h(s0 , t)| < ²s for all t ∈ [−1, 1].
Thus ρi ◦h(s0 , −1) < ρi ◦h(s, −1)+²s ≤ (ρi ◦h(s, −1)+ρi ◦h(s, 1))/2 ≤ ρi ◦h(s, 1)−²s <
ρi ◦ h(s0 , 1). Similarly, if ρi ◦ h(s, −1) > ρi ◦ h(s, 1) then ρi ◦ h(s0 , −1) > ρi ◦ h(s0 , 1)
for each s0 ∈ Cs . Thus (ρ2 |I ) ◦ (ρ1 |I )−1 : ρ1 (I) → ρ2 (I) is increasing for each match
in the matchbox h(Cs × [−1, 1]) or (ρ2 |I ) ◦ (ρ1 |I )−1 : ρ1 (I) → ρ2 (I) is decreasing
for each match in matchbox h(Cs × [−1, 1]). So Cs ⊂ C+ or Cs ⊂ C− for each
23
s ∈ C. Since C is compact and {Cs }s∈C is an open covering of C there exists a finite
subcollection {Cs1 , . . . , Csk } that covers C. Thus C+ and C− are a finite union of
elements of {Cs1 , . . . , Csk } which means they are clopen subsets of C. If C+ 6= ∅ then
M+ = h(C+ × [−1, 1]) is a matchbox. If C+ = ∅ then M+ = h(C+ × [−1, 1]) = ∅.
Similarly M− is either a matchbox or empty.
If U is a directed matchbox in a continuum then the direction for U naturally
induces a direction for any arc contained in U .
Suppose U and V are directed matchboxes in a continuum. If the directions for
each nondegenerate path component of U ∩ V induced by the directions for U and V
are the same then U and V are said to be coherently directed.
Suppose ρ1 is a directing map on a matchbox U1 and ρ2 is a directing map on a
matchbox U2 . If x ∈ Int(U1 ) ∩ Int(U2 ) and there exists a matchbox neighborhood of x
contained in U1 ∩ U2 such that ρ1 |V and ρ2 |V are equivalent then ρ1 and ρ2 are said to
be equivalent near x. This is an equivalence relation that partitions the collection of all
directing maps on all matchbox neighborhoods of x into two equivalence classes, each
of which is called a direction at x. The direction for a directed matchbox naturally
induces a direction at each point in the interior of that matchbox.
If x ∈ Int(U )∩Int(V ) and there exists a matchbox neighborhood W of x contained
in U ∩V such that the directions for W induced by the direction for U and the direction
for V are equal then we say U and V are coherently directed near x. If U and V are
24
coherently directed near x for each x ∈ Int(U ) ∩ Int(V ) then U and V are coherently
directed.
Suppose M is a matchbox continuum. We say M is orientable if there exists
a collection of coherently directed matchboxes whose interiors cover M . If M is
orientable then a choice of coherently directed matchboxes whose interiors cover M
induces a direction at each point M . This continuous choice of directions at each
point of M is called an orientation of M. If M is orientable then there exist exactly
two distinct orientations of M .
Proposition 2.14. G is orientable if and only if lim{G, f } is orientable.
←−
Proof. By Proposition 2.11, we can choose a pairwise disjoint collection of sets
in G, {U1 , . . . , Ur }, such that Ui is a star neighborhood of the branch point bi and
πk−1 (Ui ) is a matchbox in lim{G, f } for each i ∈ {1, . . . , r}. We can then choose a
←−
pairwise disjoint collection of sets in G, {Ur+1 , . . . , Ur+m }, such that Ur+i is an arc
contained in Ji for each i ∈ {1, . . . , m} and G\
Sr
i=1
Int(Ui ) ⊂
Sm
i=1
Int(Ur+i ).
Assume G is orientable and choose an orientation for G. Since G =
Sr+m
i=1
Int(Ui ),
we can choose a directing map ρi on Ui for each i ∈ {1, . . . , r + m} such that if
U is a nondegenerate path component in Ui ∩ Uj then ρj |I ◦ ρi |−1
: ρi (I) → ρj (I)
I
is an increasing map for each I ∈ A(U ). Let ρi be the directing map on πk−1 (Ii )
defined by ρi = ρi ◦ πk . For each match I of πk−1 (Ui ), πk |I is a homeomorphism
onto πk (I) ∈ A(Ui ) and ρi |I = ρi |πk (I) ◦ πk |I is a topological embedding. If I is a
nondegenerate path component of πk−1 (Ui ) ∩ πk−1 (Uj ) = πk−1 (Ui ∩ Uj ) and U is the
25
nondegenerate path component of Ui ∩ Uj containing πk (I) then πk (I) ∈ A(U ). Since
−1
−1
−1
−1
ρj |I ◦ρi |−1
I = ρj |πk (I) ◦πk |I ◦πk |I ◦ρi |πk (I) = ρj |πk (I) ◦ρi |πk (I) , ρj |I ◦ρi |I : ρi (I) → ρj (I)
is an increasing map. So the matchboxes πk−1 (Ui ) and πk−1 (Uj ) with directions [ρi ] and
[ρj ], respectively, are coherently directed. Since lim{G, f } = πk−1 (
←−
Sr+m
−1
{G, f } is orientable.
i=1 Int(πk (Ui )), lim
←−
Sr+m
i=1
(Int(Ui ))) =
Assume lim{G, f } is orientable and choose an orientation of lim{G, f }. For i ∈
←−
←−
{1, . . . , r}, let ρi be a directing map representing the direction for πk−1 (Ui ) and Ii be
the match in πk−1 (Ui ) containing bi ≡ (bi , bi , . . .). Choose a directing map ρi on Ui
such that (ρi ◦ πk )|Ii ◦ ρi |−1
Ii : ρi (Ii ) → ρi ◦ πk (Ii ) is an increasing map. Let U i,+ be all
matches I in πk−1 (Ui ) such that (ρi ◦ πk )|I ◦ ρi |−1
I : ρi (I) → ρi ◦ πk (I) is an increasing
map. Then bi ∈ Ii ⊂ U i,+ . Since ρi ◦ πk and ρi are directing maps on πk−1 (Ui ),
U i,+ is a matchbox in lim{G, f } contained in πk−1 (Ui ) by Proposition 2.13. Choose
←−
² > 0 such that Bd (bi , ²) ⊂ U i,+ and positive integer ` such that diam(G)/2k+2` < ².
−1
({bi }), d(bi , x) =
Then for x ∈ πk+2`
P∞
j=k+2`+1
d(bi , πj (x))/2j < ². Choose a star
neighborhood Vi of bi contained in Ui such that f 2` (Vi ) ⊂ Ui . Since f is an immersion,
ρi |f 2` ◦πk+2` (I) ◦ f 2` ◦ ρi |−1
πk+2` (I) : πk+2` (I) → πk (I) is an increasing map for each match
−1
−1
−1
I in πk+2`
(Vi ). Since πk+2`
({bi }) ⊂ Bd (bi , ²) and f 2` (Vi ) ⊂ Ui , πk+2`
(Vi ) ⊂ U i,+ . Thus
−1
(ρi ◦ πk )|I ◦ ρi |−1
I : ρi (I) → ρi ◦ πk (I) is an increasing map for each match I in πk+2` (Vi )
and [(ρi ◦ πk )|π−1
k+2` (Vi )
] = [ρi |π−1
k+2` (Vi )
−1
]. For each match I in πk+2`
(Vi ), consider the map
(ρi ◦ πk )|I ◦ (ρi ◦ πk+2` )|−1
I : πk+2` (I) → πk (I), which is an increasing map since (ρi ◦
−1
−1
2`
πk )|I ◦(ρi ◦πk+2` )|−1
I = ρi |πk (I) ◦πk |I ◦πk+2` |I ◦ρi |πk+2` (I) = ρi |f 2` ◦πk+2` (I) ◦f ◦ρi |πk+2` (I) .
26
So [ρi |Vi ◦ πk+2` ] = [(ρi ◦ πk )|π−1
k+2` (Vi )
] = [ρi |π−1
k+2` (Vi )
]. Since fˆ is a homeomorphism,
ρi |fˆ−2` (I) ◦ fˆ−2` ◦ ρi |−1
: ρi (I) → ρi ◦ fˆ−2` (I) is an increasing map for each match I
I
: ρi (I) →
in πk−1 (Vi ). For each match I in πk−1 (Vi ), consider the map (ρi ◦ πk )|I ◦ ρ−1
I
−1
ˆ2`
ρi ◦πk (I), which is increasing since (ρi ◦πk )|I ◦ρi |−1
I = (ρi ◦πk )|I ◦f ◦ρi |fˆ−2` (I) ◦ρi |fˆ−2` (I) ◦
−1
ˆ−2` ◦ρ |−1 ). So π −1 (Vi ) ⊂ U .
fˆ−2` ◦ρi |−1
i,+
I = ((ρi ◦πk+2` )|fˆ−2` (I) ◦ρi |fˆ−2` (I) )◦(ρi |fˆ−2` (I) ◦ f
k
i I
Since πk−1 ({bi }) is a cross-section of both πk−1 (Ui ) and πk−1 (Vi ), πk−1 (Ui ) = U i,+ . Thus
[ρi ◦ πk ] = [ρi ]. For j ∈ {r + 1, . . . , r + m}, let ρj be a representative of the directing
map for πk−1 (Uj ) and Ij be a match in πk−1 (Uj ). Choose a directing map ρj on
Uj such that (ρj ◦ πk )|Ij ◦ ρj |−1
Ij : ρj (Ij ) → ρj ◦ πk (Ij ) is an increasing map. If
Ui ∈ {U1 . . . , Ur } and U is a path component of Ui ∩Uj then each match in πk−1 (Ui ∩Uj )
maps onto the arc U . Let I be the match in πk−1 (Ui ∩ Uj ) that is contained in Ii . The
−1
map ρi |πk (I) ◦ ρj |−1
πk (I) : ρj ◦ πk (I) → ρi ◦ πk (I) is increasing since ρi |πk (I) ◦ ρj |πk (I) =
−1
−1
= ((ρi ◦ πk )|I ◦ ρi |−1
(ρi ◦ πk )|I ◦ (ρj ◦ πk )|−1
I ) ◦ (ρi |I ◦ ρj |I ) ◦ (ρj |I ◦ (ρj ◦ πk )|I ).
I
So the directing maps [ρi ] and [ρj ] coherently direct the stars Ui and Uj . Thus G is
orientable.
27
CHAPTER 3
MAIN RESULTS
Orientable Bouquets of Circles
Our goal in this section is to prove the following theorem utilizing techniques
similar to those used by Mioduszewski [16] and by Barge and Diamond [4] to construct
“nearly” commuting infinite diagrams from homeomorphic inverse limit spaces.
Theorem 3.1. Let G and G0 be orientable bouquets of circles with branch points
b and b0 , respectively, let f : G → G and f 0 : G0 → G0 be collapsing surjective
immersions such that f (b) = b and f 0 (b0 ) = b0 , let Mf be the transition matrix for
f relative to an ordering of the components of G\{b}, and let Mf 0 be the transition
matrix for f 0 relative to an ordering of the components of G0 \{b0 }. If lim{G, f } is
←−
homeomorphic to lim{G0 , f 0 } then Mf is weakly equivalent to Mf 0 .
←−
We know begin the proof of the above theorem. Let {J1 , . . . , Jm } be an ordering
of the components of G\{b} and let {J10 , . . . , Jn0 } be an ordering of the components
of G0 \{b0 }. Since two matrices that are similar by a permutation matrix are weakly
equivalent, without loss of generality, we assume Mf is the transition matrix for f
relative to {J1 , . . . , Jm } and Mf 0 is the transition matrix for f 0 relative to {J10 , . . . , Jn0 }.
28
Assume lim{G, f } is homeomorphic to lim{G0 , f 0 }. Let φ be a homeomorphism
←−
←−
from lim{G, f } to lim{G0 , f 0 }. We will spend the rest of this section showing that
←−
←−
Mf is weakly equivalent to Mf 0 .
Since G and G0 are connected, f 2 : G → G and (f 0 )2 : G0 → G0 are orientation preserving collapsing surjective immersions. Since lim{G, f } is homeomorphic
←−
to lim{G0 , f 0 }, lim{G, f 2 } is homeomorphic to lim{G0 , (f 0 )2 }. If we can show that
←−
←−
←−
the transition matrix Mf 2 of f 2 relative to {J1 , . . . , Jm } is weakly equivalent to the
transition matrix M(f 0 )2 of (f 0 )2 relative to {J10 , . . . , Jn0 } then Mf is weakly equivalent
to Mf 0 since Mf 2 = (Mf )2 and M(f 0 )2 = (Mf 0 )2 . Thus, without loss of generality, we
assume that f and f 0 are orientation preserving.
Choose Menger-convex metrics, d and d0 , compatible with the topologies on G and
G0 , respectively, and let d and d0 be the induced metrics on lim{G, f } and lim{G0 , f 0 },
←−
←−
respectively. Let πk be the k-th projection map acting on lim{G, f } and let πk0 be
←−
the k-th projection map acting on lim{G0 , f 0 }.
←−
Choose an orientation of G. Since lim{G, f } is orientable by Proposition 2.14,
←−
choose the orientation of lim{G, f } induced by the orientation of G using the map
←−
πk . Since f is orientation preserving this orientation is independent of which π k is
used. Since lim{G0 , f 0 } is orientable by Proposition 2.14, choose the orientation of
←−
lim{G0 , f 0 } induced by the orientation of lim{G, f } using the map φ. Then choose the
←−
←−
orientation of G0 induced by the orientation of lim{G0 , f 0 } using the map πk0 . Since
←−
f 0 is orientation preserving this orientation is independent of which πk0 is used.
29
For each Ji ∈ {J1 , . . . , Jm }, let Ci be the zero-dimensional compactum and
hi,k : Ci × Ji → πk−1 (Ji ) the homeomorphism defined in Proposition 2.9. For each
i ∈ {1, . . . , m}, choose an si ∈ Ci . For k ≥ 1, let ek : G → lim{G, f } be the function
←−
defined by ek (x) = hi,k (si , x) for x ∈ Ji and ek (b) = b. For each Ji0 ∈ {J10 , . . . , Jn0 },
let Ci0 be the zero-dimensional compactum and h0i,k : Ci0 × Ji0 → (πk0 )−1 (Ji0 ) the homeomorphism defined in Proposition 2.9. For each i ∈ {1, . . . , n}, choose an s0i ∈ Ci0 .
For k ≥ 1, let e0k : G0 → lim{G0 , f 0 } be the function defined by e0k (x0 ) = h0i,k (s0i , x0 )
←−
for x0 ∈ Ji0 and e0k (b0 ) = b0 . By definition ek |G\{b} and e0k |G0 \{b0 } are orientation preserving topological embeddings. Let t`,k : G → G0 and s`,k : G0 → G be defined by
t`,k (x) = π`0 ◦ φ ◦ ek (x) and s`,k (x0 ) = π` ◦ φ−1 ◦ e0k (x0 ).
Given ² > 0, a metric space (Y, d), and two functions f : X → Y and g : X → Y ,
we will use the notation f = g when d(f (x), g(x)) < ² for all x ∈ X.
²
Lemma 3.2. Given ² > 0 and a positive integer n, there exists a positive integer
M such that sn,m ◦ tm,` = f `−n for all integers m ≥ M and ` > n.
²
Proof. Let ² > 0 and let n be a positive integer. Since πn ◦ φ−1 is uniformly
continuous on lim{G0 , f 0 }, there exists a δ > 0 such that if x0 , z 0 ∈ lim{G0 , f 0 } with
←−
←−
d0 (x0 , z 0 ) < δ then d(πn ◦ φ−1 (x0 ), πn ◦ φ−1 (z 0 )) < ². Choose a positive integer M such
that diam(G0 )/2M < δ. Let m and ` be integers such that m ≥ M and ` > n. If x ∈ G
0
then d0 (e0m ◦πm
◦φ◦e` (x), φ◦e` (x)) =
Pm
j=1
P∞
j=1
0
d0 (πj0 ◦e0m ◦πm
◦φ◦e` (x), πj0 ◦φ◦e` (x))/2j =
0
d0 ((f 0 )m−j ◦ πm
◦ φ ◦ e` (x), πj0 ◦ φ ◦ e` (x))/2j +
e` (x), πj0 ◦ φ ◦ e` (x))/2j ≤
P∞
j=m+1
P∞
j=m+1
0
d0 (πj0 ◦ e0m ◦ πm
◦φ◦
diam(G0 )/2j = diam(G0 )/2m < δ. By definition of
30
0
e` , f `−n = πn ◦ e` . Thus d(sn,m ◦ tm,` (x), f `−n (x)) = d(πn ◦ φ−1 ◦ e0m ◦ πm
◦ φ ◦ e` (x), πn ◦
φ−1 ◦ φ ◦ e` (x)) < ².
Lemma 3.3. Given ²0 > 0 and a positive integer m, there exists a positive integer
N such that tm,n ◦ sn,` =0 (f 0 )`−m for all integers n ≥ N and ` > m.
²
Proof. As above.
Lemma 3.4. Given ²0 > 0 and a positive integer m, there exists a positive integer
L such that for each integer ` ≥ L there exists a ∆` > 0 such that if x, z ∈ G with
d(x, z) < ∆` then d0 (tm,` (x), tm,` (z)) < ²0 .
0
Proof. Let ²0 > 0 and let m be a positive integer. Since πm
◦ φ is uniformly
continuous on lim{G, f } there exists a σ > 0 such that if x, z ∈ lim{G, f } with
←−
←−
0
0
◦ φ(x), πm
◦ φ(z)) < ². Choose a positive integer L such
d(x, z) < σ then d0 (πm
that diam(G)/2L < σ/2. Let ` be a integer such that ` ≥ L. Since f `−j is uniformly continuous on G for 1 ≤ j ≤ `, there exists a δj > 0 such that if x, z ∈ G
with d(x, z) < δj then d(f `−j (x), f `−j (z)) < σ/2. Let ∆` = min{δ1 , . . . , δ` }. If
x, z ∈ G with d(x, z) < ∆` then d(e` (x), e` (z)) =
P`
j=1
P∞
d(f `−j (x), f `−j (z))/2j +
j=`+1
P∞
j=`+1
P∞
j=1
d(πj ◦ e` (x), πj ◦ e` (z))/2j =
d(πj ◦ e` (x), πj ◦ e` (z))/2j <
P`
j=1
σ/2j+1 +
0
diam(G)/2j < σ/2 + diam(G)/2` < σ. Thus d0 (tm,` (x), tm,` (z)) = d0 (πm
◦φ◦
0
e` (x), πm
◦ φ ◦ e` (z)) < ².
Lemma 3.5. Given ² > 0 and a positive integer n, there exists a positive integer
M such that for each integer m ≥ M there exists a ∆0m > 0 such that if x0 , z 0 ∈ G0
with d0 (x0 , z 0 ) < ∆0m then d(sn,m (x0 ), sn,m (z 0 )) < ².
31
Proof. As above.
Since d is a Menger-convex metric, there exists an E > 0 such that Bd (x, E) is an
open star neighborhood of x for each x ∈ G and the shortest arc between any two
distinct points in Cl(Bd (x, E)) is contained in Cl(Bd (x, E)). Similarly, there exists
an E0 > 0 such that Bd0 (x0 , E0 ) is an open star neighborhood of x0 for all x0 ∈ G0
and the shortest arc between any two distinct points in Cl(Bd0 (x0 , E0 )) is contained
in Cl(Bd0 (x0 , E0 )). Given two distinct points in a star there exists a unique arc,
contained in the star, between the points. By Lemma 3.4, given 0 < ²0 < E0 and a
positive integer m, there exists a positive integer L such that for each integer ` ≥ L
there exists a ∆` > 0 such that if d(x, z) < ∆` then d0 (tm,` (x), tm,` (z)) < ²0 . For 0 <
δ < min{∆` , E}, we define a map tm,`,δ : G → G0 such that tm,`,δ (x) = tm,` (x) for all
x∈
/ Bd (b, δ)\{b}. If x ∈ Bd (b, δ)\{b} then we define tm,`,δ (x) in the following manner.
Let z be the end point of the star Cl(Bd (b, δ)) such that x is on the arc between b
and z contained in Bd (b, E). If tm,` (b) = tm,` (z) then define tm,`,δ (x) to be tm,` (b). If
tm,` (b) 6= tm,` (z) then define tm,`,δ (x) to be the point x0 on the arc between tm,` (b)
and tm,` (z) contained in Bd0 (tm,` (b), E0 ) such that d0 (tm,` (b), x0 )/d0 (tm,` (b), tm,` (z)) =
d(b, x)/d(b, z). By definition, d0 (tm,`,δ (b), tm,`,δ (x)) < ²0 for all x ∈ Bd (b, δ). Define the
map sn,m,δ0 : G0 → G similarly.
Lemma 3.6. Given 0 < ²0 < E0 and a positive integer m, there exists a positive
integer L such that for each integer ` ≥ L there exists a 0 < ∆` < E such that
32
tm,`,δ =0 tm,` and if x, z ∈ G with d(x, z) < ∆` then d0 (tm,`,δ (x), tm,`,δ (z)) < ²0 for
²
0 < δ < ∆` .
Proof. Let 0 < ²0 < E0 and let m be a positive integer. By Lemma 3.4, there
exists a positive integer L such that for each integer ` ≥ L there exists a 0 < ∆` < E
such that if x, z ∈ G with d(x, z) < ∆` then d0 (tm,` (x), tm,` (z)) < ²0 /3.
Let x ∈ G and 0 < δ < ∆` . If x ∈
/ Bd (b, δ)\{b} then d0 (tm,`,δ (x), tm,` (x)) =
d0 (tm,` (x), tm,` (x)) = 0 < ²0 . If x ∈ Bd (b, δ)\{b} then d0 (tm,`,δ (x), tm,` (x)) ≤
d0 (tm,`,δ (x), tm,`,δ (b)) + d0 (tm,` (b), tm,` (x)) < ²0 /3 + ²0 /3 < ²0 .
Let x, z ∈ G with d(x, z) < ∆` and 0 < δ < ∆` . If x, z ∈
/ Bd (b, δ)\{b} then
d0 (tm,`,δ (x), tm,`,δ (z)) = d0 (tm,` (x), tm,` (z)) < ²0 /3 < ²0 . If x, z ∈ Bd (b, δ)\{b} then
d0 (tm,`,δ (x), tm,`,δ (z)) ≤ d0 (tm,`,δ (x), tm,`,δ (b)) + d0 (tm,`,δ (b), tm,`,δ (z)) < ²0 /3 + ²0 /3 <
²0 .
If x ∈ Bd (b, δ)\{b} and z = b then d0 (tm,`,δ (x), tm,`,δ (z)) < ²0 /3 < ²0 .
If
x ∈ Bd (b, δ)\{b} and z ∈
/ Bd (b, δ) then x ∈ Bd (z, ∆` ). Let w be the end point of
Cl(Bd (b, δ)) on the arc between x and z contained in Bd (z, ∆` ). So d0 (tm,`,δ (x), tm,`,δ (z))
< d0 (tm,`,δ (x), tm,`,δ (b)) + d0 (tm,`,δ (b), tm,`,δ (w)) + d0 (tm,`,δ (w), tm,`,δ (z)) =
d0 (tm,`,δ (x), tm,`,δ (b)) + d0 (tm,` (b), tm,` (w)) + d0 (tm,` (w), tm,` (z)) < ²0 /3 + ²0 /3 + ²0 /3 =
²0 .
Lemma 3.7. Given 0 < ² < E and a positive integer n, there exists a positive
integer M such that for each integer m ≥ M there exists a 0 < ∆0m < E0 such that
sn,m,δ0 = sn,m and if x0 , z 0 ∈ G0 with d0 (x0 , z 0 ) < ∆0m then d(sn,m,δ0 (x0 ), sn,m,δ0 (z 0 )) < ²
²
for 0 < δ 0 < ∆0m .
33
Proof. As above.
Lemma 3.8. Given 0 < ² < E and positive integers n and M , there exists an
integer m ≥ M , a 0 < ∆0 < E0 , and an integer L > n such that for each integer
` ≥ L there exists a 0 < ∆` < E such that sn,m,δ0 ◦ tm,`,δ = f `−n for 0 < δ 0 < ∆0 and
²
0 < δ < ∆` .
Proof. Let 0 < ² < E and let n and M be positive integers. By Lemma
3.2, there exists a positive integer M1 such that for integers m ≥ M1 and ` > n,
d(sn,m ◦ tm,` (x), f `−n (x)) < ²/3 for all x ∈ G. By Lemma 3.7, there exists a positive
integer M2 such that for each integer m ≥ M2 there exists a 0 < ∆0m < E0 such that
if d0 (x0 , z 0 ) < ∆0m then d(sn,m,δ0 (x0 ), sn,m,δ0 (z 0 )) < ²/3 and d(sn,m,δ0 (x0 ), sn,m (x0 )) < ²/3
for 0 < δ 0 < ∆0m and x0 , z 0 ∈ G0 . Choose an integer m ≥ max{M, M1 , M2 }. By
Lemma 3.6, there exists a positive integer L1 such that for each integer ` ≥ L1 there
exists a 0 < ∆` < E such that d0 (tm,`,δ (x), tm,` (x)) < ∆0m for 0 < δ < ∆` and
x ∈ G. Let L = max{L1 , n + 1}. If ` is an integer such that ` ≥ L then d(sn,m,δ0 ◦
tm,`,δ (x), f `−n (x)) ≤ d(sn,m,δ0 ◦ tm,`,δ (x), sn,m,δ0 ◦ tm,` (x)) + d(sn,m,δ0 ◦ tm,` (x), sn,m ◦
tm,` (x)) + d(sn,m ◦ tm,` (x), f `−n (x)) < ²/3 + ²/3 + ²/3 = ² for 0 < δ 0 < ∆0m and
0 < δ < ∆` .
Lemma 3.9. Given 0 < ²0 < E0 and positive integers m and N , there exists an
integer n ≥ N , 0 < ∆ < E, and an integer L > m such that for each integer ` ≥ L
there exists a 0 < ∆0` < E0 such that tm,n,δ ◦ sn,`,δ0 =0 (f 0 )`−m for 0 < δ < ∆ and
²
0 < δ 0 < ∆0` .
34
Proof. As above.
Lemma 3.10. Given 0 < ² < min{E, E0 } there exist two strictly increasing se∞
0 ∞
∞
quences {ni }∞
i=1 and {mi }i=1 of positive integers and two sequences {δi }i=1 and {δi }i=2
of positive numbers such that sni ,mi ,δi0 ◦ tmi ,ni+1 ,δi+1 = f ni+1 −ni and tmi ,ni+1 ,δi+1 ◦
²
0
sni+1 ,mi+1 ,δi+1
= (f 0 )mi+1 −mi for each i ∈ N.
²
Proof. Let 0 < ² < min{E, E0 }. Choose positive integers n1 and M . By Lemma
3.8, there exist an integer m1 ≥ M , 0 < ∆0 < E0 , and an integer N > n1 such that
for each integer n ≥ N there exists a 0 < ∆n < E such that sn1 ,m1 ,δ0 ◦ tm1 ,n,δ = f n−n1
²
for 0 < δ 0 < ∆0 and 0 < δ < ∆n . Choose 0 < δ10 < ∆0 . Assume ni , mi , δi0 , and
N > ni are given such that for each integer n ≥ N there exists a 0 < ∆n < E such
that sni ,mi ,δi0 ◦ tmi ,n,δ = f n−ni for 0 < δ < ∆n . By Lemma 3.9, there exists an integer
²
ni+1 ≥ N , 0 < ∆ < E, and an integer M > mi such that tmi ,ni+1 ,δ ◦ sni+1 ,m,δ0 =
²
(f 0 )m−mi for 0 < δ < ∆ and 0 < δ 0 < ∆0m . Choose 0 < δi+1 < min{∆ni+1 , ∆}. Then
sni ,mi ,δi0 ◦tmi ,ni+1 ,δi+1 = f ni+1 −ni . Assume mi , ni+1 , δi+1 and M > mi are given such that
²
for each integer n ≥ M there exists a 0 < ∆0m < E0 such that tmi ,ni+1 ,δi+1 ◦sni+1 ,mi+1 ,δ0 =
²
(f 0 )m−mi for 0 < δ 0 < ∆0m . By Lemma 3.8, there exists an integer mi+1 ≥ M ,
0 < ∆0 < E0 , and an integer N > ni+1 such that sni+1 ,mi+1 ,δ0 ◦ tmi+1 ,n,δ = f n−ni+1
²
0
for 0 < δ 0 < ∆0 and 0 < δ < ∆n . Choose 0 < δi+1
< min{∆0mi+1 , ∆0 }. Then
0
tmi ,ni+1 ,δi+1 ◦ sni+1 ,mi+1 ,δi+1
= (f 0 )mi+1 −mi .
²
The collections of 1-cycles {Cl(J1 ), . . . , Cl(Jm )} and {Cl(J10 ), . . . , Cl(Jn0 )} form
bases for the free abelian groups H1 (G; Z) and H1 (G0 ; Z), respectively. Let F and F 0
35
be the matrices of f∗ and f∗0 , respectively, relative to the above bases for H1 (G; Z) and
H1 (G0 ; Z). Since f and f 0 are orientation preserving, F and F 0 are square nonnegative
integer matrices.
Lemma 3.11. F and F 0 are weakly equivalent.
Proof. Let 0 < ² < min{E, E0 }. By Lemma 3.10, there exist two strictly increas∞
0 ∞
ing sequences {ni }∞
i=1 and {mi }i=1 of positive integers and two sequences {δi }i=1 and
ni+1 −ni
{δi }∞
and tmi ,ni+1 ,δi+1 ◦
i=2 of positive numbers such that sni ,mi ,δi0 ◦ tmi ,ni+1 ,δi+1 = f
²
0
sni+1 ,mi+1 ,δi+1
= (f 0 )mi+1 −mi for each i ∈ N. Let H : G × [0, 1] → G be the map defined
²
in the following manner. Given x ∈ G, there exists a unique arc Ix , possibly degenerate, contained in Bd (f ni+1 −ni (x), E) between the points f ni+1 −ni (x) and sni ,mi ,δi ◦
0
tmi ,ni+1 ,δi+1
(x). Let H(x, t) be the unique point on Ix such that d(f ni+1 −ni (x), H(x, t)) =
0
t · d(f ni+1 −ni (x), sni ,mi ,δi ◦ tmi ,ni+1 ,δi+1
(x)). Since H(x, 0) = f ni+1 −ni (x) and H(x, 1) =
0
0
(x).
(x) for all x ∈ G, f ni+1 −ni is homotopic to sni ,mi ,δi ◦ tmi ,ni+1 ,δi+1
sni ,mi ,δi ◦ tmi ,ni+1 ,δi+1
0
0
)∗ .
)∗ = (sni ,mi ,δi )∗ ◦ (tmi ,ni+1 ,δi+1
Thus (f∗ )ni+1 −ni = (f ni+1 −ni )∗ = (sni ,mi ,δi ◦ tmi ,ni+1 ,δi+1
0
)∗ ◦ (sni+1 ,mi+1 ,δi+1 )∗ . For i ∈ N, let Si and Ti
Similarly, (f∗0 )mi+1 −mi = (tmi ,ni+1 ,δi+1
0
)∗ , respectively, relative to the bases
be the matrices of (sni ,mi ,δi )∗ and (tmi ,ni+1 ,δi+1
for H1 (G; Z) and H1 (G0 ; Z). Then Si Ti = F ni+1 −ni and Ti Si+1 = (F 0 )mi+1 −mi . Since
sni ,mi ,δi is orientation preserving on G\Bd (b, δi ) and sni ,mi ,δi (Bd (b, δi )) is contained in
Bd0 (sni ,mi ,δi (x), E0 ), Si is a nonnegative integer matrix. Similarly, Ti is a nonnegative
integer matrix. Thus F and F 0 are weakly equivalent.
36
Since Mf = F and Mf 0 = F 0 , Mf is weakly equivalent to Mf 0 , which concludes
the proof of the Theorem 3.1.
Orientable Finite Connected Graphs without Endpoints
We first prove the following lemma utilizing techniques developed by Williams
[25].
Lemma 3.12. If G is an orientable finite connected graph with branch points but
without end points and f : G → G is an aperiodic collapsing surjective immersion
such that f (B(G)) = B(G) then there exist a positive integer n, an orientable bouquet
of circles G∗ with branch point b∗ , and a collapsing surjective immersion f∗ : G∗ → G∗
such that f∗ (b∗ ) = b∗ and f∗ is shift equivalent to f n .
Proof. If #B(G) = 1 then G is an orientable bouquet of circles whose branch
point is fixed by f . Thus by letting n = 1, G∗ = G and f∗ = f , f∗ is trivially shift
equivalent to f n .
If #B(G) ≥ 2 then we first construct a positive integer n, an orientable finite connected graph G1 without end points where #B(G1 ) = #B(G) such that there exists a
branch point of G1 that intersects the closure of every component of G1 \B(G1 ), and
a collapsing surjective immersion f1 : G1 → G1 such that f1 (b) = b for all b ∈ B(G1 )
and f1 is shift equivalent to f n .
37
If there exists a branch point of G that intersects the closure of every component of
G\B(G) then by letting n be a positive integer such that f n (b) = b for each b ∈ B(G),
G1 = G, and f1 = f n , f1 is trivially shift equivalent to f n .
If there does not exist a branch point of G that intersects the closure of every
component of G\B(G) then we construct n, G1 , and f1 as follows.
Let m be a positive integer such that f m (b) = b for all b ∈ B(G). Since there
does not exist a branch point of G that intersects the closure of every component of
G\B(G), we choose a b ∈ B(G) and let {V1 , . . . , Vj } be the components of G\B(G)
such that b ∈
/ Cl(Vi ). If U is a star neighborhood of b then there exists an I ∈ A(U )
such that f m (V ) ⊂ I for all sufficiently small star neighborhoods V of b since f m is
a collapsing immersion. For each Vi ∈ {V1 , . . . , Vj }, there exists a positive integer ki
and an xi ∈ Vi such that f ki m (xi ) = b and f ki m (V ) ⊂ I for sufficiently small star
neighborhoods V of xi since f m is an aperiodic collapsing immersion. Let n = km
where k = max{k1 , . . . , kj }.
Let G1 be the partition of G consisting of the finite set b1 ≡ {b, x1 , . . . , xj } and
the one-point sets {x} for x ∈
/ {b, x1 , . . . , xj } and let r : G → G1 be the surjective
function that takes each point of G to the element of G1 containing it. Assume G1 is
given the quotient topology determined by r. Since only a finite number of points of
G are identified to obtain G1 , G1 is a finite connected graph without end points and
#B(G1 ) ≥ 2. By the construction of G1 , b1 intersects the closure of every component
of G1 \B(G1 ).
38
We now proceed to define an arc structure system for G1 . Each star in G1 which is
an arc will have the natural arc structure an arc possesses. If x ∈ B(G)\{b} then there
exists a star neighborhood U of x in G such that U ∩ b1 = ∅. Since r is one-to-one on
U , r(U ) is a star neighborhood of the branch point {x} in G1 . Then A(r(U )) = {r(I) |
I ∈ A(U )} is the arc structure on r(U ) induced by the arc structure on U . This arc
structure on r(U ) then induces a coherent arc structure on every star neighborhood of
{x}. To construct an arc structure on every star neighborhood of b1 in G1 , we choose
a star neighborhood U0 of b and a star neighborhood Ui of each xi ∈ {x1 , . . . , xj }
such that Ui is an arc and {U0 , . . . , Uj } are pairwise disjoint. Then r(
Sj
i=0
Ui ) is a
star neighborhood of b1 in G1 . For each Ui ∈ {U0 , . . . , Uj }, let E` (Ui ) be the left end
points of Ui and Er (Ui ) be the right end points of Ui determined by an orientation
of G. Then E1 (r(
partition of E(r(
structure on r(
Sj
i=0
Sj
i=0
Sj
i=0
Ui )) =
Sj
i=0
r(E` (Ui )) and E2 (r(
Sj
i=0
Ui )) =
Ui )) into two nonempty subsets. Let A(r(
Sj
i=0
Sj
i=0
r(Er (Ui )) is a
Ui )) be the arc
Ui ) induced by this partition. This arc structure on r(
Sj
i=0
Ui )
then induces a coherent arc structure on every star neighborhood of b1 in G1 . Thus
we have defined an arc structure system for G1 . By definition of the arc structure
system for G1 , r is a surjective immersion which is not collapsing and an orientation
of G induces an orientation of G1 .
Let s : G1 → G be the function defined by s(x) = f n ◦ r|−1
G\b1 (x) for x ∈ G1 \{b1 }
and s(b1 ) = b. Since r|G\b1 is a topological embedding, f n ◦ r|−1
G\b1 : G1 \{b1 } → G
is continuous. So s is continuous at each x ∈ G1 \{b1 }. The image under s of every
39
sequence of points converging to b1 converges to b = s(b1 ). So s is continuous at b1 .
By definition, s is a collapsing surjective immersion such that s(b1 ) = b.
Let f1 : G1 → G1 be the map defined by f1 = r◦s. Since s is a collapsing surjective
immersion such that s(b1 ) = b and r is a surjective immersion such that r(b) = b1 , f1
is a collapsing surjective immersion such that f1 (b1 ) = b1 . Since f n = s ◦ r, f1 = r ◦ s,
r ◦ f n = f1 ◦ r, and s ◦ f1 = f n ◦ s, f1 is shift equivalent to f n .
Now we construct an orientable bouquet of circles with branch point b∗ and a
collapsing surjective immersion f∗ : G∗ → G∗ such that f∗ (b∗ ) = b∗ and f∗ is shift
equivalent to f1 in the following manner.
Let {U1 , . . . , Um } be the components of G1 \{b1 }. Then each Ui is an open arc
or simple open n-od (n ≥ 3) in G1 . For each Ui , there exists a unique collection of
distinct open arcs {Ii,1 , . . . , Ii,`i } such that there exists a component of G1 \f1−1 ({b1 })
whose image under f1 is Ii,j . Since f1 is surjective,
S`i
j=1 Ii,j
= Ui .
For each 1 ≤ i ≤ m and 1 ≤ j ≤ `i , let Oi,j be the set of ordered triples
(x, i, j) such that x ∈ Cl(Ii,j ) and let pi,j : Oi,j → G1 be the function defined by
pi,j (x, i, j) = x. Since pi,j is one-to-one, the topology on G1 naturally induces a
topology on Oi,j such that pi,j is a topological embedding. Since Cl(Ii,j ) is a circle
in G1 and pi,j (Oi,j ) = Cl(Ii,j ), Oi,j is a finite connected graph without end points or
branch points, that is, a circle.
Assume G2 =
S
1≤i≤m
1≤j≤`i
Oi,j is given the topology generated by the union of the
topologies on each space Oi,j . Then G2 is a finite graph without end points or branch
40
points, that is, a finite union of disjoint circles. Let p : G2 → G1 be the map defined
by p(x, i, j) = pi,j (x, i, j).
Let G∗ be the partition of G2 consisting of the one-point sets {(x, i, j)} for x 6= b1
and the finite set b∗ = {(b1 , i, j) | 1 ≤ i ≤ m and 1 ≤ j ≤ `i } and q : G2 → G∗ be
the surjective function that takes each point of G2 to the element of G∗ containing
it. Assume G∗ is given the quotient topology determined by q. Since only a finite
number of points in G2 are identified to obtain G∗ , G∗ is a bouquet of circles with
branch point b∗ .
Let r : G∗ → G1 be the function defined by r(x∗ ) = p◦q|−1
G2 \b∗ (x∗ ) for x∗ ∈ G∗ \{b∗ }
and r(b∗ ) = b1 . Since q is one-to-one on G2 \b∗ , p ◦ q|−1
G2 \b∗ is continuous. So r is
continuous at each x∗ ∈ G∗ \{b∗ }. The image under r of every sequence of points
converging to b∗ converges to b1 = r(b∗ ). So r is continuous at b∗ .
We now proceed to define an arc structure system for G∗ using the arc structure
system for G1 . Each star in G∗ which is an arc will be given the natural arc structure
an arc possesses. Let U be a star neighborhood of b1 in G1 . Then r−1 (U ) is a star
neighborhood of b∗ in G∗ . The arc structure on U naturally induces an arc structure
on r−1 (U ) in the following manner. The end points of U , E(U ), can be partitioned into
two subsets, E1 (U ) and E2 (U ), such that the end points of each arc of A(U ) are not
contained in the same partition and given a point from each partition the arc between
the points contained in U is an element of A(U ). Then E1 (r−1 (U )) = r −1 (E1 (U ))
and E2 (r−1 (U )) = r −1 (E2 (U )) is a partition of E(r −1 (U )) into two nonempty disjoint
41
subsets. Let A(r −1 (U )) be the arc structure on r −1 (U ) induced by this partition.
This arc structure on r −1 (U ) then induces a coherent arc structure on every star
neighborhood of b∗ . Thus we have defined an arc structure system for G∗ .
By definition of the arc structure system, r is a surjective immersion such that
r(b∗ ) = b1 which is not collapsing and an orientation of G1 induces an orientation of
G∗ .
Let {V1 , . . . , Vn } be the components of G1 \f1−1 ({b1 }). For each Vi , f1 (Vi ) is an
open arc such that f1 (Cl(Vi )) is a circle in G1 . By the construction of G∗ , there
exists one and only one open arc Ii of G∗ \{b∗ } such that r(Ii ) = f1 (Vi ) and for each
open arc I of G∗ \{b∗ } there exists a V ∈ {V1 , . . . , Vn } such that r(I) = f1 (V ). Let
s : G1 → G∗ be the function defined by s(x) = (r|Ii )−1 ◦f1 (x) for x ∈ Vi and s(x) = b∗
for x ∈ f1−1 ({b1 }). Since r|Ii is a topological embedding, (r|Ii )−1 ◦ f1 |Vi : Vi → Ii is
continuous. So s is continuous at each x ∈ G1 \f1−1 ({b1 }). If x ∈ f1−1 ({b1 }) then the
image under s of every sequence of points converging to x converges to b∗ = s(x).
So s is continuous at each x ∈ f1−1 ({b1 }). By definition, s is a collapsing surjective
immersion such that s(b1 ) = b∗ and f1 = r ◦ s.
Let f∗ : G∗ → G∗ be the map defined by f∗ = s ◦ r. Since r is a surjective
immersion such that r(b∗ ) = b1 and s is a collapsing surjective immersion such that
s(b1 ) = b∗ , f∗ is a collapsing surjective immersion such that f∗ (b∗ ) = b∗ . Since
s ◦ f1 = s ◦ r ◦ s = f∗ ◦ s and r ◦ f∗ = r ◦ s ◦ r = f ◦ r, f∗ is shift equivalent to f1 .
Thus f∗ is shift equivalent to f n .
42
Theorem 3.13. Let G and G0 be orientable finite connected graphs with branch
points but without end points and let f : G → G and f 0 : G0 → G0 be aperiodic
collapsing surjective immersions such that f (B(G)) = B(G) and f 0 (B(G0 )) = B(G0 ).
If lim{G, f } is homeomorphic to lim{G0 , f 0 } then there exist Perron numbers α and
←−
←−
n
β and positive integers m and n such that αλf = λm
f 0 and βλf 0 = λf .
Proof. By Lemma 3.12, we can choose positive integers, ` and `0 , orientable
bouquets of circles, G∗ and G0∗ , with branch points b∗ and b0∗ , respectively, and collapsing surjective immersions, f∗ : G∗ → G∗ and f∗0 : G0∗ → G0∗ , such that f∗ (b∗ ) = b∗ ,
0
f∗0 (b0∗ ) = b0∗ , f∗ is shift equivalent to f ` , and f∗0 is shift equivalent to (f 0 )` . Then
log(λf∗ ) = htop (f∗ ) = htop (f ` ) = `htop (f ) = log(λ`f ) and log(λf∗0 ) = htop (f∗0 ) =
0
0
0
htop ((f 0 )` ) = `0 htop (f 0 ) = log(λ`f 0 ) by Lemma 2.5. So λf∗ = λ`f and λf∗0 = λ`f 0 .
Since f∗ is shift equivalent to f ` , lim{G∗ , f∗ } is homeomorphic to lim{G, f ` }
←−
←−
which in turn is homeomorphic to lim{G, f }. Similarly, lim{G0∗ , f∗0 } is homeomorphic
←−
←−
to lim{G0 , f 0 }.
←−
Let Mf∗ be the aperiodic transition matrix for f∗ relative to some ordering of
the components of G∗ \B(G∗ ) and Mf∗0 be the aperiodic transition matrix for f∗0
relative to some ordering of the components of G0∗ \B(G0∗ ). Suppose lim{G, f } is
←−
homeomorphic to lim{G0 , f 0 }. Then lim{G∗ , f∗ } is homeomorphic to lim{G0∗ , f∗0 } and
←−
←−
←−
Mf∗ is weakly equivalent to Mf∗0 by Theorem 3.1. By Theorem 2.6, there exists a
Perron number a and a positive integer η such that aλf∗ = ληf∗0 . Let α = aλf`−1 ,
which is a Perron number since λf is a Perron number and Perron numbers are
43
closed under multiplication [13], and let m = `0 η, which is a positive integer. Then
0
αλf = aλ`f = aλf∗ = ληf∗0 = (λ`f 0 )η = λm
f 0 . Similarly, there exists a Perron number β
and a positive integer n such that βλf 0 = λnf .
Non-Orientable Finite Connected Graphs without Endpoints
Our goal in this section is to prove the following theorem utilizing techniques
similar to those used by Fokkink [12] to analyze matchbox manifolds.
Theorem 3.14. Let G and G0 be non-orientable finite connected graphs with
branch points but without end points and let f : G → G and f 0 : G0 → G0 be aperiodic
collapsing surjective immersions such that f (B(G)) = B(G) and f 0 (B(G0 )) = B(G0 ).
If lim{G, f } is homeomorphic to lim{G0 , f 0 } then there exist Perron numbers α and
←−
←−
n
β and positive integers m and n such that αλf = λm
f 0 and βλf 0 = λf .
We begin the proof of the above theorem by constructing an orientable double
e be the set of all ordered pairs (x, Dx ) such that
cover for each finite graph. Let G
eD be the set of all
x ∈ G and Dx ∈ Dx . Given a directed star (U, DU ) in G, let U
U
e such that x ∈ Int(U ) and Dx is the direction at x induced by DU .
(x, Dx ) ∈ G
e naturally inherits a topology from the topology on G in the following
The set G
e induced by all directed stars in
manner. Let SeG be the collection of all subsets of G
e then there exists a directed star (U, DU ) such that x ∈ Int(U ) and
G. If (x, Dx ) ∈ G
eD ∈ SeG . If U
e and Ve are
Dx is the direction at x induced by DU . So (x, Dx ) ∈ U
U
elements of SeG then there exist directed stars (U, DU ) and (V, DV ) in G that induce
44
O+PSfrag replacements
O
R+
L−
L+
L
R
O−
R−
G
e
G
e
Figure 2. A non-orientable graph G with corresponding orientable covering space G
ertw wert wert wert wret wert wert wert wert wert wert wert wert tre.
e and Ve , respectively. If (x, Dx ) ∈ U
e ∩ Ve then (U, DU ) and (V, DV ) are coherently
U
directed near x. So there exists a directed star neighborhood (W, DW ) of x such that
W ⊂ U ∩ V , (W, DW ) and (U, DU ) are coherently directed, and (W, DW ) and (V, DV )
fD ⊂ U
e ∩ Ve . Therefore SeG is a basis for a
are coherently directed. Thus (x, Dx ) ∈ W
W
e We will assume the set G
e is endowed with the topology generated by
topology on G.
SeG .
e → G be the open surjective two-to-one map defined by pr1 (x, Dx ) = x.
Let pr1 : G
eD and
Let x ∈ G and (U, DU ) be a directed star neighborhood of x in G. Then U
U
e−D are two disjoint open sets in G
e whose union is pr1−1 (Int(U )) and pr1 | e and
U
U
UD
U
e is a two-fold covering of G with
pr1 |Ue−D are homeomorphisms onto Int(U ). Thus G
U
covering map pr1 .
e is a finite graph with branch points but without end points.
Lemma 3.15. G
45
e → G is
Proof. Since G is a compact second-countable metric space and pr1 : G
e is a compact second-countable metrizable space. Since
a two-to-one covering map, G
e is a finite graph with branch points but
pr1 is a two-to-one local homeomorphism, G
without end points.
e be an element of SeG and (U, DU ) be the directed star in G that induces
Let U
e ) is a star in G
e and pr1 | e is a topological embedding. Let A(U ) be
it. Then Cl(U
Cl(U )
e )) = {pr1 |−1 (I)|I ∈ A(U )}. Since pr1 | e is a
the arc structure on U and A(Cl(U
e)
Cl(U )
Cl(U
e )) is an arc structure on the star Cl(U
e ). Thus an
homeomorphism onto U , A(Cl(U
e When
arc structure system for G naturally induces an arc structure system for G.
e we will assume it is endowed with the arc structure system induced
considering G,
by the arc structure system for G.
e is orientable.
Lemma 3.16. G
e1 , . . . , U
en } be a finite collection of elements of SeG that cover G.
e
Proof. Let {U
ei . Let ρi be
For i = 1, . . . , n, let (Ui , DUi ) be the directed star in G that induces U
ei ) → R be the map defined by
a directing map for Ui representing DUi and ρ̃i : Cl(U
ei )) then pr1 (I)
˜ ∈ A(Ui ) and ρ̃i | ˜ = ρi | ˜ ◦ pr1 | ˜
ρ̃i = ρi ◦ pr1 |Cl(Uei ) . If I˜ ∈ A(Cl(U
I
pr1 (I)
I
ei ). Let (Cl(U
ei ), D e )
is a topological embedding. So ρ̃i is a directing map for Cl(U
Cl(Ui )
ei ∩ U
ej then (Ui , DU )
be the directed star with direction DCl(Uei ) = [ρ̃i ]. If (x, Dx ) ∈ U
i
and (Uj , DUj ) are coherently directed near x. So there exists a star neighborhood V
of x in G contained in Ui ∩ Uj such that ρi |V is equivalent to ρj |V . Let (V, DV ) be
ei ∩ U
ej .
the directed star with direction DV = [ρi |V ] = [ρj |V ]. Then (x, Dx ) ∈ VeDV ⊂ U
46
˜ ∈ A(V ). Since ρ̃j | ˜ ◦ ρ̃i |−1 = ρj | ˜ ◦ pr1 | ˜ ◦ pr1 |−1 ◦
If I˜ ∈ A(Cl(VeDV )) then pr1 (I)
I
pr1 (I)
I
I˜
I˜
−1
−1
˜
˜
ρi |−1
˜ ◦ ρi |pr (I)
˜ = ρj |pr1 (I)
˜ , ρ̃j |I˜ ◦ ρ̃i |I˜ : ρ̃i (I) → ρ̃j (I) is an increasing map. So
pr (I)
1
ρ̃i |Cl(VeD
1
V
)
is equivalent to ρ̃j |Cl(VeD
V
)
ei ), D e ) and (Cl(U
ej ), D e ) are
and (Cl(U
Cl(Ui )
Cl(Uj )
e1 ), D e ), . . . , (Cl(U
en ), D e )} is a
coherently directed near (x, Dx ). Thus {(Cl(U
Cl(U1 )
Cl(Un )
e whose interior covers G.
e Therefore G
e is
collection of coherently directed stars in G
orientable.
e is connected.
Lemma 3.17. G
e is not connected. Since pr1 is
Proof. For purposes of contradiction, assume G
a two-to-one clopen map and G is connected, there exist two disjoint clopen subsets,
e1 and G
e2 , of G
e such that G
e =G
e1 ∪ G
e2 and pr1 (G
e1 ) = pr1 (G
e2 ) = G. Since G
e is
G
e1 is homeomorphic to G, G is orientable. But this contradicts the
orientable and G
e is connected.
fact that G is non-orientable. Thus G
e →S
Let pr2 : G
x∈G Dx be the surjective function defined by pr2 (x, Dx ) = Dx .
Assume
S
x∈G
Dx is endowed with the quotient topology induced by pr2 .
Then
e )|U
e ∈ SeG } is a basis for the topology on S
e
{pr2 (U
x∈G Dx . Let (x, Dx ) ∈ G and
V be a star neighborhood of f (x) in G. Since f is a collapsing immersion, there
exists a star neighborhood U of x in G such that the restriction of f to each arc in
A(U ) is a topological embedding into an arc of A(V ) and f (U ) is contained in an arc
of A(V ). Let DU be the direction for U that induces Dx and let ρV be a directing map
on V . Then ρV ◦ f |U : U → R is a directing map on U . Let f+ :
S
x∈G
Dx →
S
x∈G
Dx
be the function defined in the following manner. If [ρV ◦ f |U ] = DU , let f+ (Dx ) be the
47
direction at f (x) induced by [ρV ]. If [ρV ◦ f |U ] = −DU , let f+ (Dx ) be the direction
at f (x) induced by −[ρV ].
Let Dx ∈
S
x∈G
e )|U
e ∈ SeG } such that f+ (Dx ) ∈ pr2 (Ve ).
Dx and pr2 (Ve ) ∈ {pr2 (U
Let (V, DV ) be the directed star neighborhood of f (x) that induces Ve . Then f+ (Dx )
is the direction at f (x) induced by DV . Since f is continuous at x there exists a star
neighborhood U of x such that f (U ) ⊂ V . Let DU be the direction for U that induces
eD ) is a neighborhood of Dx such that f+ (pr2 (U
eD )) ⊂ pr2 (Ve ). Thus
Dx . Then pr2 (U
U
U
f+ is continuous at Dx . Hence f+ is continuous.
e → G
e be the function defined by f˜(x, Dx ) = (f ◦ pr1 (x, Dx ), f+ ◦
Let f˜ : G
pr2 (x, Dx )). Then pr1 ◦ f˜ = f ◦ pr1 by definition.
Lemma 3.18. f˜ is an aperiodic collapsing surjective immersion such that
e = B(G).
e
f˜(B(G))
e and (V, DV ) be a directed star neighborProof. f˜ is surjective: Let (z, Dz ) ∈ G
hood of z such that DV is coherent with Dz . Since f is a surjective map, there exists
an x ∈ G such that f (x) = z and a star neighborhood U of x such that f (U ) ⊂ V . Let
DU = [ρV ◦ f |U ] where ρV is a representative of DV , and Dx be the direction at x induced by DU . Then f+ (Dx ) = Dz . Thus f˜(x, Dx ) = (f ◦pr1 (x, Dx ), f+ ◦pr2 (x, Dx )) =
(z, Dz ).
f˜ is a collapsing immersion: Since f ◦ pr1 and f+ ◦ pr2 are continuous, f˜ is
e and Ve ∈ SeG containing f˜(x, Dx ). Then Cl(Ve ) is
continuous. Let (x, Dx ) ∈ G
a star neighborhood of f˜(x, Dx ) = (f (x), f+ (Dx )) and V ≡ pr1 (Cl(Ve )) is a star
48
neighborhood of f (x). Let DV be the direction for V that induces f+ (Dx ). Since f is
a collapsing immersion at x there exists a star neighborhood U of x and a J ∈ A(V )
such that the restriction of f to each arc in A(U ) is a topological embedding into
J. Then pr1 |−1
◦ f |I is a topological embedding into pr1 |−1
(J) ∈ A(Cl(Ve )) for
Cl(Ve )
Cl(Ve )
eD )
each I ∈ A(U ). Let DU be the direction for U that induces Dx . Then Cl(U
U
eD )), pr1 (I)
˜ ∈ A(U ) and f˜| ˜ =
is a star neighborhood of (x, Dx ). For I˜ ∈ A(Cl(U
U
I
−1
(J). Thus f˜ is a
◦ f |pr1 (I)
pr1 |−1
˜ ◦ pr1 |I˜ is a topological embedding into pr1 |
Cl(Ve )
Cl(Ve )
collapsing immersion at (x, Dx ).
f˜ is aperiodic: Let {J1 , . . . , Jn } be the components of G\B(G). Since f is aperiodic, there exists a positive integer M such that f m (Ji ) = G for each Ji and m ≥ M .
e
e such that Ji,1 ∪ Ji,2 = pr1−1 (Ji ). So
Let {Ji,1 , Ji,2 } be the components of G\B(
G)
e
e
pr1 ◦ f˜m (Ji,` ) = f m ◦ pr1 (Ji,` ) = f m (Ji ) = G for each component Ji,` of G\B(
G)
and m ≥ M . By definition of f˜, if Ji,1 ⊂ f˜(Ji,` ) then Ji,2 ⊂ f˜(Ji,3−` ). So either
Ji,1 ⊂ f˜m (Ji,` ) or Ji,2 ⊂ f˜m (Ji,` ). Choose m ≥ M such that f˜m (b) = b for each
e and Ji,` ⊂ f˜m (Ji,` ) for each Ji,` . Since the number of branch points and
b ∈ B(G)
e
e are finite there exists a k such that f˜km (Ji,` ) = f˜(k+1)m (Ji,` )
components of G\B(
G)
for each Ji,` . Let A1 = f˜km (J1,1 ) and A2 = f˜km (J1,2 ). For purposes of contradiction,
e then there exists a Ji,` containing x that is conassume A1 ∩ A2 = ∅. If x ∈ A1 \B(G)
e
tained in A1 . So Ji,` is an open neighborhood of x contained in A1 . If x ∈ A1 ∩ B(G)
then given a star neighborhood W of x there exists a star neighborhood V of x and
an I ∈ A(W ) such that f˜m (V ) ⊂ I. Since x ∈ A1 , there exists a z ∈ J1,1 and a star
49
neighborhood U of z such that f˜(k+1)m (U ) ⊂ I. So I ⊂ A1 . Since f˜m (V ) ⊂ I and
A1 ∩ A2 = ∅, V ⊂ A1 . So Int(V ) is an open neighborhood of x contained in A1 . Thus
A1 is an open set. Similarly, A2 is an open set. Since A1 and A2 are two disjoint open
e G
e is not connected. But this contradicts the fact that G
e is
sets whose union is G,
e then there exists a Ji` containing x
connected. Thus A1 ∩A2 6= ∅. If x ∈ A1 ∩A2 \B(G)
that is contained in A1 ∩ A2 . If J1,1 ⊂ f˜km (Ji,` ) then A1 = f˜km (J1,1 ) ⊂ A1 ∩ A2 . Thus
e Similarly, if J1,2 ⊂ f˜km (Ji,` ) then A1 = A2 = G.
e If x ∈ A1 ∩ A2 ∩ B(G)
e
A1 = A2 = G.
then given a star neighborhood W of x there exists a star neighborhood V of x and
an I ∈ A(W ) such that f˜m (V ) ⊂ I. Since x ∈ A1 , there exists a z1 ∈ I1,1 and a star
neighborhood U1 of x contained in I1,1 such that f˜(k+1)m (I1,1 ) ⊂ I. Since x ∈ A2 ,
there exists a z2 ∈ I1,2 and a star neighborhood U2 of x contained in I1,2 such that
e
e such that
f˜(k+1)m (I1,2 ) ⊂ I. So I ⊂ A1 ∩ A2 . If Ji,` is a component of G\B(
G)
e by earlier argument. If Ji,` is a
Ji,` ∩ I 6= ∅ then Ji,` ⊂ A1 ∩ A2 . So A1 = A2 = G
e
e then J1,1 or J1,2 is a subset of f˜m (Ji,` ). If J1,1 ⊂ f˜m (Ji,` ) then
component of G\B(
G)
G = f˜km (J1,1 ) ⊂ f˜(k+1)m (Ji,` ). Similarly, if J1,2 ⊂ f˜m (Ji,` ) then G = f˜(k+1)m (Ji,` ).
Thus f˜ is aperiodic.
e f˜} → lim{G, f } be the ladder map induced by pr1 .
Let P : lim{G,
←−
←−
Lemma 3.19. P is a two-to-one covering map.
Proof. Since pr1 is two-to-one covering map and f˜|pr1−1 ({x}) is a one-to-one map
e = G, P is an open two-to-one surjective map by Theorem 2.8. Let
for each x ∈ pr1 (G)
e f˜}, respectively.
πi and π
ei be the i-th projection maps acting on lim{G, f } and lim{G,
←−
←−
50
Let x ∈ lim{G, f } and (U, DU ) be a directed star neighborhood of π1 (x) in G. Since
←−
Int(U ) is an open neighborhood of π1 (x) in G, π1−1 (Int(U )) is an open neighborhood
eD and U
e−D are two disjoint open sets in G
e such
of x in lim{G, f }. By definition, U
U
U
←−
eD ∪ U
e−D and both pr1 | e and pr1 | e
that pr1−1 (Int(U )) = U
U
U
UD
U−D are homeomorphisms
U
U
eD ) and π
e−D ) are two disjoint open sets in lim{G,
e f˜} such
onto Int(U ). So π
e1−1 (U
e1−1 (U
U
U
←−
eD ) ∪ π
e−D ). Since P is an
e1−1 (U
that P −1 (π1−1 (Int(U ))) = π
e1−1 (pr1−1 (Int(U ))) = π
e1−1 (U
U
U
open map, P |πe−1 (UeD
1
U
)
and P |πe−1 (Ue−D
1
U
)
are open maps. Let z ∈ π1−1 (Int(U )). As a
result of the construction of P −1 ({z}) given an element z of lim{G, f }, in Theorem
←−
2.8, if {z̃ 1 , z̃ 2 } = P −1 ({z}) then {e
π1 (z̃ 1 ), π
e1 (z̃ 2 )} = pr1−1 ({π1 (z)}). Since pr1 |UeD and
U
pr1 |Ue−D are homeomorphisms onto Int(U ), one element of pr1−1 ({π1 (z)}) is contained
U
eD while the other element is contained in U
e−D . Thus one element of P −1 ({z}) is
in U
U
U
eD ) while the other element is contained in π
e−D ). So P | −1 e
contained in π
e1 (U
e1 (U
U
U
π
e (UD
1
and P |πe−1 (Ue−D
1
U
)
U
)
are homeomorphisms onto π1−1 (Int(U )). Then π1−1 (Int(U )) is evenly
covered by P . Thus P is a two-to-one covering map.
e f˜} induces a
Since P is a local homeomorphism the direction at each x̃ ∈ lim{G,
←−
direction at P (x̃).
Lemma 3.20. The directions at each x ∈ lim{G, f } induced by the directions at
←−
the two elements of P −1 ({x}) are different.
Proof. Let lim{G, f }+ be the set of all x ∈ lim{G, f } such that the directions
←−
←−
at x induced by the two elements of P −1 ({x}) are the same and lim{G, f }− be the
←−
set of all x ∈ lim{G, f } such that the directions at x induced by the both elements
←−
51
e f˜} would
of P −1 ({x}) are different. If lim{G, f }− = ∅ then the orientation of lim{G,
←−
←−
induce an orientation of lim{G, f }. Since lim{G, f } is non-orientable, lim{G, f }− 6= ∅.
←−
←−
←−
For purposes of contradiction, assume lim{G, f }+ 6= ∅. Let x ∈ lim{G, f }+ . Since P
←−
←−
is a two-to-one covering map, there exists a matchbox neighborhood M of x such that
P −1 (M ) is the union of two disjoint matchboxes M̃1 and M̃2 such that the restriction
of P to each is a homeomorphism onto M . Let ρ1 be a directing map on M̃1 and ρ2 be
e f˜}. Then ρ1 ◦P |−1 and
a directing map on M̃2 coherent with the orientation of lim{G,
M̃1
←−
ρ2 ◦ P |−1
are two directing maps on M . Let M+ be the union of all matches I in M
M̃
2
−1
−1
such that (ρ2 ◦P |−1
)|I ◦(ρ1 ◦P |−1
)|−1
I : ρ1 ◦P |M̃ (I) → ρ2 ◦P |M̃ (I) is increasing. Then
M̃
M̃
2
1
1
2
M+ is a matchbox neighborhood of x contained in lim{G, f }+ by Proposition 2.13. So
←−
lim{G, f }+ is open. Similarly, lim{G, f }− is open. Since lim{G, f }+ and lim{G, f }−
←−
←−
←−
←−
are two disjoint open sets whose union is lim{G, f }, lim{G, f } is not connected. But
←−
←−
this contradicts the fact that lim{G, f } is a continuum. Thus lim{G, f }+ = ∅.
←−
←−
e0 , pr10 : G
e0 → G0 , f˜0 : G
e0 → G
e0 , and P 0 : lim{G
e0 , f˜0 } → lim{G0 , f 0 }
We construct G
←−
←−
similarly.
Assume lim{G, f } and lim{G0 , f 0 } are homeomorphic. Let φ : lim{G, f } →
←−
←−
←−
e f˜}, let φ̃(x) be the
lim{G0 , f 0 } be a homeomorphism. For each element x in lim{G,
←−
←−
e0 , f˜0 } such that the direction at φ ◦ P (x) induced by the direction at
element of lim{G
←−
x is the same as the direction at φ ◦ P (x) induced by the direction at φ̃(x).
Lemma 3.21. φ̃ is a homeomorphism.
52
e0 , f˜0 }. By Lemma 3.20 and the fact that φ
Proof. φ̃ is surjective: Let x0 ∈ lim{G
←−
is a homeomorphism, there exists one and only one x ∈ (φ ◦ P )−1 ({P 0 (x0 )}) such that
the direction at φ ◦ P (x) induced by the direction at x is the same as the direction
at P 0 (x0 ) induced by the direction at x0 . Then φ̃(x) = x0 .
φ̃ is injective: Suppose that φ̃(x) = φ̃(z). Then φ ◦ P (x) = φ ◦ P (z) and the
direction at φ ◦ P (x) induced by the direction at x is the same as the direction at
φ ◦ P (z) induced by the direction at z. Then x = z by Lemma 3.20.
e f˜} and V be an open neighborhood of φ̃(x). Since
φ̃ is continuous: Let x ∈ lim{G,
←−
φ ◦ P and P 0 are two-to-one covering maps, there exists a matchbox neighborhood M
of φ ◦ P (x) contained in P 0 (V ) such that (φ ◦ P )−1 (M ) is the union of two disjoint
matchboxes such that φ ◦ P restricted to each is a homeomorphism onto M and
(P 0 )−1 (M ) is the union of two disjoint matchboxes such that P 0 restricted to each
is a homeomorphism onto M . Let M̃ be a matchbox neighborhood of x such that
φ ◦ P |M̃ is a homeomorphism onto M and M̃ 0 be a matchbox neighborhood of φ̃(x)
such that M̃ 0 ⊂ V and P 0 |M̃ 0 is a homeomorphism onto M . Let ρ be a directing
e f˜} and ρ0 be a directing map on
map on M̃ coherent with the orientation of lim{G,
←−
e0 , f˜0 }. Then ρ ◦ (φ ◦ P | )−1 and ρ0 ◦ P |−10
M̃ 0 coherent with the orientation of lim{G
M̃
M̃
←−
are the induced directing maps on M . Let M+ be the union of the matches I in M
(I) → ρ ◦ (φ ◦ P |M̃ )−1 (I) is
)|−1 : ρ0 ◦ P |−1
such that (ρ ◦ (φ ◦ P |M̃ )−1 )|I ◦ (ρ0 ◦ P |−1
M̃ 0
M̃ 0 I
increasing. By definition of φ̃(x), the match in M containing φ ◦ P (x) is contained
in M+ . Let M̃+ = (φ ◦ P |M̃ )−1 (M+ ) and M̃+0 = P 0 |−1
(M+ ). Then M̃+ is a matchbox
M̃ 0
53
neighborhood of x such that φ̃(M̃+ ) = M̃+0 . Since M̃+ ⊂ V , φ̃ is continuous at x. Thus
e f˜} is compact and lim{G
e0 , f˜0 } is
φ̃ is continuous at each x ∈ lim{G̃, f˜}. Since lim{G,
←−
←−
←−
Hausdorff, φ̃ is a homeomorphism.
e and G
e0 are orientable finite connected graphs with branch points but
Since G
e → G,
e f˜0 : G
e0 → G
e0 are aperiodic collapsing surjective
without end points and f˜ : G
e = B(G)
e and f˜0 (B(G
e0 )) = B(G
e0 ), and lim{G,
e f˜} is
immersions such that f˜(B(G))
←−
e0 , f˜0 }, there exist Perron numbers α and β and positive intehomeomorphic to lim{G
←−
e→G
and βλf˜0 = λnf˜ by Theorem 3.13. Since pr1 : G
gers m and n such that αλf˜ = λm
f˜0
e0 → G0 are two-to-one covering maps such that pr1 ◦ f˜ = f ◦ pr1 and
and pr10 : G
pr10 ◦ f˜0 = f 0 ◦ pr10 , htop (f˜) = htop (f ) and htop (f˜0 ) = htop (f 0 ) by Theorem 2.4. So
n
λf˜ = λf and λf˜0 = λf 0 . Thus αλf = λm
f 0 and βλf 0 = λf , which concludes the proof of
the theorem.
Finite Connected Graphs without Endpoints
Theorem 3.22. Let G and G0 be finite connected graphs with branch points but
without end points and let f : G → G and f 0 : G0 → G0 be aperiodic collapsing surjective immersions such that f (B(G)) = B(G) and f 0 (B(G0 )) = B(G0 ). If lim{G, f } is
←−
homeomorphic to lim{G0 , f 0 } then there exist Perron numbers α and β and positive
←−
n
integers m and n such that αλf = λm
f 0 and βλf 0 = λf .
Proof. Assume lim{G, f } is homeomorphic to lim{G0 , f 0 }. Then either lim{G, f }
←−
←−
←−
and lim{G0 , f 0 } are both orientable or both non-orientable. So either G and G0 are
←−
54
both orientable or both non-orientable. If G and G0 are both orientable then then
there exist Perron numbers α and β and positive integers m and n such that αλf = λm
f0
and βλf 0 = λnf by Theorem 3.13. If G and G0 are both non-orientable then then there
exist Perron numbers α and β and positive integers m and n such that αλf = λm
f0
and βλf 0 = λnf by Theorem 3.14.
Barge and Diamond [4] have shown that given two Perron numbers λ and γ, if
there are Perron numbers α and β and positive integers m and n such that αλ = γ m
and βγ = λn then Q(λ) = Q(γ). This results in the following corollary to the above
theorem.
Corollary 3.23. Let G and G0 be finite connected graphs with branch points but
without end points and let f : G → G and f 0 : G0 → G0 be aperiodic collapsing surjective immersions such that f (B(G)) = B(G) and f 0 (B(G0 )) = B(G0 ). If lim{G, f }
←−
is homeomorphic to lim{G0 , f 0 } then Q(λf ) = Q(λf 0 ).
←−
It should be noted that the algebraic extension relationship between spectral
radii is weaker than the Perron relationship between the spectral radii. To see this,
let λf = 2 and λf 0 = 3. Then Q(λf ) = Q(λf 0 ) but there do not exist Perron numbers,
n
α and β, and positive integers, m and n such that αλf = λm
f 0 and βλf 0 = λf .
Two Examples
Our first example shows that we can use weak equivalence of matrices to determine
when certain inverse limits of collapsing surjective immersions on orientable bouquets
55
of circles which fix branch points are not homeomorphic. Our second example shows
that weak equivalence of transition matrices is not a complete topological invariant
for inverse limits of collapsing surjective immersions on orientable bouquets of circles
which fix branch points. The second example requires a slight understanding of
substitution tilings. For a discussion of this topic, consult Anderson and Putnam [2].
Example 1
Let G be an oriented bouquet of two circles S1 and S2 . Let f : G → G be an
immersion which fixes the branch point such that S1 → S1 S1 S2 S1 S1 and S2 → S1 . Let
f 0 : G → G be an immersion which fixes the branch point such that S1 → S1 S2 S2 S1 S1
and S2 → S1 S2 S1 . Then f and f 0 are collapsing surjective immersions. The transition
matrices for f and f 0 relative to the natural ordering of the components of G\B(G)
¶
µ
¶
µ
3 2
4 1
. Swanson and Volkmer [23] show that these
and Mf 0 =
are Mf =
2 1
1 0
primitive matrices are not weakly equivalent by an ideal argument. Thus lim{G, f }
←−
and lim{G, f 0 } are not homeomorphic by Theorem 3.1.
←−
√
Since λf = 2 + 5 = λf 0 , there trivially exists Perron numbers α and β and
n
positive integers m and n such that αλf = λm
f 0 and βλf 0 = λf . Thus we can not use
Theorem 3.22 or Corollary 3.23 to determine whether lim{G, f } and lim{G, f 0 } are
←−
←−
homeomorphic or not.
56
Example 2
Let ϕ be the substitution on two letters given by ϕ(1) = 11221, ϕ(2) = 1 and Tϕ
be the associated tiling space. Let χ be the substitution on two letters given by χ(1) =
12121, χ(2) = 1 and Tχ be the associated tiling space. Using techniques developed
by Anderson and Putnam [2], there exist orientable bouquets of two circles Gϕ and
Gχ and collapsing surjective immersions fϕ : Gϕ → Gϕ and fχ : Gχ → Gχ , which fix
branch points, such that Tϕ is homeomorphic to lim{Gϕ , fϕ } and Tχ is homeomorphic
←−
to lim{Gχ , fχ }. We can order the components of Gϕ \B(Gϕ ) and Gχ \B(Gχ ) such that
←−
µ
¶
3 1
transition matrices for fϕ and fχ relative to these orderings are Mϕ =
= Mχ .
2 0
These transition matrices are, trivially, weakly equivalent. But, Barge and Diamond
[6] show that Tϕ and Tχ are not homeomorphic using weak equivalence of associated
substitutions. Thus lim{Gϕ , fϕ } is not homeomorphic to lim{Gχ , fχ }.
←−
←−
One-Dimensional Hyperbolic Attractors
We first state the following result due to Williams [24], [25].
Theorem 3.24. If g : X → X is a diffeomorphism of a manifold having a connected one-dimensional hyperbolic attractor Ω then there exist two positive integers k
and `, a compact branched one-manifold G which is topologically a bouquet of circles,
and an expansive immersion f : G → G where the branch point of G is fixed under f ,
every point of G is non-wandering under f , and every point of G has a neighborhood
57
whose image under f ` is an arc such that g|kΩ is topologically conjugate to the shift
homeomorphism induced by f .
Let g : X → X be a diffeomorphism of a manifold with one-dimensional connected
hyperbolic attractor Ω. By Theorem 3.24, there exist two positive integers k and `,
a compact branched one-manifold G which is topologically a bouquet of circles, and
an expansive immersion f : G → G where the branch point of G is fixed under f ,
every point of G is non-wandering under f , and every point of G has a neighborhood
whose image under f ` is an arc such that the shift homeomorphism induced by f is
topologically conjugate to g|kΩ .
Assume G is endowed with the arc structure system that the differentiable structure on G naturally induces. Williams [25] showed that if I is an arc in G then there
exists a positive integer j such that G ⊂ f j (I), f as above. So f ` is a collapsing
surjective immersion that fixes the branch point of G.
Let g 0 : X 0 → X 0 be a diffeomorphism of a manifold with one-dimensional connected hyperbolic attractor Ω0 . By Theorem 3.24, there exist two positive integers
k 0 and `0 , a compact branched one-manifold G0 which is topologically a bouquet of
circles, and an expansive immersion f 0 : G0 → G0 where the branch point of G0 is fixed
under f 0 , every point of G0 is non-wandering under f 0 , and every point of G0 has a
0
neighborhood whose image under (f 0 )` is an arc such that the shift homeomorphism
0
induced by f 0 is topologically conjugate to g 0 |kΩ0 .
58
Assume G0 is endowed with the arc structure system that the differentiable struc0
ture on G0 naturally induces. Then (f 0 )` is a surjective collapsing immersion that
fixes the branch point of G0 .
Assume Ω is homeomorphic to Ω0 . Since g|kΩ is topologically conjugate to the shift
0
map fˆ induced by f and g 0 |kΩ0 is topologically conjugate to the shift map fˆ0 induced
by f 0 , lim{G, f } is homeomorphic to lim{G0 , f 0 }. Thus lim{G, f ` } is homeomorphic
←−
←−
←−
0
to lim{G0 , (f 0 )` }.
←−
Let Mf be the transition matrix for f relative to an ordering of the components
of G\B(G) and let Mf 0 be the transition matrix for f 0 relative to an ordering of the
components of G0 \B(G0 ).
Proposition 3.25. If Ω and Ω0 are orientable then Mf is weakly equivalent to
Mf 0 .
0
Proof. Since Ω and Ω0 are orientable, lim{G, f ` } and lim{G0 , (f 0 )` } are ori←−
←−
entable. So Mf ` is weakly equivalent to M(f 0 )`0 by Theorem 3.1. Since Mf ` = (Mf )`
0
and M(f 0 )`0 = (Mf 0 )` , Mf is weakly equivalent to Mf 0 .
Assume htop (g|Ω ) = log(ω) and htop (g 0 |Ω0 ) = log(ω 0 ). Using results of Williams
[24], there exists a connected finite graph K and an aperiodic collapsing surjective
immersion h : K → K where h(B(K)) = B(K) such that the shift homeomorphism
ĥ induced by h is topologically conjugate to g|Ω . Bowen [10] showed that the the
topological entropy of the induced shift homeomorphism on an inverse limit of a
compactum with single surjective bonding map is equal to the topological entropy of
59
the bonding map. By Theorem 2.4 and above result, htop (h) = htop (ĥ) = htop (g|Ω ) =
log(ω). If Mh is the transition matrix for h relative to an ordering of the components
of K\B(K) then the spectral radius of Mh is ω. Since Mf is aperiodic, ω is a Perron
number. Similarly, ω 0 is a Perron number.
Proposition 3.26. There exist Perron numbers, α and β, and positive integers,
m and n, such that αω = (ω 0 )m and βω 0 = ω n .
Proof. By Theorem 2.4, htop (fˆ) = htop (g|kΩ ) = k log(ω) = log(ω k ) and htop (fˆ0 ) =
0
0
htop (g 0 |kΩ0 ) = k 0 log(ω 0 ) = log((ω 0 )k ). Since the topological entropy of the induced shift
homeomorphism on an inverse limit of a compactum with single surjective bonding
map is equal to the topological entropy of the bonding map, htop (f ) = htop (fˆ) =
0
log(ω k ) and htop (f 0 ) = htop (fˆ0 ) = log((ω 0 )k ). Then htop (f ` ) = ` log(ω k ) = log(ω k` )
0
0
0 0
0 0
and htop ((f 0 )` ) = `0 log((ω 0 )k ) = log((ω 0 )k ` ). So λf ` = ω k` and λ(f 0 )`0 = (ω 0 )k ` .
By Theorem 3.22, there exists a Perron number a and a positive integer η such that
aλf ` = λη(f 0 )`0 . Let α = aω k`−1 , which is a Perron number since ω is a Perron number, and let m = k 0 `0 η, which is a positive integer. Then αω = aλf ` = λη(f 0 )`0 =
0 0
((ω 0 )k ` )η = (ω 0 )m . Similarly, there exists a Perron number β and positive integer n
such that βω 0 = ω n .
The above Perron relationship implies the weaker result that Q(ω) = Q(ω 0 ).
60
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