Embeddings of inverse limit spaces
by James Lee Kassebaum
A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
Mathematical Sciences
Montana State University
© Copyright by James Lee Kassebaum (1995)
Abstract:
Barge and Martin have shown that the inverse limit of any interval map may be topologically realized
as an attractor of a planar homeomorphism H in such a way that H restricted to the attractor is
conjugate to the induced homeomorphism on the inverse limit. They extend the interval map to a near
homeomorphism of a disk containing the interval and use a theorem of Morton Brown to obtain their
result. We present similar results for degree -1, 0, or 1 circle maps and give necessary conditions for a
map to be extended to a near homeomorphism. Then, it is shown that for a given finite, connected,
planar, graph, G, containing a branch point, the set of surjective, continuous maps of G which cannot
be extended to a near homeomorphism on any neighborhood of G are open and dense in the set of all
such maps, with the C^0-topology. However, an example of such a map for which the inverse limit can
be embedded as an attractor in the plane is given. Next, we prove the inverse limit of any n-od (n ≥ 3)
can be embedded as an attractor in 3-space. We then give necessary and sufficient conditions for the
inverse limit of a finite, connected, planar, graph with a surjective bonding map to be chainable. Since
any chainable continuum may be embedded in the plane, such an inverse limit is planar. Finally, we
give an example of a chainable continuum for which there exists a homeomorphism which is not
essentially extendable. EMBEDDINGS OF
INVERSE LIMIT SPACES
by
James Lee Kassebaum
A thesis submitted in partial fulfillment
of the requirements for the degree
of
' Doctor of Philosophy
in
Mathematical Sciences
MONTANA STATE UNIVERSITY
Bozeman, Montana
October 1995
3 )3 ^
ii
A PPR O V A L
of a thesis submitted by
James Lee Kassebaum
This thesis has been read by each member of the thesis 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.
// / / / ^ r
Date
Marcy B a^e
'
Chairperson, Graduate Committee
Approved for the Major Department
Date
i ,
S
/ f f j"
Joh^i Lund
Head, Mathematical Sciences
Approved for the College of Graduate Studies
Date
Robert Brown
Graduate Dean
iii
ST A T E M E N T OF P E R M IS S IO N TO U SE
In presenting this thesis 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 thesis is
allowable only for scholarly purposes, consistent with “fair use” as prescribed in the
U.S. Copyright Law. Requests for extensive copying or reproduction of this thesis
should be referred to University Microfilms International, 300 North Zeeb Road, Ann
\
Arbor, Michigan 48106, to whom I have granted “the exclusive right to reproduce
and distribute my dissertation for' sale in and from microform or electronic format,
along with the right to reproduce and distribute my abstract in any format in whole
or in part.”
Signature
Date
iv
ACKNOW LEDGEM ENTS
I
would first like to thank Jesus Christ for His leading me to Montana State
University to work with such an impressive, yet accessible, group of mathematicians.
I count myself most fortunate to have had the opportunity to work with Marcy
Barge, without whose help the following work would not have been possible. Detailing
his efforts in full would require more paper than the thesis itself. The members
of my Graduate Committee, Marcy Barge, Russell Walker, Richard Swanson, Gary
Bogar, Richard Gillette, and Ray Ditterline, also deserve much praise for their efforts.
Finally, I would like to thank my parents, Tim'Snyder, Mary Oman, and the many
friends that I have made while at Montana State University for their support.
V
TABLE. OF CONTENTS
,
L IST O F F I G U R E S ..................
P age
vi
A B S T R A C T ..................................'....................... '............................................
viii
1. IN T R O D U C T IO N .........................................................................................
I
2. E M B E D D IN G C IR C L E -L IK E C O N T IN U A AS A T T R A C T O R S .
5
3. E M B E D D IN G G -L IK E C O N T IN U A AS A T T R A C T O R S ..............
22
4. E M B E D D IN G n-O D -L IK E C O N T IN U A AS A T T R A C T O R S . . .
48
5. C H A IN A B IL IT Y A N D E X T E N D IB IL IT Y ' ............... ■......................
71
R E F E R E N C E S C IT E D
93
...............
Vl
' LIST OF FIGURES
Figure
Page
1
A ..................................................................................................................
9
2
P -1 (A )...................: ..................................................................................
10
3
a o P ^ ( A ) ............... .......................................; .................. ..................
10
4
T o a o R - i( A ) ......................................................................
ii
5
H o T o a o R -1,( A ) ...................................................................................
11
6
T -i o R o T o a o R -i(A ) ..................................... ' ...................................
12
7
Ck-1 o T -1 o R o T o a o R _1(A) ............................................................. ' 12
8 ■ R o a -1 o T -1 o R o T o a o R - 1( A ) .......................................................
13
9
15
10 image of A: . ....................................................
30
11
31
12
32
13 /L u ,, ....................................................... : ..............................................
34
14 image of / ..................................................................................................
35
15
36
16
37
17
40
18
41
19
42
20 Disk D ........................................................................................................
43
21 D U Di U D2 U D3 U D4 ..........................................................................'.
45
22 K ..................................................................................................................
50
23 / 3 ..................................................................................................................
51
24 n = 3 : ........................................................................................................
51
25 Ti = 4 : ..........................................................
52
53
26 n > 4, n odd: ..................... •.....................................................................
27 n > 4, rt e v e n :.............................................................................
54
28 R , . . .........................................................................................................
56
29 y = gz(x) ..........................................
57
30 % ...............................................................................................................
58
31 y — vz(x) ..................................................................................................
59
32
62
33
..................................... ... . ...................... .................................................
63
vii
35
36
37
38
. .............................................................
39
......................‘..............................................................................................
40
............................: . . ...............................................................................
41
42 /Iz1U2 ............................................................................................... ’ . . .
•43 Image of / ..........................................................................................
44 im age(f) ................................................................
45 /k ,* = 1,2,3................................ . / .......................................................
46 K ...................................................................
47 /Iz1UZ2 ..............................................
48
......... ' ....................' . . ' ............ ................................................................
49 ' .....................................................................................................................
67
68
69
72
74
74
75
79
80
82
83
86
88
89
90
viii
ABSTRACT
Barge and Martin have shown that the inverse limit of any interval map may
be topologically realized as an attractor of a planar homeomorphism H in such a
way that H restricted to the attractor is conjugate to the induced homeomorphism
on the inverse limit. They extend the interval map to a near homeomorphism of
a disk containing the interval and use a theorem of Morton Brown to obtain their
result. We present similar results for degree -I, 0, or I circle maps and give necessary
conditions for a map to be extended to a near homeomorphism. Then, it is shown that
for a given finite, connected, planar, graph, G, containing a branch point, the set of
surjective, continuous maps of G which cannot be extended to a near homeomorphism
on any neighborhood of G. are open and dense in the set of all such maps, with the
Cf0Topology. However, an example of such a map for which the inverse limit can
be embedded as an attractor in the plane is given. Next, we prove the inverse limit
of any n-od (n > 3) can be embedded as an attractor in 3-space. We then give
necessary and sufficient conditions for the inverse limit of a finite, connected, planar,
graph with a surjective bonding map to be chainable. Since any chainable continuum
may be embedded in the plane, such an inverse limit is planar. Finally, we give an
example of a chainable continuum for which there exists a homeomorphism which is
not essentially extendable.
I
CHA PTER I
IN T R O D U C T IO N
.We begin our discussion with the following preliminary definitions:
D efinition 1.1 Let Z Ie a compact metric space and g : Z ^
Define Iim g = {(z0, z±,.. .)|
Z be continuous.
6 Zand g(zi+i) = z*}. Then Iim g is a subset of the
product space {(z0, Z1, .. .)|z% 6 Z } with the product topology.
D efinition 1.2 Let h : X
X be a map and Lf C X be relatively compact with
h{cl{U)) C Lf. Then A = nn>oZfi1(W) is called an attractor for h.
Inverse limit spaces provide interesting examples of continua. The pseudo-arc^ a
homogeneous indecomposable plane continuum (Bing, [5]), is homeomorphic to the
inverse limit of an interval map ([11], [18]). Much attention has been paid to the
embeddability of inverse limit spaces in Euclidean space. Ralph Bennett ([4]), R.
H. Bing ([6], [7]), J. R. Isbell ([12]), James Keesling ([13], [14]), Michael C. McCord
([17]), and David C. Wilson ([14]) have all published on this topic. In addition, inverse
limit spaces have been used to model attractors of homeomorphisms. For example,
the solenoid (Smale, [23]) is homeomorphic to the inverse limit space of a circle map.
D efinition 1.3 Define the induced hom eom orphism g : Iim g —> Um g by
<X(zo, Zr, - - -)) =
zo, Z1, ...).
2
D efinition 1.4 Let G be a finite, connected, planar graph and f : G
continuous map. If there exists a homeomorphism H : X ^
G be a
X such that K is
an attractor of H, K is homeomorphic to Iim f , and H \K is -conjugate to the shift
homeomorphism, f , on Iim f ,■ then we will say that Iim / can be embedded as an
attractor in X .
Barge and Martin ([I]) have shown that for any interval map f , Iim f can be embed­
ded as an attractor in IR2. They have thus insured a plethora of interesting planar
attractors. In the proof of their result, they use the following definitions and theorem
of Morton Brown ([10]).
D efinition 1.5 (Brown [10]) Let X be a metric space. A map f : X
X
is a n ear
hom eo m o rp hism if for any e > 0 there is a homeomorphism He of X onto itself
such that 11JTe —/I] < e.
D efinition 1.6 (Brown [10]) Let Xi be a sequence of compact metric spaces, and for
i > 2 let fi map X i into X i^ . Let fij = / i+x/i+2.../,- and f u = I. If z is a point
of X i then Zi will always denote the Jtil coordinate of z. Hence z = (zf). Then the
subspace S — {z E Hi 0
=
° f Hi0 X i is the limit space of the inverse
system (Xi,fi).
T h eo rem 1.1 (Brown [10]) Let S = Lim (X i, ff) where the X i are all homeomorphic
to a compact metric space X , and for all i, fi is a near homeomorphism. Then S is
homeomorphic to X .
The method of Barge and Martin is to extend the interval map / to F-a, near home­
omorphism of a disk containing the interval in its interior-and use the theorem of
Morton Brown.
3
In chapter 2, we obtain similar results for degree -1,0, or I circle maps. Also, two
results on extendibility to a near homeomorphism are given. In chapter 3, we show
that if / is a continuous, onto map of a triod which can be approximated by planar
embeddings, hm / is at least planar. Now let G be a finite, connected planar graph.
If G contains a branch point, the set of continuous maps, / , of G onto i/self such that
/ cannot be approximated by planar embeddings is open and dense in the set of all
maps of G onto itself. Since such a map / cannot be extended to a near homeomor­
phism on any planar neighborhood of Gr, the Barge-Martin construction cannot be
used to embed Iim / as an attractor in IR2. However, an example of a map / : G —> G
(onto, with G a triod) which cannot be approximated by planar embeddings, but for
which Iim / can be embedded as an attractor in IR2 is supplied. Sanford and Walker
have shown that “a positive entropy map of the product of a Cantor Set and an arc
(which covers a homeomorphism) cannot be “embedded” into a near, homeomorphism
of the 2-disk. Thus, a theorem of M. Brown cannot be used to embed the induced
shift map on the corresponding inverse limit space into a 2-disk homeomorphism”
([22]).
In chapter 4, the inverse limit of any n-od (n > 3) is embedded as an attractor in IR3
and in chapter 5 we return to the question of which maps / : G —> G have the prop­
erty that Iim / can be embedded -as an attractor in the plane. As a first step toward
that goal, we give necessary, and sufficient conditions that Iim / , / : G —» G(onto),
be snake-like or chainable-a condition insuring that hm / is planar. Finally, after a
few examples, propositions, and a corollary, we turn to the question of extending a
homeomorphism of a given planar continuum to a homeomorphism of all of IR2. For,
once we embed Iim / in the plane, we need to be able to extend / to a homeomor-
4
phism of all of IR2 in order to embed Inn f as an attractor in the plane. Brechner ([9]')
has shown that there exists a chainable continuum M C S 2 and a homeomorphism
h :M
M {onto) such that h is not “essentially extendable”. We give a similar
example of a chainable continuum K and a homeomorphism f : K —>K {onto) such
that / is not essentially extendable to all of IR2. Barge and Walker also have such an
example ([2]). Brechner ([8]) and Wayne Lewis ([15];) both have results that imply
that if K is any chainable continuum and f : K —>K {onto) is any homeomorphism, /
is essentially extendable to all of IR3. We then close with an example that shows that
even if Iim / is planar the shift homeomorphism, / , may not be essentially extendable
to IR2.
j
5
CHA PTER 2
E M B E D D IN G C IR C L E -L IK E C O N T IN U A AS A T T R A C T O R S
In the following, it is shown that for a circle map / , Iim / can be embed­
ded as an attractor in the plane for deg(/) = O and the plane less the origin for
deg(/) e {—1 ,1}. Dr. James Keesling has pointed out that Michael C. McCord has
shown the following related result ([17], pg. 323):
P ro p o sitio n 2.1 (McCord [17]) A circle-like continuum X can be embedded in the
plane if and only if H 1(X) = O or H 1(X) % Z.
( H1(X) denotes the 1-dimensional Cech cohomology Z -module of X.)
Finally, two theorems on extending a map to a near homeomorphism are given. The
importance of these results lies in the fact that in order to embed Iim / as an attractor
in a space Y, we first need to extend / : S'1 —> S'1 to a near homeomorphism of Y.
D efinition 2.7 Let S'1 = {z G C | \z] = 1} arid A = {z G C | | < |z| < |} and
define the covering m aps:
TT : IR —> S'1 by ^(x) = e2mx and P : IR x [|, |] —» A fa/ P(x, y) = ye2™.
D efinition 2.8 Let f : S 1
/ : IR —»IR such that
tt o /
S 1 be continuous. A lift of f is a continuous function
= / o tt.
D efinition 2.9 deg(f) = /( I ) —/(0), where f is a lift of f.
L em m a 2.1 Let f : S 1
S 1 be continuous with deg(f) = 0. Then there is an
interval I and a map g : I
I so that Iim / is homeomorphic to Iim g.
6
P roof:
Let / be a lift of / , a = min{/(()}, b = max{a + I , max{/(f)}}, and
9 — f\[a,b]- Note that since deg(f) = 0, f ( x + k) = f (x) for each real number x and
integer k. Define IT : Iim g
Iim / via n (t0,ti, • • •) = (7r^o), 7r(ti),...), where
tt is
the covering map ir(x) = e27rix. (t0, t i ,...) € hin (?=> / ( t n+1) = <n
TTO/ ( t n+i) =
tt(tn)
=4> /(7r(t„+1)) = 7r(tn), so (7r(to), Tr(^1) , ...) 6 Iim / .
H is one-to-one: Suppose that n ( t 0, h , . . . ) = !!(ty, t [ , . . .). For each positive integer
n, 7r(tn+1) = 7r(^+1)
tn+i = t'n+1 + k, k <m integer.
So, f ( t n+i) = tn = f(t'n+1 + k) = f(t'n+1) = t'n. Therefore, In = Hn, for each positive
integer n.
II is onto: Let (s0, s i , ...) G lira / and Tn = {7r-1(sn)} Pi [a, b\, for each positive
integer n. Then for n > I, /(T n) G Tn_i, since f { s n) = sn_i,
tt o
/ = / o tt,
and {7r-1(sn)} = {tn + k\tn G [0,1], A; an integer}. Construct an element of Iim g
by letting A0 = /( T i) ,Zi = /(T 2)1Z2 = /(T 3) ,....
Then (Z0,Zi ,...) G Iim y and
n((Z0, Z i , . . .)) == ( sq, Si , ...).
I! is continuous: Since the metric topology on Iim / is equivalent to the prod­
uct topology and for each Z G Iim / , If(Z) = (ir(TTj(t)))jeN (itj is projection in the j th
coordinate), H is continuous.
Iim g is compact and Iim / is Hausdorff, so H is a homeomorphism.
L em m a 2.2 ( Iim g,g) is conjugate to ( Iim / , f) .
7
P roof: H : Iim y —>Iim / is a homeomorphism and (fo n )((t0, h , ...)) = f( e 2wito, e2nitl,.
= (eW(^o),
...) = n(/((o), fo, ( i ,...) = (n o
( i , ...))
T h eo rem 2.2 If deg(f) = 0, then Iim / can be embedded as an attractor in the plane.
Proof:
By a result of Barge and Martin [1], there is an embedding h from Iim g
into the plane, IR2, a homeomorphism k = h o g o h~l : A(Iimg) —> Zi(limg), and
a homeomorphism, K of the plane, so that ^ l ^ m g ) = ^ and Zi(Iimg) is a global
attractor for K . So, we have the following commutative diagram:
Iim /
> Iim g
Zi(lim g)
—^
{<Is
lim /
> lim g
Zi(lim g)
IR2
Ii
—^
b
IR2
where i is the inclusion map. (Zi o II-1)(lim / ) = Zi(lim g) embeds lim / in IR2 and
Zi o r r 1 o / o n o Zi-1 — h Og O/I-1 = &, so that the shift map on this embedding is
just k and the result follows.
8
T h eo rem 2.3 Let f : S 1
S 1 be continuous with deg(f) G {—1,1}. Then lira / can
be embedded as an attractor in the plane less the origin.
P roof:
Let / : IR —> IR be a lift of / . Since /(a; + I) = f{x) 4 -1 for every x G IR,
there is a 6 > 0 so that graph(f) C S = {(%,y) C TR?\y £ JR,y — b < x < y + b}. For ■
every y £ IR, define a homeomorphism hy : [y - b,y + b\ x {y}
V ’
I (/0 ) +
[y — b,y -Lb] x {y}
(s - y), y),x £ [ y , y + b]
Now, define H : cl(S) —> d{S) by H(x,y) = hy{x,y). Note that H(x, x) = (f(x),x),
for every a: G IR. H is one-to-one and onto, since each hy is..Let
S = {(a;,y) G IR2Iy G [0,l],y - b < x < y + b}. Then, let H = H\g.
H is continuous: Fix (x, y) G S. Since any two norms on a finite dimensional
vector space are equivalent, we’ll use the one-norm, Le., ||(a:,2/)|| = |rr| + \y\.
C ase I: x < y . If ||(a:,y) - [X^y1)]] is small, X1 < 2/1 and \\H{x,y) - H( X u y 1)W
= Il(Ly) +
(z - y), y) - ( / ( » ) + SnfcB tiifo1 - m ) ,^ ) || < |/(y) - / f o f l
_j_ Iy(3:-y)-6(a:-y)-/fa)(a:-y)-yi(a:i-yi)+fc(a:i-i/i)+/(yi)(a:i-yi) j -)- |y _
< \f(y) —/(yi))
+ \ \ y ( x - y ) - y i ( x 1- y 1) \ + \ ( x - y ) - ( x i - y i ) \ + \ \ f ( y 1)(x1- y 1) - f ( y ) ( x - y ) \ + \ y - y 1\
< \f(y) - /(s/i)| + \(\y\\(x - y ) - {x\ - s/i)l + k i - yi\\y - y i \ ) +
- y ) - (xi - m)l
+ l(\f(yi)\\(xi - y i ) - ( x - y ) \ + \ x - y\\f(yi) - f(y)\) + \ y - y i \ which can be made
small by the continuity of / and boundedness of the region cl (S).
C ase II: x = y and without loss of generality Xi > yu \\H(x,y) —H(xi,yi)\\
= W(f(y),y) ~ (f(yi) + m±^
m l (xi -yi),yi)W < \f(y) - f(yi)\ + \yi+b~bf~{yiM\xi - m l
+ \y - yi| is small if \\(x ,y) ~ (ah,2/i)|| is small.
9
C ase III: x > y.
Similar to case I.
Therefore, JT : S —> 5 is continuous.
Since H is bijective continuous and S is
closed and compact, JT is a homeomorphism. Since d(S) can be partitioned into
closed sets {S',} so that H\g. is a homeomorphism for every %, JT : cl(S) —* cl(S) is a
homeomorphism by the pasting lemma ([19], pg. 108).
deg(f) = I : Let T : IR2 —> IR2 be the linear transformation on IR2 induced by the
I 0
matrix
. Also, let a : IR x [I, |] —>IR x [—6, b] be the homeomorphism defined
I I
by a(x,y) = (x, 2b(y — I)). Define JT : A —» A by K (ye2nix)
= (P o a~l o T-1 o H o T o a o P ^ 1)^ye2nzx).
Figure I: A
10
X - I
Figure 2: P X(A)
6
x —2
y —0
-b
Figure 3: a o P 1(A)
11
(x + 2,1 + 2)
(x + l , x +
(x —2, x —2)
Figure 4: T o a o P 1(J4)
( / ( x + l ) , x + I)
(f(x),x)
( f ( x - I), X
Figure 5: H o T o a o P 1(A)
-
I)
12
Figure 6: T 1 O H o T o a o P 1(A)
(/W-^C1-ZW) + 1)
Figure 7: a 1 o T 1 O H o T o a o P 1(A)
13
270
i ( z -/(*)) + 1Ie2-ZM
Figure 8: P o ct-1 o T -1 o H o T o a o P ^ 1(A)
K is a hom eom orphism :
well-defined: Suppose that yie2nixi = Tj2C2mx2 € A. Then ?/i = y2 and
Xi = x + fci,x2 = x + fc2,x € [0,1), &i, A2 E Z .
K (y ie2mXl) - ( P o a -1 o T -1 o H o T o a)({(x + A, yi)|A E Z})
= (P o a " 1 o T - 1 o tf)({(x + A, x + A + 2byi - 2b)\k E Z})
(Without loss of generality, x + A + 2by\ —3 6 < x + A < x + A + 2by\ — 2b. Let
w = x + 2by\ — 2b)
= (P o a - i o T -i)({(/(w ) + A + (/M -"+ '')(26 _ 261/1), w + A)|A E Z})
14
= -P({(/M + h + (& t«±i)(26 - 2bVl), 2 ^ 1 - (& t«±»)(i _ to) + i)|t € Z})
= (st^
- (S2l=H±5)(i - 9l) + x) e x p [ 2 » ( / » + (£!«1=«±»)(26 - 26m))] e A.
^ ( 2 / 2 6 ^ ) = (P o a-io T -io # o ro a)({ (a ;+ & ,2 /i)|& E Z}) =
Therefore,
AT : A —> A is well-defined.
O ne-to one: ill-1 = P o a -1 0 T~10 iJ _1 o T 0 a o'P_1 is well-defined and K 0 PT-1 =
PT-1
0
K = idy.
O nto: P -1 (A) = IR x [ |,|] . P , T, and a are homeomorphisms and P is a cover­
ing map. Therefore PT(A) = A.
C ontinuous: Let P be an open set in A. Then PT-1 (P)
= ( P o a - 1O T-1O P -1O T o a )(P -^ P )) = P((C)i- 1T - 1O P -1O T o a )(P -^ P ))) which
is open in A, since P , being a covering map is open. Therefore, K is continuous.
Since A is compact Hausdorff, and K is bijective continuous, K is a homeomorphism. Also, for any x G IR, P (e 27m:) =
P" : A = {z E (0
< |z| <
+ l)e27rzP ^ . Extend K to
> A by letting P = idy on A —A. Define a near
homeomorphism G : A —> A piecewise linearly along radial line segments as follows:
15
V= X
I
4
I
2
I
1
3
2
7
4
Figure 9:
Let F — G o K . F is a near homeomorphism, so that X = Iim F is homeomorphic
to A. Also, F |s i = / so that F |^ m y is a global attractor for int(X) under F. The
result follows.
deg(f) = —1 : Choose S = {(x,y) e IR2|y G IR, —y — 5 < a; < —y + b} so that
graph(f) C S and a homeomorphism H : cl(S) —» cl(S) so that
H ( x , —x) = ( f ( x ) , —x). Finally, let T : IR2 ^ IR2 be the linear transformation on
I 0
and proceed as in the deg(f) = I case.
IR2 induced by the matrix
-I I
<&■ 16
D efinition 2.10 (Munkres [19]) Let P : E ^ B be a map. I f f is a continuous
mapping of some space X into B, a lifting of f is a map f : X ^
L em m a 2.3 Let f : S 1 ^ S 1 be continuous and assume that
of embeddings, K : S 1 ^
E so that
is a sequence
A = {z e C \ \ < \z\ < \} , with \\hn - /H00 -» 0 oa
n —> oo. Let S 1 = IR x [|, |] and P : E ^ A be defined by P(x,y) = ye2mx e A.
Define / : / = [0,1] —> S 1 by f(t) = f ( e 2mt) and Jin : I ^ A by hn(t) = Ln^
Let f : I —>E be a lifting of f . Then there exists a sequence
it).
so that hn is a
lifting of K71 for each n and \\Jin — /H00 —> 0 as n ^ oo.
P roof:
Let e > 0 be given. Cover A with open sets {Ua}ae<p where each Ua is
evenly covered by P. Since /([0 ,1]) is compact, there exists a compact set K C E
so that /( [ 0 ,1]) C int{K) and P(int(K)) = A. P ^ (U aetpUa) = Uj3efiVe where Vg is
open and P\v0 :
—> A is a homeomorphism, for each /? 6 fi. (IZ3) j3efi is an open
cover of K . Therefore, there is a Lebesgue number 6 for this open cover. Choose
n G Z+, n > 2, so that ^ < <5. The balls of rational radii less than
the coor­
dinates of whose centers are rational, form a countable basis for E. Cover int(K )
with these balls, say int(K) — U^L1B 7i(Xi), each B 7i(Xi) C int(K ) and open in E.
P IBr-(Kf) : B 7i(Xi) -» A is a homeomorphism for every i. / -1 [(Ug1Bri(^i)) n /(/)]
is an open cover of I. Choose a Lebesgue number £ for this cover and a partition ■
S1 < S2 < ... < fife of / with S1 = 0, s& = I, and s1+1 —Sj- < | for I < / < A:—I. Then
for each I < j < k - I, f ( [ sj , Sj +1\) C Brij(Xij) and f ( [ s j , s j+1]) C P (B7ij(Xij)), for
some B 7ij(Xij). Since only finitely many B 7ij (Xij) jS are required and ||Zij —/H00 —»■0
as ^ > oo, there exists an N so that,2 > N implies that for every I < j < k —1, there
exists a B 7ij (Xij) so that f([sj, sJ+1]), Jifilsj , sJ+1]) C P (B 7ij (Xij)) and
~
_
f([sj, sj+1]) C B 7ij (Xij). Fix i0 > N. Construct the lifting hio of hio as follows:
17
^lollsj.sj+i]
=
(-PIBrij (X ij. ) )
1
° ^iol[s,,Sj+i] for each I < j < k — I, where for each j,
B rij(Xij) is chosen so that / ( [ s , , s^+i]),^„([5, , s,-+i]) C P ( B rij(Xlj)) and
f([sj, Sj+i]) C B rij (Xij). The fact that Jiio is a lifting follows exactly as in the proof of
the path lifting lemma ([19], pg. 337). The fact that | | / - A ioU00 < e follows from the
fact that Jii0([sj, sj+i\), f([sj, sj+i\) C B rij(Xij) for each j. Therefore, W f - J i iW00 < e,
for every i > N. Thus, || / — Ai Ij00 —» 0 as %^ oo.
T h eo rem 2.4 Let f : S 1
S 1 be continuous with deg(f) = n 6 Z — {—1,0,1}.
Then there does not exist a near homeomorphism G : D 2 ^ D 2 so that (T|gi = f ,
where D 2 = {z G C | |z| < |} .
Proof:
Suppose that G is such a near homeomorphism. Let / and P : E ^ A
be defined as in the previous lemma. Let / be a lifting of / and by the previous
lemma, choose a homeomorphism A : S 1 •—> A so that ||A —/H00 < f , A a lifting of A.
P(A(0)) = A(O) = A(I) = P(A(1)), so that 7Ti (A(1)) —7ri(A(0)) G Z , where
Tr1 : IR2 —> IR is defined by TTi (z, y) = x. Since deg(f) = n, Tr1(/(I)) —TTi(f(0)) = n.
Therefore, 7Ti (A(1)) —TTi(A(0)) = n. Let P : IR2 ^ IR2 be defined by T(x,y)
— (x + I,?/). Then Tn(Ji(I)) A Ji(I) / (j). Therefore, by [16], T(Ji(I)) A Ji(I) ^ 0.
So, there exists (x,y) G Ji(I) so that (z + 1,2/) G Ji(I)- Say, A(si) = (z, y) and
A(s2) = (z + 1,2/), where Si, S2 G I. Since A(I) — Tn(A(0)), it can’t be the case that
Si = 0 and s2 = I or vice versa. Then A(si) = 7r(A(si)) = 7t(A(s 2)) = A(s2), so that
A(e2irtSl) = h(e27rtS2), which contradicts the fact that A is a homeomorphism. Thus,
there does not exist a near homeomorphism G : D2 ^ D2 so that GrIsi — f-
18
R em ark : This result also follows from Proposition 2.8. For, if such a near homeomorphism G exists, then Iim / C Iim G ry D 2. So, Um / is embedded in the plane,
contradicting Proposition 2.8.
19
P ro p o sitio n 2.2 Let x E X , X a topological space. If Vx is a neighborhood of x
such that i :
tti(Vx, x )
> tti( X , x ) is trivial (i is the inclusion map and TtiiVx, x)
represents the fundamental group of Vx relative to the base point x, [19], pg. 326.)
and Ux is a path connected neighborhood of x such that Ux C Vx, then for every
V E Ux, [P] € TtiiVx, y) implies that [P] = [ey], as an element of TtiiX, y).
Proof:
Let y E U x and [P] 6 TtifVx, y) be given. Since Ux is path connected, there
exists a path a in Ux from x to y. [a] * [P] * [a] e TtiiVx, x), so [a] * [P] * [a] = [ex] in
Tti(X, x). Thus, [P] = [a] * [ex] * [a] = [ey] in Tti(X, y).
T h eo rem 2.5 Suppose that X is a compact, semi-locally simply connected, path con­
nected, locally path connected metric space such that TtiiX, X0) is finitely generated for
some Xo G X . Let F : X ^ X be a near homeomorphism (F is surjective, since X
is compact). Then P* : TTi(Xj X0) —» Trx(X, P (x 0)) is an isomorphism of fundamental
groups.
P roof:
F a near homeomorphism implies that there exists a sequence
of
homeomorphisms of X so that \\Hk — PH00 —> 0 as /c —> oo. Let [^r1], . . . , [gm] gen­
erate 7Ti(X,x0). For every x E X , there is an open neighborhood Vx of x so that
i : TtifVx, x) c—»
tti(X, x)
is trivial, where i is the inclusion map. Since X is lo­
cally path connected, for every x, there exists a path connected neighborhood Ux
of x so that Ux C Vx. X compact implies that there exists a Lebesgue number 6
for {Ux}xeX- Denote F*([&]) = [/j],l < i < m. f T 1UiiI) H (Uz6x P |(x ))) is an
open cover of I, for every I < z < m, so for every i, there exists a Lebesgue num­
ber Tji. Let rj = \ min {%}. Partition I, say f0 = 0 < ti < ... <
* l< »< m
= I, with
20
tJ ~ tJ-I < VA < j < rn. Then for each %and fe-i, ^ ] ,/iCfe-i, ^]) C Bs_(x) some
x e X . About each point /<(0) = F(X0) = y0, f ^t j ) = j/y, I < i < m, I < j < re - I
choose a path connected neighborhood W0, Wij, respectively with diameter less than
f . Since \\Hk — T1H00 —>0 as k —> oo, we can choose a homeomorphism H of X such
that for each I < i < m and [tj-^tj], I < j < n, f ^ t j ^ t j ]) and
Ti ])) are
contained in the same f-ball and F(x0) , H( x 0) € W0^yij, H (gi(tj))
G Wij(I
<
j
<
n
—
I). Denote
-E o
by Jii and H(gi(tj)) by
Zi j ,
I < i < m and
1 < J < ^ —I- Let z0 = H( x 0). Choose paths a0, a>ij with diameters less than | from
z0 to y0 and % to
respectively. Then for each I < i <
2 < j < n - I, diam(ai^ - 1) * hi\[s._us.] *
proposition,
and
* fi\[Sj_ltSj] < $■ So by the previous
~ p Qiji *
~ p ®n * ^Ibo.si] * «0 and when j = n, /i|[s„_1)Sra]
for every I < i < m, fc «P ./i|[Sn_1|S„] *
to
When j = I, /z|[so,si]
a0 * ^i|[an_1>an] * Cfyn-I). Hence,
fi\[s0,Sl]
~P OJq * hi|[Sn_ljSm] ~klOZi^n—l) * Qii(ra—I) * ^i|[sm-2,sm-i] * d^(n_2) *•••.* <^il * ^i|[so,si] * dig
~ p Qi0 * hj * dg That is [/j] = [a;0] * [hi] * [d0] for every i. Thus, since Tr1(A, X0) is
finitely generated, if [P] G tt^ X , £0), iAQP]) = [ckq] * E*([P]) * [do]. Since E is a
homeomorphism, E* VTr1(X ,x 0) —>7r1(X, P ( x 0)) is an isomorphism ([19], pg. 330):
Fir is o n e-to -o n e: Let [P], [Q] G 7Ti(X, x 0). Suppose that X*([E]) = E*.([Q])
G Tr1(X, E (^ 0)), then [ao] * E*([E]) * [do] = .[ao] * E*([Q]) * [do], which implies that
Hir(IPj) = E*([Q]) so that [P] = [QJ 1
F* is onto: Let [Q] E n ( X , F(X0)), then [a0] * [Q]* N
€ tt^ X , H (x0)). H* onto im­
plies that there exists a [P] E tt^ X , x 0) such that tf*([P]) = [a0] * [Q] * [a0]. Whence,
[q:o]*F*([P])*[ q:o] = [a0] * [Q\ * [ao] which implies that P*([P]) = [Q\. Thus, P* is an
isomorphism.
C H A PTER 3
E M BED D IN G G-LIKE C O N TIN U A AS ATTRACTORS
In the following we give three results concerning inverse limits of maps of fi­
nite, connected, planar graphs (G). The first shows that if f is any triod map (or a
particular type of n-od map) which can be uniformly approximated by an embedding,
Iim / is planar, and, in some instances, may be embedded as an attractor of a homeomorphism of the plane extending / . Secondly, we show that there exists an open and
dense class of maps in the set of all maps of a given finite, connected, planar graph
for which the Barge-Martin construction cannot be used to embed their inverse limits
as planar attractors. Finally, we close with an example of a triod map for which the
Barge-Martin construction does not apply, but the inverse limit can be embedded as
a planar attractor.
Let T denote a triod with edges Zi , Z2, and Z3.. Let O be the intersection of the
edges and a, b, and c be the extreme points of the edges Zi, Z2, and Z3 respectively. Let
D be a closed disk and h : T C int(D)
int(D) be an embedding. Our first goal
is to extend h to a homeomorphism H 1 : D' ^ D ', where D 1 is a closed disk with
D C Znt(Dt) and H' \ qdi — idy.
L em ina 3.4 D —(Zi(T) U dD} is connected.
P roof: D —(dDU{h(0)}) is connected, since it is an open annulus. Zi(Zi ) is an end-cut
in D —(dDU{h(0)}) and so by Theorem 11.6, pg. 118, [20], [D —(dDU{h(0)})} —h(li)
is a domain. Zi(Z2) is an end-cut in this domain, and so removing it does not destroy
connectivity. Similarly, Zi(Z3) is an end-cut in this domain, and so removing it does
not
23
destroy connectivity. Therefore, D - {h(T) U dD) is connected.
PTU
D efinition 3.11 (Whyburn [26]) A locally compact connected set is called a gener­
alized continuum.
L em m a 3.5 There exists a sequence [Li)
of locally connected generalized con-
tinua contained in int(D) such that h(a) 6 OLiyLi A h(T) — 0, and Li+i C Li for
every i. Also diam{Lj) —>0 as %—» oo.
P roof:
Choose e > 0 so that Bt (h(a)) A OD = 0. Choose x E Ii — {0, a} so
that h([x,a]) C B e(h(a)). Consider ip = B e(h(a)) — h(T — (x, a])-an open subset
of IR2. Consider the components of <p. h((x, a]) is connected and h((x,a\) C ip.
So, h ((x , o]) C K y where K is a component of <p. Also, K is open in IR2. Then
h((x,a]) is an end-cut in K and so K — h((x,a\) is open in IR2 and connected
(Theorem 11.6, pg.
118, [20]).
Therefore, Li = K — h({x, a]) is a locally con­
nected generalized continuum such that h(a) E OLiyLi A (Zi(T) U OD) — 0 and
diam(Li) < 2e. Bs.{h{a)) A OD = 0. Choose y > x so that y E h — {0, a} and
h((yya\) C Bi(/i(o)). Let (p — B^(Zi(a)) — h(T — (y,a}) and h((y, a]) E K y Ka ,
component of
C
an open subset of IR2 implies that each component of <p is
contained in exactly one component of <p. Also, h{{yya]) C K implies that K E K.
Therefore, L2 = K — h((y, a]) E L1, where L2 is a locally connected generalized con­
tinuum such that h(a) E OL2yL2 A (Zi(T) U OD) = 0, and diam(L2) < e. Continue
inductively to get the sequence (L1) ^ 1.
24
L em m a 3.6 Let x G dD. Then there exists a sequence
generalized continua C int(D) such that x G
J f j+ 1 C
of locally connected
K i, and K i Pl h(T) = 0, / o r
every i. Also, diam(Ki) —>0 as z —>oo.
Proof:
Choose e > 0 such that B e(x) Pl Zt(T) = 0. Let A/be a component of
(Be(x) (I D) — dD such that x G dKi, K 2 be a component of (B^(x) D D) - dD such
that x G d K 2 and K 2 C A / , ___
L em m a 3.7 TeZ z G dD. Then there exists a simple arc from h(a) to x lying entirely
within {D — (h(T)\J dD))VJ {h(a),x}.
P roof:
D — (Zt(T) U dD) is a locally connected generalized continuum and so is
arc-wise connected (Corollary 4.11, pg. 27, [25]). Also, by the preceding two lemmas,
there exist sequences (T i)-^1 and [Ki^ 1 of locally connected generalized continua
contained within int(D) such that K i Pt Zt(T) = Ti Pl Zt(T) = 0, K i+i C K i,
Ti+1 C Li, h(a) G dLi, and x G dK^, for every i. Also, diam(Li), diam^Kf) —>0 as
i —►oo. Choose points x\ G A /, tq G T1. Let T1 be a simple arc in D —(Zt(T) U dD)
connecting aq and ^1. Select a sequence of points {xn}^L2 such that xn G K n, for
every n and a sequence of points
such that yn G Tn, for every n. Obtain
sequences of simple arcs (S1n )^T2 and (Tn)^T2 so that 5n C Tfn_1, Tn C Tn. ^ S1n
connects xn and xn-i, and Tn connects r/n and yn-i. Let
M = {x, Zt(a)} U T1 U (U^L2Tn) U (U^z2S1n). Then (*) M is a locally connected
generalized continuum in (D - (Zt(T) U dD)) U {x, h(a)}. Therefore M and hence also
25
(D - (Zi(T) U dD)) U {x, h(a)} contains a simple arc joining x and h(a).
PTL
(*) M — {x, h(a)} U Ti U (U^T2 Tra) U (U™=2Sn) is a locally connected continuum.
Proof:
[°’ 1I
=
Define / : [0,1] —>M as follows:
[3 ’ 3 ]
u (Un<=2(n+2’ ^+l]) u
>1 ~ ^ ] ) U {0, !}• T1 a simple arc
implies that there exists a homeomorphism f x : [|, |] —> T1. Similarly, there exist
homeomorphisms f n : [ ^ 5, ^ rr] -> Tn, n > 2 and gn : [I —
I—
—»• Sn, n > 2.
Define
/ i ( z ) ,z e [ |, |]
e [ ^ 2, ^ i ] , n > 2
1 - ^ 2], n > 2
/(() = ' 9 n {t), Z E [I -
Zi(a), Z = O
z, Z — I.
/ is continuous at any Z 6 (0,1), by the pasting lemma ([19], pg. 108). Suppose
that (Zi) ^ 1 C [0,1] and f,
0 as Z-> oo. /(Zi) € cZ(U^L2Tn) = U^T2Tn U {/i(a)},
i large enough. For every j, there exists an Nj such that i > Nj implies that
/(Zi) E U^LjTn U {/i(a)} C d(Lj). diam(cl(Lj)) —> 0 as j
oo, so
/(Zi) -> ?/ E D ^ 1CZ(Tj ) = {/i(a)}. Therefore, /(Zi) —> Zi(a). Thus, / is continuous
at Z = 0. Similarly, / is continuous at Z = I. So, by Theorem 8.2, pg. 89, [20], M
is compact and locally connected. Also T1 U (U^L2Tn) U (U^L2Sn) is connected, since
it’s the union of overlapping connected sets. Whence, its closure M is connected.
T h eo rem 3.6 There exists a homeomorphism H' : D1 ^ D' such that H'\ t = h and
H'\dD' = idy.
Proof:
Let a be a simple arc in {D — (Zi(T) U dD)} U {Zi(a),z} connecting Zi(o) and
x, where x E dD. Then U = D - (Zi(T) U dD U a) is an open, connected set in IR2.
26
Similarly, we can choose a simple arc P m U from h{b) to y e dD, y ^ x. The points
x and y split dD into two disjoint simple arcs J1 and J2- K l 1) U a U 7i U /? U Zt(Z2)
and Zt(Zi) U a U 72 U /? U Zt(Z2) are two simple closed curves in D which intersect
along a U Zi(Zi ) U Zi(Z2) U (3. This splits D into two disks, say D11 and Dg, respectively.
K h - {0}) must lie entirely within the interior of one of these two disks. Without loss
of generality, Zi(Z3 - {0}) C ZnZ(Dj). Consider D - (Dg UdDj) = ZnZ(Dj). ZijZ3 - {0})
is an end-cut in ZnZ(Dj), so, by Theorem 11.6, pg. 118, [20], (ZnZ(Dj))-ZijZ3 - {0})
is connected. Then let z G 71 — (a U /?). z G <9Dj. Choose a simple arc 7 from
Zi(c) to z in (ZnZjDj))-ZijZ3 —{0}). Write D as the union of three disks D j, Dj, and
Dj,, where Dj and Djs are obtained from the disk previously labeled Dj by removing
2, and D1
3 correspond to the disks Dl l D2, and D3 in D obtained by
Zi(Z3) U 7o- D111D1
joining a, b, and c to OD with radial line segments. We choose three homeomorphisms
Zi1 : DD1 —►OD111 Zi2 : OD2 —» <9Dj and Zi3 : OD3 —> OD1
31 which agree with one another
on the common boundaries. By Corollary 3, pg 173, of [20], we can extend these to
homeomorphisms H1 \ D1 ^ D111H2 : D2 —> Dj, and D3 : D3 —> D1
3 which agree
with one another on the common boundary points. Therefore, by the pasting lemma
([19], pg. 108), we get a homeomorphism D : D —» D which extends Zi. Recall
that D C int(D'). Then, by the annulus theorem ([21], pg.
11), there exists a
homeomorphism D ' : D' —» D1 such that H'\D = H and H'\dDi = idy.
The previous Theorem does not hold in general for any n-od, n > 4, as noted by
Barge. For example, consider a homeomorphism of a 4-od which fixes two of the legs
and switches the other two.
D efinition 3.12 L etG be any finite, connected, planar graph. Define
C(G) = { / : G —>G \f is continuous and surjective} and Tj(G) = { / G C(G) | / cannot
27
be uniformly approximated by planar embeddings).
T h eo rem 3.7 Let f 6 C(T) — r](T) and
hn ■T
be a sequence of embeddings,
int(D) such that ||/in~/||oo —*• 0 as n —> oo. Let H'n : D' ^ D' be a sequence
of homeomorphisms such that HI1It = K and H ^ 9di = idy. Then if { # 'J lc^ 1 is a
subsequence of {H'n}™=1 such that
converges uniformly to F : D1 —> D',
Iim / can be topologically realized as a global attractor in the plane.
Proof:
F is a near homeomorphism. F\T = Iim H ' \ T = Iim hn. = / . Let D i and
D2 be two closed disks such that D' C Int(Dx) and D x C int(D2). let p : D2
D2
be a near homeomorphism such that p\T = idy,p(D') = T,p(D x) = D', and if
x G int(D2), there exists an n G Z+ such that pn(x) G D'. Extend F to F : D2 —* D2
by
F (x ),x e D' )
x , x <£ D1
j "
Let G : D 2 ^ D2 be defined by G(x) = p(F(x)). Then G is a near homeomorphism
F(z) = j
such that G\t = f and if z G int(D2), there exists and n G Z+ such that Gn(x) G T.
The proof would then proceed as in the construction of Barge and Martin ([I]).
If it is not the case that the sequence of homeomorphisms ( F n)^L1 of the preceeding
theorem have a uniformly convergent subsequence, we can use the following theorem
by McCord [17] to embed Iim / in IR2. First, we need a definition of McCord’s.
D efinition 3.13 (McCord [17]) Let (E,d) be a locally compact metric space and let
f : X —> Y be a map where X and Y are compact subsets of E.
Then f can be
approximated by embeddings in E, provided that for each e > O there exists an open
set U with X C U C E and a I — I map p, : U ^ E such that d(p(x), f(x)) < e for
all x in X .
28
T h eo rem 3.8 (McCord [17]) Let (E,d) be a locally compact metric space and let
( X, f ) be an inverse limit sequence of compact subsets of E such that each bonding
map / ”+1 : X n+l —> X n can be approximated by embeddings in E. Then the limit X 00
can be embedded in E.
Note that / G C(T) — Tj(T) is distinctly different from the case where f can be
approximated by embeddings in IR2. / € C(T) - rj(T) implies that for every e > 0,
there exists an embedding g : T ^
IR2 such that ||/ - g||oo < e. however, / can
be approximated by embeddings in IR2 implies that for every e > 0, there exists an
embedding /r of a neighborhood of T into IR2 such that
||/-A t|r ||o o
< e. The example,
given by Barge, of a homeomorphism of a 4-od, Gr, which switches two of the legs and
fixes the other two is an example of a map which is an element of C(G) — rj(G) but
which cannot be approximated by embeddings in IR2.
T h eo rem 3.9 If f E C(T) —r](T), then Iim / is planar.
P roof:
H 1 : D'
We have shown that for every e > 0, there exists a homeomorphism
D' such that H'\ qDi = idy and ||/ — H'\T\\ < e. Hence, / can be
approximated by embeddings in IR2 and by McCord’s theorem, Iim / is planar.
Next, we show that there exists an open and dense class of maps in the set of all maps
of a given finite, connected, planar graph for which the Barge-Martin construction
cannot be used to embed their inverse limits as planar attractors. We begin with
some notation.
29
Denote elements of I1 by {s|0 < s < 1}, elements of I2 by {g'|0 < s' < I), and
elements of /3 by {s"|0 < s" < 1}.
P ro p o sitio n 3.3 t](T) is open in (C(T), || • H00).
P roof:
Suppose not. Then there exists an / e r](t) such that for every c > 0, there
is an /t G C(T) — r](T) with ||/ —AH00 < e. Since f G t/(T), there is an e0 > 0 so that
there is no embedding # : T —> IR2 with ||/ - ^||oo < e0. Choose h G C(T) - j](T)
so that \\h — /Il00 < ^ and g : T —* IR2 an embedding with ||/i —^H00 <
Then
H/-2II00 < e0, a contradiction.
L em m a 3.8 Let e > 0 be given. Let I = [a, b] C k — {0},% G {1, 2, 3}. Let
to = a < ti < t2 < ts < t^ < t^ = b be a partition of [a, b] into five equal subintervals.
Define a piecewise linear function k : I —>T as follows:
k(to) = f , k(t\) = 0, k(k(ti + t2)) — I", k(t2) = 0, k (\( t2 + t3)) =
k(tf) = 0,
k(^(t3 + t4)) = |, k(t4) = 0, k(t5) = I". Then there does not exist a sequence of
embeddings {fin}™=!,Pn : / —> IR2 such that,
\\Pn
— ^H00 —>0 as n —> 00.
30
Zi
Figure 10: image of k
P roof:
Assume that there exists such a sequence of embeddings {,SrJ^L v Choose
one, say /3, such that \\f3 - ^ 00 <
Then let N = B^([0, |] U [0, f'] U [0, §"]),
2,1 = cZ(W) n cZ (B ^ (|)),l2 = Cf(Af) n CZ(B^(|"))), and
= cf(Af) n c f(B ^ (|')).
/3(t0) 6 B t i (I). Let Xi E L1 and 71 : [0,1] -> cZ(B^(|)) be defined by
T1(Z) = (I —t ) x i
+ t/3(to).
Consider 7 f 1(cZ(B^(|)) fl
/3([to,
Z4]))-a closed subset of
[0,1] not containing 0. Therefore, the least element of this set, say
Oil
= 7 i ( [ 0 , t*])
is a simple arc from X1 to an element, say P ( s i ) €
t*
is positive. Then
P ( [ t o , Lt]),
such that
ain/?([fo,bt]) = {/3(si)} and Ct1 C cZ(B^(|)). Let X2 € L2lX3 E L3 and choose simple
arcs Qi2 and a3 from P(S2) E /?([Z0,Li]) to
X2
and (3(s3) E P ( [ t 0 , U ] ) to
X3
respectively
31
such that a ; n /3([to, (4 ]) = {^(32)}, «3 n
( 4 ]) = {^(33)}, « 2 C B
j L
and
a 3 C B jl (I'). Then S1 < s2 < S3 and TV—((X1U /)^ !, S2^ U a2) consists of the two disks
B 1 and D2 as follows:
Figure 11:
P((s2, f4]) is connected and doesn’t intersect BDiLidD2 and so must lie entirely within
D 1 or D2. But, P (\(t2 + is)) E B
j l
( |') ,
so
/3((
s
2,
^4]) C D2.
32
21
%
Figure 12:
As pictured, /^([S2jS3]) U 0:3 splits D2 into two sub-disks D21 and D22. /3((s3, £ 4 ] ) must
lie entirely within one of these disks. But, /?(|(t3 + f4))
6
S
j_
( | ) ,
so
y9((s3,t 4]) C D22. Since /?((£!,f5]) n /? ([s i,s 3]) = 0 and /?(t4) G B^(O) A D22, there is
an s G (t4,ts] so that either /l(s) 6 B ^ ( |) or /3(s) G
(|'), a contradiction.
T h eo rem 3.10 i/(T) zs open and dense in C(T).
P roof:
The fact that rj(T) is open in C(T) is shown in Propositon 3.32.
Let
h G C(T) — ri(T) and e > O be given. There exists an s G T — {0} and a 5 > 0
33
such that O ^ Bs(s) and ||h(t)|| < |, for every t e Bg(s). Without loss of generality,
s E Ii - {0}. Define h piecewise linearly on cl(B6(s)) as follows:
h(s - 6 ) = h(s - 6), h(s - f<5) = 0, h(s - f ) = f , h(s - f f ) = 0, h(s - §) = f ",
h(s -
h
=
) = °> ^(s) =
Ms + ^ ) = 0, h ( s + f ) = f, h(s + f f ) = 0, h(s + |)
= 0, and h(a + f) = h(a + 6). If f 6 T - cZ(Bg(a)), let h(t) = h(f). h
is continuous by the pasting lemma ([19], pg. 108). h(cl(Bs(s))) D h(cl(Bg(s))), so
h is onto. Also, by the way that h is defined on d (B i(s)) and the previous lemma,
h e r)(T). I f t E cl(B6(s)), \\h(t) - h(t)|| < e. Therefore, ||h - All00 < |. Since, e > 0
was arbitrary, Tj(T) is dense in C(T).
C orollary 3.1 If G is any finite, connected, planar graph containing a branch point,
Tj(G) is open and dense in C(G).
Proof:
The previous proposition holds to show that Tj(G) is open in C(G) and the
above construction can be used at a branch point of G to show that Tj(G) is dense in
C(G).
34
Note that if / 6 r](T), the Barge-Martin ([I]) construction cannot be used to
embed Iim / as an attractor in the plane. However, as the following example shows,
there is a map / 6 Tj(T) such that Iim / can be embedded as an attractor in the plane.
E xam ple 3.1 Let f : T ^ T be defined piecewise linearly on Z1 U Z2 as follows:
h
h
Figure 13: /It1U2
35
D e f i n e f h 3 = idy.
h
Figure 14: image of /
Then f G Ty(T) and Iim J can be embedded as a global attractor in the plane.
P roof:
/ G Ty(T) : Suppose that / G C(T) — Ty(T) and let /i : T —» Bs-(T) be
an embedding such that ||/i — /H00 <
AT =
Note that /i([0, |']) fl B ^ ( |') = 0. Let
(Zi U Zg U [0 ,|']),T i = cZ(AT) H cZ(BjL(l)),T2 = cZ(AT) H cZ(B^(§')), and
L3 = cl(N) D d (B ^ (l" )). Let p\ G Lll^2 6 L2, and p3 G L3 and as in the proof that
Ty(T) is open and dense in C(T), choose simple arcs ou, Ct2, O3 from pi to h(si), L(g2)
to p2, and L(s3) to p3 respectively such that cti Pl L(T) = {L(si)},
Ct1 C CZ(Bjl(I ))1ct2nL(ZiU/3U[0,1']) = {L(s2)}, Ct2 C CZ(B^(T))1^ n L (T ) = {L(s3)},
Ct3 C c l ( B s - ( l " ) ) , S i , s 2 G Zi1O < s2 <
S1
< I and S3 G Z3. Ct1 U LQs1, S2]) U Ct2 divides
N into two sub-disks B 1 and B 2. L([0, S2)) and L([0, |']) are entirely contained within
one of these two disks. But, L(Z3) C B 1, so L([0, S2)), L([0, |']) C B 1.
36
Figure 15:
37
/i([0, si]) U /t([0, s3]) U Ot3 breaks Di up into two sub-disks Dn and D n -
Figure 16:
Since /i(|') 6 D i and /i(|') E B ^ ( |) , /i(|') E B 11. A([|', I']) cannot intersect 9 B U,
so /&([!', I']) C B 11 but Zi(F) = I', a contradiction. Thus, / E //(T).
Let x E [§, I'], then by choosing all pre-images of x from the segment labeled “a” on
the graph of / | ZlUz2, we get a simple arc S0 — {(z, U1, a2, a3, .. .)|z E [|, 1']} in Iim f
with endpoints (I', I ', ...) and ( |, I', a^, O3,...) . In similar fashion, we complete the
following table:
38
S IM P L E ARC
S0 = { ( % , U 1 , a 2 , o3, . ..)|a; <5 [ | , 1']}
E N D P O IN T S
( I ' , i ' , I ' , . . 0
(g, 2 , Cl2, ® 3 ,
S i =
{(% , h , a 2,
o3, . .
.)|®
e
[ |, I']}
( f ,
{ ( x , C x , a 2, a 3, . . . ) \ x
(% ',
( 4 ,
S3 =
{ 0 , V1 ,
o2, o3, . . 0 1 ®
E
[J, /(§)]}
S5 =
{ (® ,% ,6 2 ,0 3,0 4 ,...)|®
{ ( % , C1 , b 2 ,
o3, o4, .
..)|®
{(® ,
C1, C2 , o3, o4,..
S7 =
{ ( % , C 1 , V2 ,
o3, o4, .
8' 2 ^ 3 , Cl4 , .-0
4 > Cl2 , O3, . . . )
G
[ |, / ( |] >
(/(§)> | , I , ° 3 , at, • • 0
( M , Z M 'G L - O
G
[J, I']}
a '
01® G
..)|®
0
4 , ^ ,- O
4, Cl2 , O g , . . 0
i' i' a 6 „6
'j
X4 J 4 J 4 , t t S , t^ 4 , • • •)
( 4 , 4 ,6 2,
S6 =
• •
( /( § ) '
( 4 ,
=
0
«2, 4 , . . . )
(g, 2 ’ ^2, Cl3 ,
S2 =
■•
[J, J']}
(4 4 4
( / ( I ) , /(§ ),§ , I', O4, O^,.
(4, 4, 4, O3, O^, Og, . . .)
(/(§)> /(!)> !, ^ , aI, of,.
( i,3 ,i,6 E ,o |,o f,...)
S8 = {(®, ci, V2 , b 3 , o4, o5, .. Olx G [J,/(§)]}
G
[J, J ' ] }
Si0'= {(®, ci, c2vc3, o4, o5, .. 01® G [J, J']}
0
,
,
, O3 , O 4 , • ■
f i l l
„7 „7
\
X4 J 4 , 4 , , t i S , t ^4 , • • v
G [J,/(§)]}
S9 = {(®,c1,c2,63,o4,o 5,..0 |x
03, o^,. . . )
■
’ I '’ 4', I', O40, of0,...)
(3,%, 1,63,04, o f,...)
(4
’ fi'
X4 , i4 ' , i4' , i4' , aio aio —v)
(I
0ir , —v)
X4, I4, I4, I4, 0t^4il, ttS
39
Replace C3 with 6^ in Sio to get Sn, replace o4 with &4 in Sn to get S42, replace U3
with c3 in S42 to get S43, replace 64 with C4 in S43 to get S44, .... By symmetry,
(h U I2, /Iz1UZ2) also contains simple arcs:
S'o = (0%, a[, a!2, a'3, .. .)\x G [I, f]}
S i = {(a:, 6 i , a ,2 ,a ^ , .. .) |a : e [ J ,! '] }
*^2
{(z> C4, d 2 , Cb3 , . . - ) | z 6 [-g, ^ ]}
In addition, it contains the simple arc:
S = {(%, c4, c2, c3, .. .)|a; G [|, I'], Ci = %, for all i}. We embed the arcs S, S0, S4, S2, ...
in.IR2 as follows:
40
Figure 17:
41
The shift map on this embedding stretches S0 across S0 U S1i U S12 U S3, S1 across
S 4 U S5, S 2 maps across S6, S3 across S7,.... Each “W” to the left of S0 U S 1 gets
shifted onto the “W” to its immediate left. In similar fashion, we embed the simple
arcs S, S 01, S[, S'2, ... in IR2 as follows:
Figure 18:
42
f on this embedding stretches S 01 U S 1 across the “M” S 21 Li S 31 U S 41 U S 51 and shifts each
“M” onto the “M” to its immediate left. We alter the above embedding, so that when
combined with the previous embedding, we have the following planar embedding of
Iim /:
Figure 19:
43
We add arcs to the previous planar embedding of Iim / to get the following disk D
and subdisks {A },~i, { A } £ i, and ( A ) z=V
.............. i )
...
Figure 20: Disk D
44
Let g be / , the shift homeomorphism, on our planar embedding of Iim / . Define a
homeomorphism H : D ^ D as follows:
Vertically folliate each Di and Di. Arrange that H takes the leaf intersecting (V0, y0)
in Di intersected with the embedding of Iim / onto the vertical leaf intersecting
g((xo,yo)), similarly for each leaf in Di. We define H\aVl : OD1
S(Z) 1 U Z)2) in
the obvious way and use Corollary 3, pg. 173, of [20] to extend to a homeomorphism
ZJId1 : D 1 —> D 1 U D2. We also need to insure that the right edge of D is invariant
under H and that it contains exactly the four labeled fixed points-the interior two
lying on our embedding of Iim / . Finally, if x lies on this edge but is not one of
the four fixed points we require that H n (x) converges monotonically to one of the
.two interior fixed points, as indicated. Define H on the limit bar to be the identity
map. ZJjDi basically shifts and horizontally shrinks each disk Di onto Di+1 for i > 2.
We add four arcs to the disk D to get the disks D1, D2, D3, and D4. Now, use the
following lemma to define H on D1, D2, and D3 and the lemma following it to define
H on D4 so that ZJ is a homeomorphism of the disk D U D1 U D2 U D3 U D4 such that
H is the identity on the boundary of this disk, our embedding of Iim / is a global
attractor, and ZJ restricted to this attractor is conjugate to / . Since the interior of
this disk is homeomorphic to IR2, the result follows.
45
Figure 21: D U D 1 U D2 U H3 U D4
46
L em m a 3.9 Let S = [0,1] x [0,1] be the unit square in the xy-plane.
Let K \dS : d S —> dS be defined by K(x, 0) = (a(z), 0),K(x, I) = (x, I),
K(Q,y) = (0,fi(y)), and K ( l,y ) = (l,/3(y)), where a , /3 : [0,1]
[0,1] are homeo-
morphisms with Ct(O) = /?(0) = 0 and Ct(I) = /3(1) = I. Then K : S -> S defined by
v)) = ((I —P(y))(a (x ) — x ) + x, f3(y)) is a homeomorphism.
Proof:
K is continuous by the continuity of ct and (3. Suppose that
K ((xi,y i)) = K ((x 2,y 2 )) for (xi,yi), (X2^y2) € S. Then ^ y l) = f3(y2), so that
2/i = 2/2- The map K can be thought of as a composition of two maps, the first of
which takes (x, y) to (x,(3(y)) and the second horizontally projects (x, (3(y)) onto the
line through the points (x, I) and (a(x), 0) in S. Since ct is an orientation preserv­
ing homeomorphism of [0,1], if Xi ^ X2, the lines Ii , through the points (^1, 1) and
(ct(xi), 0), and I2, through the points (^2, 1) and (ct(z2), 0), do not intersect. This
contradicts the assumption that K (x i,y i) = K (x 2,y 2). Therefore, Xi = X2 and K is
one-to-one. By definition, K maps OS onto OS and K (S) C S. Therfore, K must
be onto, otherwise one could construct a continuous map of S onto OS that is homotopically nontrivial when restricted to OS. Since K is bijective continuous and S is
compact Hausdorff, Li is a homeomorphism.
L em m a 3.10 Let S = {(r, 0)|O < r < I, # = |t , 0 < t < 1} 2>e a quarter disk in
the xy-plane and a, /3 : [0,1] —» [0,1] be homeomorphisms with Ct(O) = P(O) = 0 and
q
;(1)
=
P(I)
= I. Then K : S —>S defined by K(r, |f ) = (I —t)a(r) + tfi(r) is a
homeomorphism.
P roof:
K is continous, since a and P are. If K (r\, |t i ) = K (r2, \ t 2), then t\ = t2,
so that (I - ti)[a(ri) - a(r2)] + ti[p(ri) - P(r2)] = 0 . If H = 0 or Z1 = I, then
47
we’re done. So, assume that (I - ^1) , > 0 and that r 2 > n . Since a and (3 are
orientation preserving homeomorphisms of [0,1], they must be strictly increasing.
Thus, o:(ri) —O^r2) ,/?(ri) —/?(r2) < 0, which contradicts the fact that
(I - t O H n ) - Oi{r2)\ +
- (3(r2)\ = 0.
We reach a similar contradiction if we assume that r 2 < Ty. Hence, K is one-to-one.
K maps OS onto itself, thus, as in the previous lemma, K is onto. Therefore, Tf is a
homeomorphism.
48
C H A PTER 4
E M B ED D IN G n-OD-LIKE CO N TIN U A AS ATTRACTORS
We show that if T is any n-odd (n > 3) and / : T —> T is continuous, then
Iim / can be embedded as an attractor in IR3. For the sake of introduction, assume
that T is a triod with legs Zi, Z2, and Z3 of unit length and that Zi n Z2 n Z3 = {0}. Also,
assume that /(0) =to G Zi,.0 < A < I. To obtain our result, we create three coordi­
nate systems F i1F2, and F3-each containing a triod with legs L1, L 12, L 13; L 1, L 22, L23;
and Li, L32, L33 respectively so that Fi
n
F2 n F3 = L 1. Next, we embed T in
M = F i U F 2 U F3 so that I1 is embedded in Fi , Z2 is embedded in F2, Z3 is embedded
in F3, and 0 gets mapped onto Z0 G L 1. In each of the coordinate systems F i1F2,
and F3 we select homeomorphisms that take I1 onto Qraph(J^1), Z2 onto graph(f\i2),
and Z3 onto graph(f\i3),. respectively. Choose near homeomorphisms in each of F1, F2,
and F3 so that Qraph(J^1) is collapsed.“properly” onto the triod Li U L 12 U L 13;
graph(J\i2) is collapsed “properly” onto L 1 U L22 U L23; and graph(J\iJ) is collapsed
“properly” onto Li U L32 U L 33, respectively. Now, pick near homeomorphisms that
map L 1 “properly” across Zi; L 12, L 22, and L32 “properly” across Z2; and L 13, L 23, and
L33 “properly” across Z3. We end up with a near homeomorphism of cZ(R3(O)) in IR3
th at restricted to T is / , and restricted to OB3(O) = idy. Finally, we choose a near
homeomorphism of cl(Bg(0)) in IR3 that restricted to T is J, and so that T is an
attractor for this near homeomorphism. The result will then follow by the BargeMartin argument ([I]).
More generally, let T denote an n-odd (n > 3) with legs, Zi1Z2, . . .,Z^1 of unit length
and let 0 denote the intersection of the legs. Orient T in the plane so that Zi makes
49
an angle, of ^ in the counter-clockwise direction from the positive x-axis, 0 corre­
sponds to the origin, and the angle between consecutive legs is
ing legs I2,. ■
Label the remain­
in the counter-clockwise direction. Let T be an n-odd with legs
L 1, L 2, . . . , Ln of length 6, situated in the plane so that L1 D Z1, L 2 D I2, ... ,L n D In.
Connect the extreme points of consecutive legs, of T with line segments to form a
regular n-gon, K. Label the edge joining L1 and L2 by C1, L2 and L3 by e2, . . . , L n '
and L1 by en.
50
Figure 22: K
Choose an embedding
(3 : K -* {(p, <^, 0) C IR3|p > 0,0 < (/> < | , 0 < 0 < ^} so that:
1. /? is a translation on Li, L2,
, Ln,
2. /3(Li) is the interval [0,6] on the z-axis,
3. P linearly contracts between consecutive legs,
51
Figure 23: /3
4. The projection onto the xy-plane of the images of C1, e2, . . . , en under /? are as
follows (e' = Trxy(Piei))):
Figure 24: n = 3 :
52
Figure 25: n = 4 :
53
TT
n
Figure 26: n > 4, n odd:
54
ZL
n
Figure 27: n > 4, n even:
55
Let K be the solid K x [0, §]. Define a homeomorphism /? : Lf
R 3 by j3(k,z) =
/?0) + zti, where k £ K , z e [Q, |], u is in the convex hull of L1, . . . , Ln, ||y|| = I, and
v is chosen so that j3(K) c { 0 ,
^)|y0 >, 0 < 0 < f , 0 < ^ < ^}. Place n - I copies
of ^(A") in the M- I sectors {(p,
0,0 < ^ < §, ^ < 6» <
g)|p > 0,0 < ^ < §, ^ < g < ^ } , {(p,
, {(p,
g)|p >
6»)|p > 0,0 < ^ < §,
by composing P(K) with a counter-clockwise rotation about the z-axis through angles
of ^ l ,
. . . , and C2ra" 2)71-; respectively. Let M be the space consisting of the union
of these n solids. Note that the intersection of these n solids is precisely the interval
[0, 6] on the z-axis and that M C B8(O) irregardless of n. Now, T C (-§ , §) x ( - § , §)
and [—2,2] x [—2,2] C K. Let f : T —>T be continuous and without loss of generality,
/(0 ) = (zo,2/o) EZ1 C R 2. Since (x0,yo) £ h ,0 < t 0 = ||(zco,2/o)|| < I, we can label the
point (x0,y 0) by t0 without ambiguity. We can do similarly for any point (x,y) £ Li,
as long as Li is specified. Let S = [-2,2] x [-2,2], and S = [ - 2 ,2] x [-2,2] x [0, §].
P ro p o sitio n 4.4 Let Z*,% G {1,2,..., n} be given. There exists a homeomorphism
H \S
S such that H (t0,z) = ( / |Zj(z),z), for z £ [0,1], and H\g§ = idy. (i.e. H
takes {t0} x [0,1] onto the graph of fhf).
P roof:
Z1 f ] .
Denote elements of Ii by t, where 0 < t < I and t = 0 corresponds to
Pl Zn = {0}. Then / | Zi : [0,1] —>T and /( I ) G Ij , j G {1,2,... ,n}. Extend f \ k
piecewise linearly to a map / : [0, §] -> T, by /( I ) = /.|Zi(l), / ( | ) = 0, and /(§ ) = t 0.
Define a homeomorphism Hz S x {z} —>S x {z}, z G [0, f] as follows.
Vertically foliate S x {z}\
56
Figure 28: H,
Hz translates each vertical line segment {x} x [—§,§] x {z} inside [—| , §]x [—§, |] x {z}
onto the vertical line segment {yz(x)} x [—| , |] x {z} inside [—|, |] x [—| , |] x {z}.
In the preceeding, gz : [—| , |] —»• [—| , |] is defined as follows:
57
Figure 29: y = gz(x)
where f( z ) = (/i(z), / 2(z)). FTz = idy on ([-2, - | ] x [-2, 2] x { z » U ([§, 2] x [-2, 2] x
{z}) and Hz(x,y,z) =
(2(y
-
2)(x
-
gz(x))
+ x , y , z) on [-§,§] x [|,2] x {z}; i.e. Ffz
linearly stretches each vertical segment {x} x [|, 2] x {z} onto the segment joining
W % ), | , z) and (x, 2, z). Similarly, on [-§ , |] x [-2, -§ ] x {z}, Ffz(x, y, z) = (2(%/ +
2)(yz(x) —x )+ x , y, z); Ffz linearly stretches each vertical segment {x} x [-2, —|] x {z}
onto the segment joining (x ,-2 ,z ) and (yz(x), - | , z ) .
Define Ff : 5 ^
S via
Ff (x, y, z) = Ffz(x, y, z). Since each Hz : S x {z} —> S' x {z} is a homeomorphism, Ff
is one-to-one and onto. FF is continuous on [—§,§] x [ - | , |] x [0, |], since Ff(x, y, z) =
Hz(x,y,z) = (gz(x),y, z), each yz is continuous, and {pz} vary continuously in z.
58
Similarly, H is continuous on ([-§ , §] x [§, 2] x [0, §]) U ([-§ , §] x [-2, -§ ] x [0, §]).
Since H — idy off the aforementioned sets, H is continuous by the pasting lemma.
Because S is compact-Hausdorff, i f is a homeomorphism.
Define a homeomorphism Vz : S x {z} —>S' x { z } , z G [0, |] as follows.
Horizontally foliate S x {z} as in figure 30.
(zo,2/o,z)
Figure 30: Vz
Vz translates each horizontal line segment [—|, |] x {y} x {z} inside [ —| , |] x [—|, |] x
{z} onto the horizontal line segment [—| , |] x {vz(y)} x {z} inside [—| , |] x [—| , |] x
{z}, where vz : [—|, |] —> [—| , |] is defined as follows as in figure 31.
59
Figure 31: y = vz{x)
Vz = idy on ([-2,2] x [- 2 ,- § ] x {z})U([-2,2] x [§,2] x {z}) and Vz(x,y,z) =
(x,2(vz(y) — y)(x + 2) + y, z); i.e. Vz linearly stretches each horizontal segment
[—2, —|] x {y} x {z} onto the segment joining (-2 , y, z) and (—|, vz(y), z). Similarly, on [§,2] x [-§,§] x
= (z,2(i/ - ^(2/))(z - 2) +2/,z); i.e.
linearly stretches each horizontal segment [|, 2] x {y} x {z} onto the segment joining
(§, vz(y), z) and (2,y, z). Define the homeomorphism Iz : S —> S' via V (x,y ,z) =
Vz(x, y, z). Then H o V is the required homeomorphism.
PTlJ
Now, label the n solids whose union is M as follows:
60
Let F1 = P(K) and label the others F2, F3, . . . , Fra counterclockwise,, where Fi =
Pi{K),i > 2,/Si = R i 0 /3, R i = counterclockwise rotation by
We can
write Pi = p. Each Ti contains a copy of T -the n-odd with legs Ll j L2,,... ,L ra,
each of length 6.
Moreover, PlJL1Fi contains a copy L1 = Pi(L1) = /J2(L1) =
... = Pn(Lx) of L1 (the interval [0,6] on the positive z-axis).
On T i, label the
copies of L 2, L3, . . . , Ln by Li2 = Pi(L2), L i 3 = Pi(L3) , ... , L in = Pi (Ln), respec­
tively.' Note that M C B8(O). Choose a homeomorphism W : cZ(B8(O)) —» cZ(B8(O))
so that 1F(O) = (0,0, £0), 1F(^O ) = AW,Z/o, W where 0 < L < !,L e Li, and
= idy on d(cZ(B8(0)). By the previous proposition, for every i = 1,2, . . . , n ,
there exists a homeomorphism Hi : [-2,2] x [-2,2] x [0, §] = B —» 5 such that
Hi(x 0 ,yo,z) = ( f Iii(Z), z), for z G [0,1] and HiIgg — idy. As S C K, we can-extend
Hi to Hi \ K ^ K, for every i, by letting Hi = idy on K — S. Choose a near home­
omorphism P : K ^ K so that P = idy on d K and P _1((a;,y, 0)) = (x,y) x [0,1],
for each (x, y) G [—2,2] x [—2,2]. Now, define a near homeomorphism Tl : M
M
as follows:
f2|ri : Fi —> Fi is defined by Q\r .(x,y,z) — Pi O P o H i O p r 1 (x ,y,z). Extend to H :
cl(B3(6)) —» Cl(Bg(O))f by letting f2 = idy on cl(B8 (6 )) —M. Then QoT : cZ(B8(O))
cZ(B8(O)) is a near homeomorphism such that if (x, y) G Zi C Li C IR2, ||(a;,y)|| = t,
then (Tl o ty)(x,y, 0 ) = fi(A(®o,JZo,0) = P i0 P ° Hi(X0 ^oH) = Pi.° P(f\k(t),t) =
Pi(f (x, y)). So, if (QoT) is composed with a near homeomorphism of cZ(B8(O)) which
“properly” takes L1 onto L1; L22, L23, . . . , L 2n onto L2; ...; and Ln2, Ln3, . . . , Lnn onto
L n, we will have succeeded in extending / ,to a near homeomorphism of cZ(B8(O)).
Toward that end, we prove the next two results.
P ro p o sitio n 4.5 Let Q — { ( p , / , 8 )
0 < Z1 < TT and
0
G
IR3|0 < p < r,0 <
< 4>x,8i < 9 < O2, where
< d2 — 8 i < 2x}, B1 = {(p, </>0, #o)|0 < p < r*, 0 < / 0 < ^ 1, 9i <
61
% < #2,r* < r,
^ere
/ZW),
% = {(A^i,%))|0 < /) < r* gi < ^ < ^2}. TTwm
0 meor AomGomorpMgm g': Q -» Q a^cA ^Aof ^|@Q = zdp, ^(^i) = ^2, ond
g{p, ^o, Oq) = 0 , (j)i, 6 0 ) G S2, for all (p, fo, 90) e S 1. Let S 3 = {(p, 0, 0)|0 < P < r*},
then there exists a near homeomorphism g* : Q ^ Q such that g*\dQ = idy, g*(Si) =
^*(p, ^o, ^o) = (p, 0,0) E 63, /or MZ (p, ^o, %) € 5 "i.
Proof:
.
First choose a homeomorphism expanding/contracting in the ^-direction
and rotating in the ^-direction, so that Si is taken onto S 1 = {(p, (J)0-, 0)|0 < p < r*}
and S 2 is taken onto S 12 = {(p, f , 0)|0 < p < r*}. Then select a homeomorphism
of this sector fixing those points on the yz-plane and expanding/contracting in the
^-direction to map onto the sector {{p, f>, #)|0 < p < r , 0 < f < f ,0 < /9 <
tt } .
62
Figure 32:
63
Figure 33:
64
Compose this homeomorphism, with a homeomorphism taking the sector {(p, 0, <9)|0 <
p < r,0 < (j) < |, 0 < ( 9 < 7 r } onto the rectangular parallelpiped [—r,r\ x [0, r] x
[0, r] and fixing S[ and S 123. Label the composition of all these homeomorphisms /3.
Figure 34: {0} x [0, r] x [0, r]
Let So = {0} x [0, r] x [0, r] and construct a continuous function G : S10 x [0,1] —> S0.
Denote G|soX{t} by Gt- We will have the following properties:
1. Gi = idy,
2. Gt is a homeomorphism if 0 < f < I;
3. for every f,G t |dSo = idy,
65
4. if (p, {6, #) e
Go((/), #)) = (/?,
0) € 5%.
Then G 0 is a near homeomorphism by property 2. Define a map G* : [-r, r] x [0, r] x
[0, r]
[-r, r) x [0, r] x [0, r] by G*(x, y, z) = G m (0, y, z) + (x, 0, 0).
C laim : Gk is a near homeomorphism.
P roof of Claim: Let e > 0 be given. Since any two norms on a finite dimensional vec­
tor space are equivalent, we will use the 1-norm, i.e. \\(x,y,z,w)\\ = |z| + |2/| + |z|-l-|w|.
Since S q [0,1] is compact, G is uniformly continuous. So, there exists 0 < 5 < I such
x
that ||(0,j/i,2i,ti)-(0,2/2,22,*2)|| <
8
implies that \\Gtl(0, yu Z1) - Gt2(0, y2, z2)\\ <
For each —r < x < r , denote {%} x [0, r] x [0, r] by Sx. Choose 0 < 77 < ^
f.
and
define H* : S = {J-r<x<rSx —> S by H*(x,y,z) = G\xi±n(0,y,z) + (x, 0,0). For fixed
r+ y
x, H*\sx is a homeomorphism of Sx, so H* is one-to-one and onto. S being compactHausdorff, in order to show H* is a homeomorphism we need only establish that H*
is continuous. So, fix (x, y,z) e S and choose 0 < <5* < f so that (max{l, y})<S* < 8 .
Then if Or1, ?/i, 21) G B s*{x, y, z) AS, \\H*(x, y, z) - H*(x 1 ,y1, Zi)|| < ||G|ai ^ ( 0 , y , z ) r+T}
G \xt\+v (0, yi, ^1)H+ 11(a:, 0,0) —(^1,0,0)11 < HG1^1+7, (0, y, z) —G \x,\+v (0, ^1, zi)[|+ f • But,
r +77
r+r}
r-\-T]
(xi,yi,zi)\\ < (max{l, y})(5* < 8 . Therefore, HGm m (0, y, 2) - G m jm (Oj Tz1j Z1)H < f
r+T?
T4-7?
and so H* is continuous. Hence, H* is a homeomorphism. Let (x, y, z) G S, then .
\\G*{x,y,z) - Lf*(o;,T/,z)|| = ||G m (0, t/ , z) - G m m (Oj Tzj Z)H < f, since
7'4-7?
<
_zz_
=
<
__ I__ =
1\
IMfctzzl _ zlMizzli _ izzlMdli
5+1-6 —
I r(r+?7)
r(r-\-rj) '
' r{r+r}). ' - r+ri
1+z
1+ ^ . 1+V
Thus,
\\G* - LPHoo < e, where || • ||oo denotes the sup norm. Hence, G* is a near homeomor­
phism, and the claim is proved.//
66
Now, (3 1 o G* o /3 is the required near homeomorphism g and the existence of g* is
proven similarly. This concludes the proof of the proposition.
PTL
The following proposition will complete our extension of / to a near homeomorphism
ofdfBsfO)).
P ro p o sitio n 4.6 There exists a near homeomorphism K : cZ(Bg(0)) —> cZ(Bg(O))
such that:
(l)
^ 'l d ( d ( B
8( o ) ) )
=
idy,
(ii) K = idy on a neighborhood of the negative z-axis intersect cZ(B8(0)) of
the form {(p,(p, 6 )\p > 0 ,0O < </> <
fixed}
n cZ(B8(0)),
(in) K \ t = idy,
AT(Bi) = Z/i,
(v) K ( L ij) = Li, ! < i < n , 2 < j < n ,
(vi) i f t e Li, K(t) = t € L1,
(vii) i f t G Lij, K(t) = t e Li.
Proof:
n= 3:
Induction on n:
tt, 0
<
< 27r, where
tt
> (f) 0 > ^ is
67
Figure 35:
Li lies along the interval [0,6] on the positive z-axis. Choose a translate Qi of
a sector of form Q (as in the previous proposition) so that Li C int(Qi),Li C
dQi,Qi n ((Ui<i<3i2<j<3l'ij) U (U2<i<3^i)) = {0}, and so that the last proposition ap­
plies to give a homeomorphism gi : Qi
Qi with
= Li, where gi(t) = t E L i ,
for every t £ L i . Extend gi to Gi : c/(B8(O)) —> cZ(B8(0)) by letting Gi = idy on
cZ(B8(0))
— Qi.
68
Figure 36:
Let Iz denote the interval [0, 6] on the positive z-axis. Select a translate Q12 of a sector
of form Q so that Iz C <9Qi2, L 12 C m£(Q12), Q12 A ((U1<*<3,2<j<3L ^) U (U1<i<3Lg)) —
L12) = {0}, and such that the previous proposition guarantees a homeomorphism
912 =Q12 -*■ Q12 with p(L12) = Iz and g(t) = (0, 0, t)
E Iz, for all f E L12.
69
Figure 37:
Extend to a homeomorphism G u : cl(B8(O)) —> cl(B8(O)) by letting Gi2 = idy on
Cl(B8 (O))-Qn- Similarly map L22 and L32 onto Iz with appropriate near homeomorphisms and then map Iz onto L2 with a near homeomorphism. Finally, use the previ­
ous proposition to map each of Lj3, L23, and L33 onto Iz with a near homeomorphism
and then Iz onto L3. Note that we can choose each of these near homeomorphisms to
be the identity on {(p, (f), 9)\p > 0, |7r < (/> <
tt, 0
< 0 < 27t}. Then the composition,
K, of all these near homeomorphisms is the required near homeomorphism. Assume
that the result is true for n = A; —I.
n = k : As in the n = 3 case, choose a homeomorphism Gi : cl(B 8 (6 )) —» cZ(B8(O))
such that Gi(Li) = Li with G i(t) = t G Li, for every t G L i,Gi = idy on
<9(d(B8(0))), G i disturbs only the leg Li, and Gi = idy on cl(B 8 (0)) fl {(p, 0 ,9)\p >
0 , ^ < (p < ir,0 < 9 < 2tt}. By the induction hypothesis, the near homeomor­
phism AT71-I = cZ(B8(O)) —> cZ(B8(0)) for the (k — l)-odd is the identity on a set
F = cZ(B8(O)) n {(p, <fi,0)\p > 0 ,(l)o < (/) < n,0 <
6
< 2 ir, where
Choose a near homeomorphism taking Li onto the ray (j) —
+
tt >
0O> | is fixed}.
—0o),9 = ^ , 0 <
70
p < 6, that is the identity on f = {(p, <^, 6 )\p > 0, </>0 +
- 0O) < ^ <
tt, 0
< 0<
27r} n cZ(B8(0)). Now select appropriate near homeomorphisms taking L k 2 onto L 2,
L k 3 onto L 3,
, Lkk onto Lk, and then Li2, L22, . . . , Lk^i 2 onto lz. Next, pick a home-
omorphism of cZ(B8(0)) that is the identity on f and so that when composed with
K n~i maps L z onto L2, Li3, L 23, . . . , Lk^ i 3 onto L3',. ..; and L ik, L 2k, . . . , Lk_i k onto
Lk. Choose a homeomorphism of cZ(B8(0)) that is the identity on <9(cZ(B8(0))) U f
that takes L 2, . . . , L k back to their original positions. Finally, choose a near homeo­
morphism taking Li back to its original position that is the identity off a sector not
containing {(p, </>, Q)\p > 0,
+ |(7T —</>0) < 0 < 7 r , 0 < 6 < 27r}. The composition of
all these maps is the required map K and the result follows by induction.
PTL
K o (Li o 'fy) is then an extension of / to a near homeomorphism of cZ(B8(O)) that
is the identity on del(B 8 (O)). Extend Tf o (fi o T) to a near homeomorphism 7 of
cl(B 9 (O)) by letting 7 = idy on cl(B9 (6 )) —cZ(B8(O)). Choose a homeomorphism 7 :
cZ(B9(O)) —> cZ(B9(O)) so that 7 = idy on 9(cZ(B9(0))),7_1(cZ(Bi (T))) = cZ(B8(O)),
and so that for any x G cZ(B9(O)), there exists an n £ Z+ so that 7n(z) G cZ(B8(0)).
Also, select a near homeomorphism 7 : cl(B 9 (O)) —> cZ(B9(O)) so that 7 — idy on
cZ(B9(O)) —B i(T ) and so that 7_1(T) = cl(Bi(T)). Let B — 7 0 7 0 7. Then F is
a near homeomorphism of cZ(B9(0)) so that F\T = f and T is a global attractor for
cl(B9 (6 )) under F. By the theorem of Morton Brown [10], Iim F is homeomorphic to
cZ(B9(O)). Finally, as in the paper by Barge-Martin ([I]), we have the following result:
Theorem 4.11 If f : T —> T is a continuous function, T an n-odd (n > 3), then
Iim / can be embedded as an attractor in IR3.
71
C H A PTER 5
CH AIN ABILITY A N D EXTEND IBILITY
As a first step in seeing which maps f : G ^ G have the property that Iim /
can be embedded as an attractor in the plane, we give necessary and sufficient condi­
tions that Iim / be snake-like or chainable. First, we need a definition and a theorem
by Bing [7].
Definition 5.14 (Bing [7]) A chain is a finite collection of open sets di,dg,... ,dj
such that di intersects dj if and only if i — j — l , j or j + I.
We do not impose
the condition that the di’s are connected. If the links of a chain are of diameter less
than e, the chain is called an e-chain. A compact continuum is called snake-like (or
chainable) if for each positive number e it can be covered by an e-chain.
Theorem 5.12 (Bing [7]) Each snake-like continuum is homeomorphic with a. plane
continuum.
Thus, if / : G —> G is such that Iim / is chainable, Iim f is planar.
Theorem 5.13 f G G(G). Iim / is chainable if and only if for every e > 0 there
exists an N E TA such that if n > N , there exist maps gn from G onto I = [0,1] and
hn : I ^ G so that the following diagram e-commutes:
fn
G
—»
9n
G
/A h n
I
72
P roof:
(=>) We prove the result for f G C(T), T a triod with each leg of unit
length, the general case being very similar. Suppose that Iim f is chainable and
let e > 0 be given.
Let L i , . . . , Lm be an |-chain of Iim / such that the dis­
tance between non-adjacent links is bounded below by some positive number and
each link intersects Iim / (Barrett, [3]).
open cover [Li, . . . , L m) of Iim / .
Let % be a Lebesque number for the
Choose iV G Z+ such that 2% #+! d'ia^ T^ <
Ew+i ;»=r < I - Then for every n > N, there exists a
t G T,diarmr~ 1 (Bs(t)) < rj.
8
(n) > 0 such that if
Fix n > N and cover T with 5-balls as follows:
Figure 38:
yj\=iBs(xi) where the distance between non-adjacent balls is bounded below by some
73
positive number and each ball intersects T.
For each z, tt" 1
C Lj , some
I < j < m. For each i, let Ji = min {j Itt" 1(B6(^j)) C Lj }. Let j* = min { jJ
and j* = max{jj}. j* < j* by definition. If j* = j*, tt”1(B6(^i)) C Lj *, for every
I < z < / which implies that d ia m fn(T) < f-which can’t happen if | < diamT, since
/ is onto. So, assuming that f < diamT, j* < j*. Let djt = V {i\ji= jt }B s (Xi ), dj t + 1 =
^{i\ji=j*+i}Bs (xi),
■■■, dj* = \J{i\ji=j*}Bs (xi).
P ro p o sitio n 5.7 If k £ rZi, j* < k < j*, there exists Ji, I < i < I, such that Ji = k.
Proof: If 4 = 0, some j* < k < j*, Iim / = ^
1
(T) C (L 1, . . . , L&_i, Lk+U . . . , Lm}.
Then, (A, B} forms a separation of fim / , where A = Ufr11Li and B = U™fc+1Lj, a
contradiction.
For ease of notation, assume that j* = I and j* = m. Chain I = [0,1] with m open
intervals as follows:
74
Ui
(O
Um
0
^
-
0
)
(
I
)
Figure 39:
i.e. each open interval intersects I and the distance between non-adjacent intervals
is bounded below by a positive number. Then U ^ 1
H «j+i) consists of m — I
disjoint open intervals, say Ui, D2, ... vm_i-ordered according to increasing left-hand
endpoints. Also, I —( U ^ 1Wj) consists of m disjoint closed intervals J i , . Jm-again
ordered according to increasing left-hand endpoints.
Ji
0
0
Vi
—
9
V2
Figure 40:
75
Similarly, VJii, ^ d iHdj ) is a finite collection of disjoint open intervals in T. T -(U i^ ( ^ i A
^ finite disjoint union of closed sets—all but one of which is a closed interval.
Figure 41:
Let
Ci
be the closed set containing 0. Label the remaining closed intervals in Z1 accord­
ing to increasing left-hand endpoints c\, d j,. . . , clp\ the closed intervals in Z2 similarly
as c\, c l , . . . ,
Cg-,
and finally those in Z3 as c|, C3 ,
, cf. Note that
exactly one dk, and that the same is true of C i1 , all i and j . Let c} =
C1
C1
is contained in
A Z1 and choose
a continuous function eL1 : Z1 —> / such that if c\ C dk,$i(c\) = Jk. cj C dk implies
that c|+1 C dk—i or dk+i. If c|+1 C dk_i, we require that dq map the open interval
in T whose closure intersects both c\ and c\+l onto Vk^ 1. Finally, if c\+l C dk+1, we
require that $ 1(cI-+1) = J k + 1 and that ^ 1 map the open interval in T whose closure
intersects both cj and c\ + 1 onto vk. Similarly, define maps $ 2 : Z2 —» 7 and d>3 : Z3 —>/
with $ 2(0) = 1L3(O) = ^L1(O). Define a continuous function gn from T onto I by
76
G /2 >. Define a map hn : I
£ G Z3
9n(f) —
T as follows:
for each I < ft < m choose one of the disjoint closed subsets Ci or
C i1
of T -
fl
dj)) contained in d& and a continuous function XErfc from Jk onto {ci or cf } C dk.
Define AnL = f n 0 ^fc- This defines Ara on
Let Ji =
= I , ... ,m,
then Uj = (6j, Uj+i), z = I , . . . , m —I. For z = z,. . . , m —I, let a, : / —> T be a path
from Ara(Bi) to Ara(oj+i) of least diameter. Define Ara|yi(t) = o:(a^ ~ L .)- The Ara thusly
defined is continuous.
P ro p o sitio n 5.8 TAe following diagram e-commutes:
r
T
9n
—>
L
T
Ara
/
Proof: case I : Suppose that t 6 Cl or c-, some i,j. Let t
the other cases being similar. Then gn(t)
9n(t) = f n{x), where x
G
G
G
Cl C dfc, some I < ft < m,
Jk and hnogn(t) = hn\jkogn(t) = / 71OTfcO
dk. But, diam'jv~1 (dk) < diamLk <
since Trra^(dfc) C L k.
Therefore, d iam fn(dk) < diam'K~1 (dk) < f . Hence, !!/"'(t) —hngn(t)\\ < | . / /
For the last two cases,, assume that t G T —(LLj-(Ajfidj)), where, without loss of gen­
erality, t is in the open interval whose closure intersects both c} and c}+{, I < z < p —I.
case 2: Assume that cj C dfc and cj+ 1 C dfc_i. Then f n(t) G f n(dk),gn(t) G %&-i,
and hn(gn(t)) = a fc_ i ( ) - w h e r e % _i is a path from An(Bfc_i) G / n(dfc_i) to
Ara(afc) e j n(dk) in T of least diameter. Also, / n(dfc_i) f i / n(dfc) ^ 0, dz&m/"(d*_i) <
| , d ia m fn(dk) < f and a fc_i of least diameter imply that ||/ n(f) - hngn(t)|j' <
d ia m fn(dk) + dzam /n(dfc_i) < e.//
77
case 3: Assume that c- C dk and c] + 1 C dk+i.
Similar to case 2 .//
Therefore, | | / n - K g nW0 0 < e. Whence, (=>) is proven.
(<=) Suppose that for every e > 0, there exists an A' G Z+ such that if n > A, there
exist maps gn from T onto I = [0,1] and hn '■I
e-commutes:
fn
T
—>
9 n nX
T such that the following diagram
T
hn
I
Let e > 0 be given. Choose m G Z+ such that E ^ to+i ^ < f and a > 0 such
that E%o % <
Choose /c > 0 such that diamA < k implies that diamfi(A) <
a , 0 < j < m. Choose A
g
Z+ such that WfN ~ ^ArflrVlIoo < f and ?? > 0 so that
diamhN(Bv(t)) < | , for every t G /. If t G / and i/, z G g ^ i B ^ t ) ) , WfN(y) ~ f N(z)W
= WfN( y ) - hNgN(y)+hNgN( y ) - h NgN(z)+hNgN( z ) - f N(z)W < IIZiv(j/)-/bvflv(z/)|| +
ll^vfl?v(y) - ^vflv(Z)II + 11ZtArfljV(z) - Ziv(Z)II < 2 | + diamhN(Bv(t)) < f . There­
fore, d i a m f N(gjf(B ri(t))) < ^ < fc. Let L x, . .. , L k be a 277-chain of /, where Li —
B ri(U ) ,..., Lk = B rt(U)-, t x, . . . , tk G /. Then C = ( tt^ ^ ^ L
is a chain of Iim f .
i )),
. . . , T r^m( ^ 1(Lfc)))
Consider 7 r^ m(fl^r1(Li)). Let y, z G TLv+nXflvX-W)- Then
flv+m, Zjv+m e fl^1(S 7)(t)) implies that ||ZiV(flv+m) - f N(zN+m)W < k which shows
that WfN+j(yN+m) - f N+J(zN+m)W < a, for every 0 < j < m. Therefore, d(y,z) <
E ^ 0 f + E ~ m+i ? < f + I = f- Thus, diarmrjv+m(flv1(Ti)) < e. Since the same
argument applies to T T ^ ^ g ^ 1 (L2)), . . . , 7r^+m(Aw1(Tfe)), C is an e-chain of Iim f .
78
Hence, Iim / is chainable.
added in proof. As noted by Piotr Mine, the following, more general result, is known:
T h eo rem 5.14 Let X n be a locally connected continuum for n = 1 ,2 ,__ Form < n,
let f mn be a map of X n to X m. Let X be the inverse limit of { X n, f mn} and let Trn
denote the projection of X into X n.
The following two conditions are equivalent:
(i) For each e > O and for each positive integer m there is an integer n > m and there
are maps y : An —> [0,1] and h : [0,1] —* X m such that \hg - f mn\ < e.
(ii) X is chainable.
However, the previously proven theorem was arrived at independently and its proof
is different than the one that I’ve seen for the above theorem.
Our goal is then to embed such an inverse limit as an attractor in IR2, which we
will return to after the following brief interlude. The following gives an example of a
map / G C(T) such that no iterate of / factors through an interval map, but Iim /
is chainable.
E xam ple 5.2 Define f piecewise linearly on l\ U Zg
in the following figure and
/IhUi3 Zn similar fashion so that image(f) is as pictured. Then Iim / is chainable,
but no iterate of f factors through an interval map.
79
h
h
Figure 42: / I ilU2
80
h
Figure 43: Image of /
81
Note : We can modify the above example to obtain an / 6 C(T) such that
Iim / is chainable; / does not factor through an interval map, but some iterate of /
factors through an interval map. We arrange that a neighborhood of the fixed point
in Ii is collapsed onto the fixed point.
Definition 5.15 Let Ch(G) = { / G C(C)| Iim / is chainable }.
By modifying the above example, we obtain a sequence { f n}^Li C C(T) - Ch(T)
such that {/„} converges uniformly to f (as in the above example). We do this
by expanding the fixed point in Z1 to a fixed interval of length ^ , n > 3, to obtain
f n- In addition, given a fixed f no e C(T) - Ch(T),n 0 > 3, we can construct a
sequence {gm}m=i C Ch(T) such that gm converges uniformly to f no. We then have
the following result:
P roposition 5.9 Ch(T) is neither open nor closed in C(T).
'
P roposition 5.10 C(T) —Ch(T) is not dense in C(T).
Proof:
Define f : T
T piecewise linearly so that /(Z11U Z2) = I'
maps across T, as pictured. Then /
\\g — / 11oo < Co, y
G
C(T), then g
G
G
Ch(T) and there exists an
Ch(T).
e0 >
G
0
Z2 and /(Z3)
such that
if
82
Figure 44: image(f)
PTL
83
Proposition 5.11 Ch(T) is not dense in C(T).
Proof:
Define / E C(T) - Ch(T) piecewise linearly along each leg li(i = 1,2,3) as
follows:
k
Figure 45: f \ u,i = 1,2,3.
We will show that if ||/ —p|| <
in each of the intervals (^f, jf) C
(M’ M)’ ^(z) <
then g 6 C(T) — Ch(T). g has a fixed point
(H',
M')
c
and (H", 55") C Z3. If x €
Let Xi = inf{% E [ ||,||] |p ( x ) = x}.
Then p(xi) = Xi .
5([0, Xi]) C [0, Cq] U [0, Cq] U [O1X1]. Similarly, define X2 E Z2 and X3 E Z3 and let
Ti = [O1X1] U [0,x2] U [0,x 3] C T . Then g(Tx) C Tl l ^(X1) =
g(x3) = x 3. If A1(T1) ^ T1, let p E T1 — ^(T1).
X l l ^ (X 2)
= x 2, and
T — {p} consists of two open
84
sets which form a separation of g(Ti)-a contradiction. Therefore, g(Tx) = T1. So,
=Iim g\Tl is a subcontinuum of Iim g. Let O1 = ( ||, 1] C h , 0 2 = ( § ', T] C Z2, and
O3 = (§ " , I"] C Z3- O1, 0 2, and O3 are open in T. So, Oi = UjL0Tr^1(Oi), I < Z < 3,
are open in Iim g. Also Oi ^ 0 for each I < z < 3, since Oi contains a fixed point of g
for each I < z < 3. Next, suppose that O1H O2 ^ 0. i.e. there exists a z G Iim 9 such
that Zj G O1 and z& G O2, where without loss of generality, j < k. gk~^{zk) = Zj G O1,
where Zfc G O2 C Z2. But, g{BjL {li)) C B jl(I1),
so
that gn(B Jl (U)) C B jl (U), for ev-
ery n G Z+ and I < z < 3. This contradicts the fact that gk~j (zk) G O1. Therefore,
O1 n O2 = 0. In similar fashion, we obtain that O1, O2, and O3 are mutually disjoint.
Finally, Um g — N C O1 U O2 U O3. Thus, Hm g — N consists of three mutually
separated sets and so Hm 9 is a tried ([24]). Whence, according to Bing ([7]), Hm 9
is not chainable.
It is clear that the above examples and results can be generalized to other finite,
connected, planar graphs. So, the following appears to be the best general result
posssible.
Corollary 5.2 Ck(G) is a Cs set in C(G).
Proof:
Let An == { / G C(G)| there exists an m G Z+ such that / m^r- factors
through an interval map }, for each n > I. Then An is open for each n > I and
Ok(G) = n-iA n.
We return to some results related to our goal. In order to embed Hm f as an attractor
in IR2, where Hm / is chainable, we need to be able to extend a homeomorphism of
an embedding of Hm / to a homeomorphism of IR2. The following discussion shows
85
that, in general, given an arbitrary drainable continuum K, embedded in IR2, and a
homeomorphism f : K ^ K, f cannot be extended to a homeomorphism of all of
IR2. We start with a definition and a theorem by Brechner ([9]).
D efinition 5.16 (Brechner [9]) We call a homeomorphism h of a planar continuum
M onto itself, essentially extendable if and only if there exists an imbedding
: M —>
E 2 such that
fh f^
1
: 4>(M) —> (f(M)
can be extended to a homeomorphism f h f -
1
:E2
E 2. (S 2 may replace E 2 in the
definition).
Theorem 5.15 (Brechner [9]) There exists a chainable continuum M C S 2 and a
homeomorphism h : M —>M such that h is not essentially extendable.
Dr. Brechner proves her theorem by “gluing” two pseudo-arcs together at a com­
mon endpoint and constructing a homeomorphism which is period two on one of the
pseudo-arcs and fixes the other. She uses some sophisticated prime end theory to
show that this homeomorphism is not essentially extendable. We construct a similar
example in the following theorem for which elementary techniques can be used to
show that the homeomorphism is not essentially extendable.
Theorem 5.16 There exists a chainable continuum K C IR2 and a homeomorphism
f : K ^ K such that f is not essentially extendable.
■
Proof: Let K be the following planar, chainable continuum:
Z= {—2 } x [—1 ,1],Z1 = { - l } x [ - l , 1], In = { -2 + ^ } x [—I, l],/n = { - ^ } x [ - i , ^ ],r =
{2} x [-1 ,1 ],T"! = {1} x [-1,1], r„ = {2 -
x [ - J .J ] , and rn = {^} x [ - J j J]-
where n > 2 at each occurrence. Also, place a double “sin 1” curve between each
set of vertical bars as pictured. Define a homeomorphism f : K
K (onto) so
86
y=x
• • * \
Figure 46: K
87
that each vertical bar and each double “sin
curve is left invariant, /(0) = 0,
/ = Uy on that part of K lying in the right half-plane, and / flips each verti­
cal bar in the left half-plane about the x-axis-i.e. if (x,y) is an element of one
of those vertical bars, f( (x,y)) = ( x , - y ) .
embedding of K.
Now, let L = a{K) be any planar
Suppose'that G : IR2 —> IR2 is a homeomorphism such that
G\ l = a f a -1. We show that G is orientation preserving. Then, in similar fash­
ion, it can be shown that G is orientation reversing-a contradiction. Let
: IR2 —>
IR2 be an orientation preserving homeomorphism such that Ha\i = U y \i. Then
H G H ^ I h ^l Y = H a f a - 1 H^ln(L), he.
H G H - 1 extends H a f a - 1 H - 1 Ij1 (L).
Let
I* — H(a(li)),i = 1 ,2 ,----Since Ha\i = Uy, H G H - 1 \H(L)(Ha(l)) flips Hd(I) — I
about its midpoint. Also, each I* is invariant under H G H -1. Choose 0 < <5 < ~ such
that H G H -1<(B$((—2,t))) C Bj_(Bs((—2, —f))), for every t G [—1,1]. Then choose
N > 2 such that (H a ) ( ( - 2 +
G
S 6((-2 ,t)), for every t G [—1,1]. Suppose,
without loss of generality, that the horizontal line ^ = O intersects
in the half­
plane {(x, y)\x > —2}, then no line y = t , t £ [—|, |] intersects I^r in the half-plane
{(%, y)\x < —2}. The line y = \ intersects '(l*N n Bs((—2, |))), by the connectivity of
1%. Let wi be the point of the intersection with least x-coordinate. Since l*N is closed,
Wi G Zjv- Similarly, let Ui2be the point of intersection of (Zjy D Bs((—2, —|))) with
least x-coordinate. Let ozi be a simple arc from (—2, —\) to (—2, |) along x = —2,
let Oz2 be a simple arc from (—2, |) to Wi along y = \, let Qz3 be a simple arc in
Zjv from wi to Ui2, and let «4 be a simple arc from u 2 to (—2, —|) along y = —
Then T = ozi * Qz2 * Qz3 * 014 is a simple closed curve in IR2 oriented clockwise so that
H G H - l (F) is oriented counterclockwise. Therefore, H G H -1 is orientation reversing.
Since H is orientation preserving, G is orientation reversing. We can do a similar
argument, using r, to show that G is orientation preserving.
88
This contradicts the existence of such a G.
The following theorem shows that even if Iim / is planar, / , the shift homeomorphism,
may not be extendable to IR2.
T h eo rem 5.17 There exist an f E C(T), T a triad, so that Iim / is planar, but f is
not essentially extendable to IR2.
Proof:
Define / E C(T) piecewise linearly as follows:
Figure 47: /Iz1UZ2
89
Define /|z3 = idy. Then Iim / | ZlUz2 can be embedded in IR,2 as follows:
Figure 48:
90
f restricted to Iim /Iz1UZ2 shifts I0 across I0 U Ii , each Ii across
i > I, Z0 across
I0 U l\,li across k+i ,i > I, and flips I about the x-axis. So, let K = o/lim / ) be
the following planar embedding of Ihn / (where the interval from (-1 ,0 ) to (0,0)
corresponds to Iim /|z3).
Figure 49:
91
Showing that / is not essentially extendable to IR2 is then equivalent to showing
that g = a fa .- 1 : /V —>
is not essentially extendable to IR2. Suppose that g
is essentially extendable to IR2. Then there exists an embedding /3 : K ^
IR2
such that /Sgp-1 extends to a homeomorphism G of IR2. Let H be and orientation
preserving homeomorphism of IR2 such that H(3\h = idy\i0. Then H G R -1 : IR2 —>
IR2 is a homeomorphism such that H G H -1\h^ k ^ = H fd a fa - ^ fd ^ H - ^ H ^ n ^ - i.e .
H G H -1 extends f on H((3(K)). Let HfS(Ii) = /*, for all i > 0, and HfS(Ii) = /*, for all
i > I- Thus H G H -1I1* — idy\i* and H G R - 1Ii0 flips I0 about the x-axis. There exists
a
6
>
0
so that H(f3([—6, 0 ) ,
H(/3([-6, 0 ) ,
t/
0 ))
0 ))
C
B _
l
((0, 0)).
Without loss of generality, assume that
is contained in the left half-plane P = {(z, y)\x <
0 }.
There exists an
> 0 so that H G H -1(Br)((0, t))) C B ^((0, —t)) for every —I < i < I. Since I* and
I* limit on Z0 as i ^ oo and H(fS([—6, 0 ) ,
0 ))
C P, there exists an N such that i > N
implies that Ii , I* C int(D), where D = cZ(B^((0,1))) U c/(Br?((0, - I ) ) ) U (cl(Bn(lf)) A
{(x,y)\x > 0}). Let
: [0, oo) —> U^L1Z* be a homeomorphism and G = sup{Z >
0|7i(Z) G dD}. Let Z* be the half ray 7i([Zi, oo))). Also, let 72 : [0,00) —» U^L1Z*
be a homeomorphism, Z2 = sup{Z > O^2(Z) G dD}, and Z* = 72((Z2, 00)). int(D) —
I* consists of two components, say Du containing (0,1) and Di containing (0 ,-1 ).
Because Z* is connected, without loss of generality, Z* C Du. H G R - 1^nt(D ) — I*)
consists of two components, say D1u containing (0,1) and D11 containing (0, —I). Since
H G H -1^O, I)) = (0, - I ) and vice versa, H G H ~1(DU) = D11 and H G H ^ (D 1) = D'u.
But, H G R - 1^*) C Z* so that H G R -1^*) A
7^ 0, contradicting the fact that
H G H -1If*) C D[.
PTL
In this chapter, we have given necessary and sufficient conditions that Iim / be
chainable for / : G —>G continuous and onto with G a finite, connected, planar graph.
We have also shown that, in general, a homeomorphism of a chainable continuum is
92
not essentially extendable to the plane. However, it is still open as to whether or not
hm / chainable implies that / is essentially extendable to the plane. Brechner and.
Lewis ([8], [15]) both have results implying that such an / is essentially extendable
to IR3. Another question is “for which f : G —>G with Iim / planar is f essentially
extendable to.the plane (or IR3)?" Finally, ii f : G —>G, can Iim / be embedded in
o
^
•
IR in such a way that / on this embedding extends to a homeomorphism of all of
IR3?
*
93
R E F E R E N C E S CITED
[1] M. Barge and J. Martin. The construction of global attractors. Proc. Amer.
M m . ^oc., 110(2):523-525, 1990.
[2] Marcy M. Barge and Russell B. Walker. Nonwandering structures at the period­
doubling limit in dimensions 2 and 3. Transactions of the American Mathematical
337(1)=259-277, May 1993.
[3] Lida Barrett. The structure of decomposable snakelike continua. D.uke Math.
Journal, 28:515-522, 1961.
[4] Ralph Bennett. Embedding products of chainable continua. Proceedings of the
American Mathematical Society, 16:1026-1027, 1965.
[5] R. H. Bing. A homogeneous indecomposable plane continuum. Duke Mathemat­
ical Journal, 15:729-742, 1948.
[6] R. H. Bing. Embedding circle-like continua in the plane. Canadian Journal of
Mathematics, 14:113-128, 1962.
[7] R.H. Bing. Snake-like continua. Duke Math. Journal, 18:653-663, 1951.
[8] Beverly Brechner. Homeomorphisms of chainable continua are essentially ex­
tendable to E3. Preprint.
[9] Beverly Brechner. On stable hdmeomorphisms and imbeddings of the pseudo
arc. Illinois Journal of Mathematics, 22(4):631-661, December 1978.
[10] Morton Brown. Some applications of an approximation theorem for inverse limits.
Proc. Amer. Math. Soc., 11:478-483, 1960.
[11] G.W. Henderson. The pseudo-arc as an inverse limit with one binding map.
Duke Math. Journal, 31:421-425, 1964.
[12] J. R. Isbell. Embeddings of inverse limits. Annals of Mathematics, 70(l):73-84,
July 1959.
[13] James Keesling. On the shape of torus-like continua and compact connected topo­
logical groups. Proceedings of the American Mathematical Society, 40(l):297-302,
September 1973.
94
[14] James Keesling. Embedding T n-Iike contimia in euclidean space. Topology and
its Applications, 21:241-249, 1985.
[15] Wayne Lewis. Extending homeomorphisms of planar subcontinua of E3. Spring
Topology Conference, University of Alabama in Birmingham, 1987.
[16] M. Schub M. Hirsch, J. Marsden, editor. Recurrent sets for planar homeomor­
phisms, New York, 1993. From Topology to Computation: Proceedings of the
Smalefest, Springer.
[17] Michael C. McCord. Embedding P-Iike compacta in manifolds. Canad. Journal
of AWt., 19(2):321-332, 1967.
[18] P. Mine and W.R.R. Transue. A transitive map of [0,1] whose inverse limit is the
pseudoarc. Proceedings of the American Mathematical Society, 111:1165-1170,
1991.
[19] James R. Munkres. Topology: a first course. Prentice-Hall Inc., New Jersey,
1975.
[20] M.H.A. Newman. Elements of the Topology of Plane Sets of Points, volume I.
Dover, New York, 1992.
[21] Dale Rolfsen. Knots and Links. Publish or Perish, Inc., 1976.
[22] Michael D. Sanford and Russell B. Walker. Extending maps of a cantor set
product with an arc to near homeomorphisms of the 2-disk. Preprint.
[23] Smale. Differentiable dynamical systems. Bulletin of the American Mathematical
,Sbcief?/, 73:747-817, 1967.
[24] R. H. Sorgenfrey. Concerning triodic continu'a. American Journal of Mathemat­
ics, 66:439-460, 1944.
'
.
[25] Gordon Whyburn and Edwin Duda. Dynamic Topology. Springer-Verlag, 1979.
[26] Gordon Thomas Whyburn. Topological Analysis. Number 23 in Princeton Math­
ematical Series. Princeton University Press, 1958.
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