Lecture 24:
d
Defn: Topological conjugacy: Given Z+
d (resp, Z ), actions T , S a
topological conjugacy from T to S is a homeomorphism φ : M → N
d
s.t. φ ◦ T = S ◦ φ i.e., φ ◦ T n = S n ◦ φ for all n ∈ Z+
d (resp, Z ).
(analogue of measure isomorphism)
Invariants of top. conjugacy:
a. Top. Trans.
b. Top, Mixing
c. Density of Periodic points
Proofs of a and b are modifications of proofs that ergodicity and
mixing are invariants of measure isomorphism.
Theorem: Full two sided shift (d = 1, |F | = 2, a Z-action) is
conjugate to horseshoe map, restricted to Ω.
Proof: Decompose the square V into a union of rectangles L, C, R
(left, center and right).
Recall: Ω := ∩n∈ZT n(V ).
Observe: Ω = ∩n∈ZT n(L∪R) because if T nx ∈ C, then T n+1(x) ∈
B and so 6∈ V .
Define φ : Ω → {R, L}Z: If x ∈ Ω, define φ(x) = y, where
yn = L if T n(x) ∈ L
yn = R if T n(x) ∈ R
Note: φ(x) is regarding as a coding of x.
Note: we are using R, L to mean sets and symbols.
−n
Note: x ∈ ∩∞
(yn)
n=−∞ T
φ ◦ T = σ ◦ φ:
Proof: “The coding of T (x) is the left shift of the coding of x.”
φ(T x)n = L if T n+1(x) ∈ L
1
φ(T x)n = R if T n+1(x) ∈ R
Thus, φ(T x)n = φ(x)n+1 = (σ(φ(x)))n. QED
Key Lemma: For all N and all sequences y−N . . . y0 . . . yN ∈
−n
(yn)
{L, R}2N +1, the set U (y−N . . . y0 . . . yN ) := ∩N
n=−N T
– is compact
– is nonempty
– has diameter < 2−N
Idea of proof: Draw pictures of 3-blocks −n
φ is 1-1: Let x ∈ Ω and y = φ(x). Then x ∈ ∩∞
(yn),
n=−∞ T
which can be written as a decreasing sequence of nonempty compact
sets whose diameters → 0. So,
−n
{x} = ∩∞
(yn)
n=−∞ T
So, x can be recovered from y = φ(x).
φ is onto: Let y ∈ {L, R}Z.
−n
(yn) is a nested decreasing sequence of compact
Then ∩N
n=−N T
sets and thus its infinite intersection is nonempty. In fact, since
diameters decrease to zero, its intersection is a single point x ∈ Ω,
and y = φ(x).
φ−1 is continuous:
If d(y, y 0) is small, then for a large N ,
0
0
y−N . . . y0 . . . yN = y−N
. . . y00 . . . yN
.
0
0
Now, φ−1(y) ∈ U (y−N . . . y0 . . . yN ) and φ−1(y 0) ∈ U (y−N
. . . y00 . . . yN
),
and so
d(φ−1(y), φ−1(y 0)) < 2−N .
So, φ is a conjugacy. 2
So, T |Ω is top. transitive, top. mixing and has dense periodic
points.
Can extend T to a homeo. of the sphere which has a contracting
fixed point in the left cap A, an expanding fixed point at ∞, and all
points in S 2 \ (Ω ∪ {p} ∪ ∞) progress from ∞ to p.
Note: The full 2-sided shift and Ω are Cantor sets; in particular,
they have both a topological basis of clopen sets (totally disconnected).
Defn: Topological factor: T ↓ S if there exists continuous, onto
φ : M → N s.t. φ ◦ T = S ◦ φ.
(analogue of measure homomorphism)
If T satisfies top trans (resp. top mixing, resp dense set of periodic
points) and S is a topological factor of T , then S satisfies the same
property.
Claim: Full one sided shift on F = {0, 1} factors onto doubling
map on circle.
φ(x0x1 . . .) = e2πi(.x0x1...)
Note: the full one sided shift and doubling map are not top. conjugate because F Z and circle are not homeomorphic (e.g., totally
disconnected vs. connected).
Thus, the doubling map on circle and horseshoe map on Ω are top.
mixing, therefore top. transitive, and have dense periodic points.
3
Lecture 25:
The horseshoe extends to a diffeomorphism of 2-sphere with one
expanding fixed point, one contracting fixed point and an invariant
set that is topologically transitive, topologically mixing and has dense
periodic points (because on Ω it is topologically conjugate to a twosided full shift).
More examples of continuous actions:
Defn: Let F be a finite alphabet and F be a collection of configurations on finite subsets of Zd (F stands for “forbidden list”).
d
XF = {x ∈ F Z : no translate of an element of F appears in x}
Defn: A (two-sided) shift space is a subset M of Zd of the form
M = XF .
Prop: A subset M of Zd is a shift space iff it is closed and shift
invariant.
Proof: Closure means that M is the complement of an open set,
equivalently the complement of a collection of cylinder sets.
Shift-invariance means that this collection of cylinder sets is shiftinvariant. Corollary: For a shift space M , σ|M is a cts Zd action.
Proof: shift invariance says that each σ i maps M to itself.
closure is equivalent to compactness. Note: We often write M when we mean σ|M .
Note: Can also consider one-sided shift spaces; gives a cts. Zd+
action.
Defn: A (two-sided) shift of finite type (SFT) is a shift space
M = XF for some finite F.
Note: WMA that no two of the finite subsets are translates of one
another and in fact WMA that there is only one subset, a square Dn
or Bn. But it will be convenient to consider other subsets.
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Example: Ledrappier 3-dot:
F = {w ∈ F {(0,1),(0,0),(1,0)} : x(0,1) + x(0,0) + x(1,0) 6= 0 mod 2}
Felipe will discuss this SFT in more detail.
Defn: A nearest neighbour SFT (nn SFT) is an SFT M = XF
where F consists of configurations on edges, i.e., on {(0, ei) : 1 ≤
i ≤ d}.
Examples of nn SFT’s in d = 2 (both are top. mixing, top.
transitive and have dense set of periodic points).
1. hard square: F = 11 in horizontal and vertical directions:
2. domino tiling: F = {R, L, T, B}.
F = {LL, LT, LB, RR, T R, BR,
T T T B L R
, , ,
,
,
}
L R T B B B
d = 1: nn SFT is called a Topological Markov Chain (TMC); it
specifies the transitions of a Markov chain that are forbidden (equivalently allowed) but not the conditional probabilities.
Any TMC is of the following form:
Defn: Let C be a square 0-1 matrix (say m × m).
Let F = {0, . . . , m − 1}.
Let MC = {x ∈ F Z : Cxi,xi+1 = 1 for all i ∈ Z}
Consider corresponding graph of C: MC is the set of all bi-infinite
paths on the graph.
Example 1: (“golden mean shift;” analogue of hard square in d =
1).
1 1
C=
1 0
Draw graph
5
Then MC is the set of all bi-infinite binary sequences such that 11
is forbidden.
Example 1 is top. transitive, top. mixing and has dense periodic
points.
Example 2: (a transient shift)
1 1
C=
0 1
MC is the set of all shifts of 0∞.1∞ together with 0∞ and 1∞.
Example 2 is not top. transitive, nor top. mixing nor has dense
periodic points.
Recall that we defined irreducibility and primitivity only for stochastic matrices, but the same concepts apply to nonnegative matrices.
WMA: C has no all zero rows and no all zero columns, because
such columns and rows have no effect on the defn. of MC .
Prop:
1. σ|MC is top. transitive iff C is irreducible.
2. σ|MC is top. mixing iff C is primitive.
3. σ|MC has dense periodic points iff C is the union of disjoint
irreducible components, i.e., no “one-way” transients from one component to another.
Proofs of 1 and 2 are nearly identical to proofs of ergodicity and
mixing for irreducible and primitive Markov chains.
6
Lecture 26:
Time to meet for HW2 and schedule for student talks.
TOPOLOGICAL ENTROPY AND PRESSURE
Defns: Given a cts Zd+ action T on compact metric M .
For an open cover α of M , let N (α) be the minimum size of a
subcover.
For open covers α, β, let
α ∨ β := {A ∩ B : A ∈ α, B ∈ β, A ∩ B 6= ∅}
Let
1
−i
log
N
(∨
T
(α))
i∈D
n
nd
The limit exists and equals the inf (by multidimensional Fekete Lemma).
h(T , α) := lim
Define the topological entropy of T :
h(T ) := sup h(T , α)
α
Entropy of a cts Zd action defined in the same way (as if it were
a Zd+ action).
Will take an alternative approach and generalize topological entropy to topological pressure as follows.
Defns: Let T be a cts Zd+ action on compact metric M and f :
M → R be continuous.
View f as weights or a potential; f := 0 is an important special
case.
For δ > 0, a subset E of M is called (n, δ)-separated if for all
x, y ∈ E, x 6= y, there exists i ∈ Dn s.t.d(T i(x), T i(y)) > δ.
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Special case d = 1: E is (n, δ)-separated if you can distinguish
points of E by observing their orbits of length n and your vision is
good enough to distinguish points that are > δ apart.
P
Snf (x) := i∈Dn f (T i(x))
P
Pn,δ (T , f ) := sup{ x∈E eSnf (x) : E is (n, δ) − separated}
Special case d = 1, f = 0:
– Pn,δ (T , f ) is the maximum size of an (n, δ)-separated set.
– so, Pn,δ (T , f ) is the maximum number of orbits of length n that
you can see if your vision is good enough to distinguish points that
are > δ apart.
Prop: Pn,δ (T , f ) is finite.
Proof: Let U be a finite cover of M by balls of radius of δ/2 (exists
by compactness).
For an (n, δ)-separated set E, we have |E| ≤ | ∨i∈Dn T −i(U)| and
so
Pn,δ (T , f ) ≤ | ∨i∈Dn T −i(U)|||eSnf ||∞
Pδ (T , f ) := lim supn→∞
log Pn,δ (T ,f )
.
nd
Note: For δ1 > δ2, Pδ2 (T , f ) ≥ Pδ1 (T , f ); δ2-vision is better than
δ1-vision.
Topological Pressure: P (T , f ) := limδ→0 Pδ (T , f ).
Meaning of pressure?
Can recover Topological entropy: h(T ) := P (T , 0)
Variational principle:
Z
P (T , f ) = sup{hµ(T ) +
8
f dµ : µ ∈ M(T )}
In particular,
h(T ) = sup{hµ(T ) : µ ∈ M(T )}
where hµ(T ) is the ergodic theory entropy of T as an MP action on
(M, A, µ).
Proof of variational principle is a delicate analysis of the interplay
between measure theory and topology.
d
Defn: Let G = Z+
d or Z . A Cts G-action is expansive with
expansive constant δ > 0 if for all x, y ∈ M , x 6= y, there exists
n ∈ G s.t. d(T n(x), T n(y)) > δ.
Note: expansiveness is a top. conjugacy invariant, though expansive constant may change.
Example: Shift action on a (two-sided) shift space.
Proof: Here G = Zd.
By defn. of the metric on a shift space M , d(x, y) > 1 iff x0 6=
y0. Thus, if x 6= y, there exists n ∈ Zd s.t. xn 6= yn and so
d(T n(x), T n(y)) > 1.
So, δ = 1 is an expansive constant. Note: an expansive Zd action need not be expansive as a Zd+ action.
Example: the full two-sided shift is expansive as a Z action but
not as a Z+ action:
x := . . . 000 . . . 000.000 . . . , y := . . . 110 . . . 000.000 . . .
Theorem: If T is expansive with expansive constant δ > 0, then
P (T , f ) = Pδ (T , f ).
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