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

Lecture 24: Defn: Topological conjugacy: Given Z + d (resp, Zd ), actions T, S a topological conjugacy from T to S is a homeomorphism φ : M N s.t. φ T = S φ i.e., φ T n = S n φ for all n Z + d (resp, Zd ). (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 Z T n (V ). Observe: Ω = n Z T n (L R) because if T n x C, then T n+1 (x) B and so V. Define φ : Ω {R, L} Z : If x Ω, define φ(x) = y, where y n = L if T n (x) L y n = 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. Note: x n= T n (y n ) 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... y 0... y N {L, R} 2N+1, the set U(y N... y 0... y N ) := N n= N T n (y n ) is compact is nonempty has diameter < 2 N Idea of proof: Draw pictures of 3-blocks φ is 1-1: Let x Ω and y = φ(x). Then x n= T n (y n ), which can be written as a decreasing sequence of nonempty compact sets whose diameters 0. So, {x} = n= T n (y n ) So, x can be recovered from y = φ(x). φ is onto: Let y {L, R} Z. Then N n= N T n (y n ) is a nested decreasing sequence of compact 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 ) is small, then for a large N, y N... y 0... y N = y N... y 0... y N. Now, φ 1 (y) U(y N... y 0... y N ) and φ 1 (y ) U(y N... y 0... y N ), and so d(φ 1 (y), φ 1 (y )) < 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(x 0 x 1...) = e 2πi(.x 0x 1...) 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 Z d (F stands for forbidden list ). X F = {x F Zd : no translate of an element of F appears in x} Defn: A (two-sided) shift space is a subset M of Z d of the form M = X F. Prop: A subset M of Z d 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 Z d 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. Z d + action. Defn: A (two-sided) shift of finite type (SFT) is a shift space M = X F 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 D n or B n. But it will be convenient to consider other subsets. 4

Example: Ledrappier 3-dot: F = {w F {(0,1),(0,0),(1,0)} : x (0,1) + x (0,0) + x (1,0) 0 mod 2} Felipe will discuss this SFT in more detail. Defn: A nearest neighbour SFT (nn SFT) is an SFT M = X F where F consists of configurations on edges, i.e., on {(0, e i ) : 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 L, T R, T T, B B, L B, R 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 M C = {x F Z : C xi,x i+1 = 1 for all i Z} Consider corresponding graph of C: M C 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 M C 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 M C 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 M C. 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 Z d + 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 } Let h(t, α) := lim 1 n d log N( i D n T i (α)) The limit exists and equals the inf (by multidimensional Fekete Lemma). Define the topological entropy of T : h(t ) := sup α h(t, α) Entropy of a cts Z d action defined in the same way (as if it were a Z d + action). Will take an alternative approach and generalize topological entropy to topological pressure as follows. Defns: Let T be a cts Z d + 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 y, there exists i D n s.t.d(t i (x), T i (y)) > δ. 7

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. S n f(x) := i D n f(t i (x)) P n,δ (T, f) := sup{ x E es nf(x) : E is (n, δ) separated} Special case d = 1, f = 0: P n,δ (T, f) is the maximum size of an (n, δ)-separated set. so, P n,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: P n,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 P n,δ (T, f) i Dn T i (U) e S nf log P P δ (T, f) := lim sup n,δ (T,f) n. n d Note: For δ 1 > δ 2, P δ2 (T, f) P δ1 (T, f); δ 2 -vision is better than δ 1 -vision. Topological Pressure: P (T, f) := lim δ 0 P δ (T, f). Meaning of pressure? Can recover Topological entropy: h(t ) := P (T, 0) Variational principle: P (T, f) = sup{h µ (T ) + fdµ : µ M(T )} 8

In particular, h(t ) = sup{h µ (T ) : µ M(T )} where h µ (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. Defn: Let G = Z + d or Zd. A Cts G-action is expansive with expansive constant δ > 0 if for all x, y M, x 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 = Z d. By defn. of the metric on a shift space M, d(x, y) > 1 iff x 0 y 0. Thus, if x y, there exists n Z d s.t. x n y n and so d(t n (x), T n (y)) > 1. So, δ = 1 is an expansive constant. Note: an expansive Z d action need not be expansive as a Z d + 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) = P δ (T, f). 9