MAGIC: Ergodic Theory Lecture 8 - Topological entropy Charles Walkden March 13th 2013 In the context of a continuous transformation of a compact metric space we study how hµ(T ) depends on µ. We also relate entropy to another important quantity: topological entropy. Throughout: metric entropy = measure-theoretic entropy = hµ(T ). Introduction Let T be an m.p.t. of a prob. space (X ; B; µ). Last time we defined the entropy hµ(T ). In this lecture we recap some basic facts about entropy. We also relate entropy to another important quantity: topological entropy. Throughout: metric entropy = measure-theoretic entropy = hµ(T ). Introduction Let T be an m.p.t. of a prob. space (X ; B; µ). Last time we defined the entropy hµ(T ). In this lecture we recap some basic facts about entropy. In the context of a continuous transformation of a compact metric space we study how hµ(T ) depends on µ. Throughout: metric entropy = measure-theoretic entropy = hµ(T ). Introduction Let T be an m.p.t. of a prob. space (X ; B; µ). Last time we defined the entropy hµ(T ). In this lecture we recap some basic facts about entropy. In the context of a continuous transformation of a compact metric space we study how hµ(T ) depends on µ. We also relate entropy to another important quantity: topological entropy. Introduction Let T be an m.p.t. of a prob. space (X ; B; µ). Last time we defined the entropy hµ(T ). In this lecture we recap some basic facts about entropy. In the context of a continuous transformation of a compact metric space we study how hµ(T ) depends on µ. We also relate entropy to another important quantity: topological entropy. Throughout: metric entropy = measure-theoretic entropy = hµ(T ). Let ζ = fAj g be a finite partition of a prob. space (X ; B; µ). Define the entropy of ζ X Hµ(ζ) = − µ(A) log µ(A): A2ζ If ζ; η are partitions then ζ _ η = fA \ B j A 2 ζ; B 2 ηg. If T : X ! X is measurable then T −1ζ = fT −1A j A 2 ζg. The entropy of T relative to ζ is 0n−1 1 1 _ −j hµ(T ; ζ) = lim Hµ @ T αA : n!1 n j=0 The entropy of T is hµ(T ) = supfhµ(T ; ζ) j ζ a finite partitiong: Recap on entropy Define the entropy of ζ X Hµ(ζ) = − µ(A) log µ(A): A2ζ If ζ; η are partitions then ζ _ η = fA \ B j A 2 ζ; B 2 ηg. If T : X ! X is measurable then T −1ζ = fT −1A j A 2 ζg. The entropy of T relative to ζ is 0n−1 1 1 _ −j hµ(T ; ζ) = lim Hµ @ T αA : n!1 n j=0 The entropy of T is hµ(T ) = supfhµ(T ; ζ) j ζ a finite partitiong: Recap on entropy Let ζ = fAj g be a finite partition of a prob. space (X ; B; µ). If ζ; η are partitions then ζ _ η = fA \ B j A 2 ζ; B 2 ηg. If T : X ! X is measurable then T −1ζ = fT −1A j A 2 ζg. The entropy of T relative to ζ is 0n−1 1 1 _ −j hµ(T ; ζ) = lim Hµ @ T αA : n!1 n j=0 The entropy of T is hµ(T ) = supfhµ(T ; ζ) j ζ a finite partitiong: Recap on entropy Let ζ = fAj g be a finite partition of a prob. space (X ; B; µ). Define the entropy of ζ X Hµ(ζ) = − µ(A) log µ(A): A2ζ If T : X ! X is measurable then T −1ζ = fT −1A j A 2 ζg. The entropy of T relative to ζ is 0n−1 1 1 _ −j hµ(T ; ζ) = lim Hµ @ T αA : n!1 n j=0 The entropy of T is hµ(T ) = supfhµ(T ; ζ) j ζ a finite partitiong: Recap on entropy Let ζ = fAj g be a finite partition of a prob. space (X ; B; µ). Define the entropy of ζ X Hµ(ζ) = − µ(A) log µ(A): A2ζ If ζ; η are partitions then ζ _ η = fA \ B j A 2 ζ; B 2 ηg. The entropy of T relative to ζ is 0n−1 1 1 _ −j hµ(T ; ζ) = lim Hµ @ T αA : n!1 n j=0 The entropy of T is hµ(T ) = supfhµ(T ; ζ) j ζ a finite partitiong: Recap on entropy Let ζ = fAj g be a finite partition of a prob. space (X ; B; µ). Define the entropy of ζ X Hµ(ζ) = − µ(A) log µ(A): A2ζ If ζ; η are partitions then ζ _ η = fA \ B j A 2 ζ; B 2 ηg. If T : X ! X is measurable then T −1ζ = fT −1A j A 2 ζg. The entropy of T is hµ(T ) = supfhµ(T ; ζ) j ζ a finite partitiong: Recap on entropy Let ζ = fAj g be a finite partition of a prob. space (X ; B; µ). Define the entropy of ζ X Hµ(ζ) = − µ(A) log µ(A): A2ζ If ζ; η are partitions then ζ _ η = fA \ B j A 2 ζ; B 2 ηg. If T : X ! X is measurable then T −1ζ = fT −1A j A 2 ζg. The entropy of T relative to ζ is 0n−1 1 1 _ −j hµ(T ; ζ) = lim Hµ @ T αA : n!1 n j=0 Recap on entropy Let ζ = fAj g be a finite partition of a prob. space (X ; B; µ). Define the entropy of ζ X Hµ(ζ) = − µ(A) log µ(A): A2ζ If ζ; η are partitions then ζ _ η = fA \ B j A 2 ζ; B 2 ηg. If T : X ! X is measurable then T −1ζ = fT −1A j A 2 ζg. The entropy of T relative to ζ is 0n−1 1 1 _ −j hµ(T ; ζ) = lim Hµ @ T αA : n!1 n j=0 The entropy of T is hµ(T ) = supfhµ(T ; ζ) j ζ a finite partitiong: Wn −j A finite partition ζ is a generator if j=−n T ζ %B. (Equiv. Wn −j j=−n T ζ separates µ-a.e. pair of points.) Theorem (Sinai) Suppose T is an invertible m.p.t. and ζ is a generator. Then hµ(T ) = hµ(T ; ζ): Let σ be the full k-shift with the Bernoulli (p1;:::; pk )-measure µ. Then ζ = f[1];:::; [k]g is a generator. k X hµ(σ) = hµ(σ; ζ) = − pj log pj : j=1 Sinai's theorem Theorem (Sinai) Suppose T is an invertible m.p.t. and ζ is a generator. Then hµ(T ) = hµ(T ; ζ): Let σ be the full k-shift with the Bernoulli (p1;:::; pk )-measure µ. Then ζ = f[1];:::; [k]g is a generator. k X hµ(σ) = hµ(σ; ζ) = − pj log pj : j=1 Sinai's theorem Wn −j A finite partition ζ is a generator if j=−n T ζ %B. (Equiv. Wn −j j=−n T ζ separates µ-a.e. pair of points.) Let σ be the full k-shift with the Bernoulli (p1;:::; pk )-measure µ. Then ζ = f[1];:::; [k]g is a generator. k X hµ(σ) = hµ(σ; ζ) = − pj log pj : j=1 Sinai's theorem Wn −j A finite partition ζ is a generator if j=−n T ζ %B. (Equiv. Wn −j j=−n T ζ separates µ-a.e. pair of points.) Theorem (Sinai) Suppose T is an invertible m.p.t. and ζ is a generator. Then hµ(T ) = hµ(T ; ζ): Sinai's theorem Wn −j A finite partition ζ is a generator if j=−n T ζ %B. (Equiv. Wn −j j=−n T ζ separates µ-a.e. pair of points.) Theorem (Sinai) Suppose T is an invertible m.p.t. and ζ is a generator. Then hµ(T ) = hµ(T ; ζ): Let σ be the full k-shift with the Bernoulli (p1;:::; pk )-measure µ. Then ζ = f[1];:::; [k]g is a generator. k X hµ(σ) = hµ(σ; ζ) = − pj log pj : j=1 Let (X ; B) be a compact metric space with the Borel σ-algebra. Let T : X ! X be continuous. Let M(X ) = fall Borel probability measuresg. Let M(X ; T ) = f all T -invariant Borel probability measuresg. ∗ A sequence µn 2 M(X ) weak -converges to µ (µn * µ) if Z Z f dµn ! f dµ 8f 2 C(X ; R): Q: How does the entropy hµ(T ) vary as a function of µ? The weak∗ topology ∗ A sequence µn 2 M(X ) weak -converges to µ (µn * µ) if Z Z f dµn ! f dµ 8f 2 C(X ; R): Q: How does the entropy hµ(T ) vary as a function of µ? The weak∗ topology Let (X ; B) be a compact metric space with the Borel σ-algebra. Let T : X ! X be continuous. Let M(X ) = fall Borel probability measuresg. Let M(X ; T ) = f all T -invariant Borel probability measuresg. Q: How does the entropy hµ(T ) vary as a function of µ? The weak∗ topology Let (X ; B) be a compact metric space with the Borel σ-algebra. Let T : X ! X be continuous. Let M(X ) = fall Borel probability measuresg. Let M(X ; T ) = f all T -invariant Borel probability measuresg. ∗ A sequence µn 2 M(X ) weak -converges to µ (µn * µ) if Z Z f dµn ! f dµ 8f 2 C(X ; R): The weak∗ topology Let (X ; B) be a compact metric space with the Borel σ-algebra.