Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Localized Pressure and Equilibrium States Tamara Kucherenko, CCNY (joint work with Christian Wolf) October 18, 2015 Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Topological Entropy Let ( X, d ) be a compact metric space and f : X → X be a continuous map. The topological entropy h top ( f ) measures the exponential growth rate of distinguishable orbits as we increase the iteration of the map. To distinguish the orbits we define a new metric on X . k =0 ,...,n − 1 d ( f k ( x ) , f k ( y )) The Bowen metric d n ( x, y ) = max F ⊂ X is ( n, ǫ ) -separated if d n ( x, y ) ≥ ǫ for any x, y ∈ F, x � = y N ( n, ǫ ) = sup { card( F ) : F is ( n, ǫ ) -separated } . 1 Topological entropy h top ( f ) = lim ǫ → 0 lim sup n log N ( n, ǫ ) n →∞ Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Measure-Theoretic Entropy Denote by M the set of all f -invariant probability measures. Let µ ∈ M . The measure-theoretic entropy h µ ( f ) measures the exponential growth rate of statistically significant orbits with respect to µ . The Variational Principle h top ( f ) = sup { h µ ( f ) : µ ∈ M} Measures where the supremum is attained are called measures of maximal entropy. Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Topological Pressure 1 Topological entropy h top ( f ) = lim ǫ → 0 lim sup n log N ( n, ǫ ) n →∞ N ( n, ǫ ) = sup { card( F ) : F is ( n, ǫ ) -separated } = sup { � 1 : F is ( n, ǫ ) -separated } . x ∈ F e S n ϕ ( x ) : F is ( n, ǫ ) -separated } N ϕ ( n, ǫ ) = sup { � x ∈ F n − 1 ϕ ( f k ( x )) . � Here ϕ : X → R is a continuous potential and S n ϕ ( x ) = k =0 1 Topological pressure P top ( ϕ ) = lim ǫ → 0 lim sup n log N ϕ ( n, ǫ ) n →∞ � The Variational Principle P top ( ϕ ) = sup { h µ + X ϕdµ : µ ∈ M} Measures where the supremum is attained are called equilibrium states. Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Localized Topological Pressure n Let Φ : X → R m be a continuous observable. Consider 1 n S n Φ( x ) = 1 Φ( f k ( x )) . � n k =1 Fix w ∈ R m . We are only interested in points x ∈ X for which 1 n S n Φ( x ) is close to w . Let X = T m be an m -dimensional torus, Intuition: f : T m → T m be continuous, F : R m → R m be any lifting of f , Φ : T m → R m be the displacement function, w Φ( x ) = F ( x ) − x n Φ( f k ( x )) = F n ( x ) − x Then 1 � is the n n k =1 average displacement of a point x ∈ R m . We are interested in points on the torus which on average move in the direction of w . Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Localized Topological Pressure n Let Φ : X → R m be a continuous observable. Consider 1 n S n Φ( x ) = 1 Φ( f k ( x )) . � n k =1 e S n ϕ ( x ) : F is ( n, ǫ ) -separated and | 1 N ϕ ( n, ǫ, w, r ) = sup { � n S n Φ( x ) − w | < r } x ∈ F 1 Localized topological pressure at w P top ( ϕ, Φ , w ) = lim r → 0 lim ǫ → 0 lim sup n log N ϕ ( n, ǫ, w, r ) n →∞ This definition is only meaningful if every neighbourhood of w contains 1 n S n Φ( x ) for arbitrarily large n . The set of such points is called the pointwice rotation set of Φ w ∈ R m : ∀ r > 0 ∀ N ∃ n ≥ N ∃ x ∈ X : | 1 � � Rot Pt (Φ) = n S n Φ( x ) − w | < r � If w ∈ Rot Pt (Φ) then there exists µ ∈ M such that w = Φ dµ . Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Localized Variational Principle � Denote M Φ ( w ) = { µ ∈ M : w = Φ dµ } the rotation class of w . � � � Question: Is it true that P top ( ϕ, Φ , w ) = sup h µ + ϕdµ : µ ∈ M Φ ( w ) Answer: No (in general) , but it is still true for a wide variety of dynamical systems. Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States � � � Example 1. P top ( ϕ, Φ , w ) < sup h µ + ϕdµ : µ ∈ M Φ ( w ) Take ϕ ≡ 0 . Idea: If X 1 and X 2 are f -invariant subsets then h top ( f | X 1 ∪ X 2 ) = max { h top ( f | X 1 ) , h top ( f | X 2 ) } , h µ ( f | X 1 ∪ X 2 ) = h µ ( f | X 1 ) µ ( X 1 ) + h µ ( f | X 2 ) µ ( X 2 ) w 1 w 2 w 3 X 1 X 2 X 3 Let X = X 1 ∪ X 2 ∪ X 3 ⊂ R . Let Φ = id X .Consider a continuous f : X → X such that X 1 , X 2 and X 3 are f -invariant and h top ( f | X 1 ) = h top ( f | X 3 ) > h top ( f | X 2 ) Let µ 1 and µ 3 be the entropy maximizing measures for f | X 1 and f | X 3 . � � Then w 1 = Φ dµ 1 ∈ X 1 and w 3 = Φ dµ 3 ∈ X 3 . Pick any w 2 ∈ X 2 . � For some α, β ∈ (0 , 1) , α + β = 1 we have w 2 = αw 1 + βw 3 = Φ d ( αµ 1 + βµ 3 ) . � � � Thus sup h µ + ϕdµ : µ ∈ M Φ ( w 2 ) = h αµ 1 + βµ 3 = h top ( f | X 1 ) . However, P top (0 , Φ , w 2 ) ≤ h top ( f | X 2 ) < h top ( f | X 1 ) . Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States � � � Example 2. P top ( ϕ, Φ , w ) > sup h µ + ϕdµ : µ ∈ M Φ ( w ) Note that P top ( ϕ, Φ , w ) depends on the behavior of the system in small neighbourhoods of w , but the supremum is taken over measures whose integrals are exactly w . Take ϕ ≡ 0 . Consider a sequence of disjoint compact intervals X n ⊂ R , X n → { 0 } . Let X = ∪ ∞ n =1 X n ∪ { 0 } and Φ = id X . Define f | X n to be topologically conjugate to the logistic map g ( x ) = 4 x (1 − x ) on [0 , 1] and f (0) = 0 . Then h top ( f | X n ) = log 2 . Let µ n be the ergodic entropy maximising measure for f | X n . Since in any neighbourhood of w = 0 there is µ n with h µ n = log 2 , we have P top (0 , id X , 0) = log 2 . However, w = 0 is a fixed point of f and extreme point of X 0 X 4 X 3 X 2 X 1 The only measure in M Φ (0) is the point-mass measure at 0. � � � We obtain sup h µ + ϕdµ : µ ∈ M Φ (0) = 0 Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Localized Variational Principle � � � To prove P top ( ϕ, Φ , w ) = sup h µ + ϕdµ : µ ∈ M Φ ( w ) we need additional assumptions. � � � � Example 2: We have w n = Φ dµ n → 0 , sup h µ + ϕdµ : µ ∈ M Φ ( w n ) = log 2 , � � � but sup h µ + ϕdµ : µ ∈ M Φ (0) = 0 � � � Assumption 1: The function v �→ sup h µ + ϕdµ : µ ∈ M Φ ( v ) is continuous at w . This condition holds if the entropy map µ �→ h µ is upper semi-continuous. � � � Example 1: sup h µ + ϕdµ : µ ∈ M Φ ( w 2 ) is not attained at an ergodic measure. � If µ is ergodic and dµ = w 2 then µ ( X 1 ) = µ ( X 3 ) = 0 and h µ ≤ h top ( f | X 2 ) < h αµ 1 + βµ 2 . � � � Assumption 2: sup h µ + ϕdµ : µ ∈ M Φ ( w ) is approximated by ergodic measures. Precisely, there is ( µ n ) -ergodic, such that � � � � � Φ dµ n → w and h µ n + ϕdµ n → sup h µ + ϕdµ : µ ∈ M Φ ( w ) Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
Classical Entropies and Topological Pressure Localized Entropies and Topological Pressure Localized Equilibrium States Localized Variational Principle Theorem Let f : X → X be a continuous map on a compact metric space X . Let Φ : X → R m and ϕ : X → R be continuous and let w ∈ Rot Pt (Φ) such that 1 v �→ sup � � � h µ ( f ) + X ϕ dµ : µ ∈ M Φ ( v ) is continuous at w ; 2 sup � � � h µ ( f ) + X ϕ dµ : µ ∈ M Φ ( w ) is approximated by ergodic measures. Then � � � � P top ( ϕ, Φ , w ) = sup h µ ( f ) + ϕ dµ : µ ∈ M and Φ dµ = w X X This theorem holds for a wide variety of systems and potentials such as expansive homeomorphisms with specification, which include topological mixing two-sided subshifts of finite type as well as diffeomorphisms with a locally maximal topological mixing hyperbolic set. Tamara Kucherenko, CCNY (joint work with Christian Wolf) Localized Pressure and Equilibrium States
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