Entropy and mixing for Z d SFTs Ronnie Pavlov University of Denver - PowerPoint PPT Presentation

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Entropy and mixing for Z d SFTs Ronnie Pavlov University of Denver www.math.du.edu/ ∼ rpavlov 1st School on Dynamical Systems and Computation (DySyCo) CMM, Santiago, Chile Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts All measures we consider will be shift-invariant probability Borel measures on A Z d Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts All measures we consider will be shift-invariant probability Borel measures on A Z d Any such µ is determined by values on cylinder sets [ w ] Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts All measures we consider will be shift-invariant probability Borel measures on A Z d Any such µ is determined by values on cylinder sets [ w ] To any such measure is assigned measure-theoretic entropy: Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts All measures we consider will be shift-invariant probability Borel measures on A Z d Any such µ is determined by values on cylinder sets [ w ] To any such measure is assigned measure-theoretic entropy: − 1 � h ( µ ) = lim µ ( w ) log µ ( w ) n d n →∞ w ∈ L ( X ) ∩ A { 1 ,..., n } d Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts All measures we consider will be shift-invariant probability Borel measures on A Z d Any such µ is determined by values on cylinder sets [ w ] To any such measure is assigned measure-theoretic entropy: − 1 � h ( µ ) = lim µ ( w ) log µ ( w ) n d n →∞ w ∈ L ( X ) ∩ A { 1 ,..., n } d Note: if µ uniformly distributed over patterns in L ( X ) ∩ A { 1 ,..., n } d : Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts All measures we consider will be shift-invariant probability Borel measures on A Z d Any such µ is determined by values on cylinder sets [ w ] To any such measure is assigned measure-theoretic entropy: − 1 � h ( µ ) = lim µ ( w ) log µ ( w ) n d n →∞ w ∈ L ( X ) ∩ A { 1 ,..., n } d Note: if µ uniformly distributed over patterns in L ( X ) ∩ A { 1 ,..., n } d : � � − 1 1 h ( µ ) = lim n d log | L ( X ) ∩ A { 1 ,..., n } d | n →∞ Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts All measures we consider will be shift-invariant probability Borel measures on A Z d Any such µ is determined by values on cylinder sets [ w ] To any such measure is assigned measure-theoretic entropy: − 1 � h ( µ ) = lim µ ( w ) log µ ( w ) n d n →∞ w ∈ L ( X ) ∩ A { 1 ,..., n } d Note: if µ uniformly distributed over patterns in L ( X ) ∩ A { 1 ,..., n } d : � � − 1 1 h ( µ ) = lim n d log = h ( X ) | L ( X ) ∩ A { 1 ,..., n } d | n →∞ Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts There usually does not exist such a uniformly distributed measure (PROVE) Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts There usually does not exist such a uniformly distributed measure (PROVE) Nevertheless, we have the classical Variational Principle: Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts There usually does not exist such a uniformly distributed measure (PROVE) Nevertheless, we have the classical Variational Principle: Theorem: (Variational Principle) sup h ( µ ) = h ( X ) (over µ with support in X ), and the sup is achieved Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts There usually does not exist such a uniformly distributed measure (PROVE) Nevertheless, we have the classical Variational Principle: Theorem: (Variational Principle) sup h ( µ ) = h ( X ) (over µ with support in X ), and the sup is achieved Measures µ for which h ( µ ) = h ( X ) are called measures of maximal entropy Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures on Z d subshifts There usually does not exist such a uniformly distributed measure (PROVE) Nevertheless, we have the classical Variational Principle: Theorem: (Variational Principle) sup h ( µ ) = h ( X ) (over µ with support in X ), and the sup is achieved Measures µ for which h ( µ ) = h ( X ) are called measures of maximal entropy Such measures are useful for studying topological entropy, since they allow the additional strength of measure theory to be brought to bear Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Any measure of maximal entropy µ for an SFT X has an interesting property Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Any measure of maximal entropy µ for an SFT X has an interesting property Theorem: (Burton-Steif/Lanford-Ruelle) For any such µ , any finite S and T ⊃ ∂ S for which S ∩ T = ∅ , and for any δ ∈ L T ( X ), µ ( x | S : x | T = δ ) is uniform over all x ∈ L S ( X ) for which x δ ∈ L ( X ). Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Any measure of maximal entropy µ for an SFT X has an interesting property Theorem: (Burton-Steif/Lanford-Ruelle) For any such µ , any finite S and T ⊃ ∂ S for which S ∩ T = ∅ , and for any δ ∈ L T ( X ), µ ( x | S : x | T = δ ) is uniform over all x ∈ L S ( X ) for which x δ ∈ L ( X ). Call such measures uniform Gibbs measures . Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Any measure of maximal entropy µ for an SFT X has an interesting property Theorem: (Burton-Steif/Lanford-Ruelle) For any such µ , any finite S and T ⊃ ∂ S for which S ∩ T = ∅ , and for any δ ∈ L T ( X ), µ ( x | S : x | T = δ ) is uniform over all x ∈ L S ( X ) for which x δ ∈ L ( X ). Call such measures uniform Gibbs measures . Example: H the Z 2 hard square shift: if µ is uniform Gibbs, Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Any measure of maximal entropy µ for an SFT X has an interesting property Theorem: (Burton-Steif/Lanford-Ruelle) For any such µ , any finite S and T ⊃ ∂ S for which S ∩ T = ∅ , and for any δ ∈ L T ( X ), µ ( x | S : x | T = δ ) is uniform over all x ∈ L S ( X ) for which x δ ∈ L ( X ). Call such measures uniform Gibbs measures . Example: H the Z 2 hard square shift: if µ is uniform Gibbs, 1 0 1 0 0 0 , fillings 0 0 0 0 , 0 0 1 0 , 1 0 conditioned on 0 0 equally probable. 0 1 1 0 0 0 Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Any measure of maximal entropy µ for an SFT X has an interesting property Theorem: (Burton-Steif/Lanford-Ruelle) For any such µ , any finite S and T ⊃ ∂ S for which S ∩ T = ∅ , and for any δ ∈ L T ( X ), µ ( x | S : x | T = δ ) is uniform over all x ∈ L S ( X ) for which x δ ∈ L ( X ). Call such measures uniform Gibbs measures . Example: H the Z 2 hard square shift: if µ is uniform Gibbs, 1 0 1 0 0 0 , fillings 0 0 0 0 , 0 0 1 0 , 1 0 conditioned on 0 0 equally probable. 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 Same conditional probabilities if changed to 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Uniform Gibbs measures are “as uniform as possible” measures on an SFT Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Uniform Gibbs measures are “as uniform as possible” measures on an SFT One way to create a uniform Gibbs measure is as weak limit of uniform measures on finite sets given boundary conditions (PROVE) Entropy and mixing for Z d SFTs Ronnie Pavlov

Measure-theoretic entropy Measures of maximal entropy Mixing for MMEs Measures of maximal entropy Uniform Gibbs measures are “as uniform as possible” measures on an SFT One way to create a uniform Gibbs measure is as weak limit of uniform measures on finite sets given boundary conditions (PROVE) An SFT can have multiple uniform Gibbs measures Entropy and mixing for Z d SFTs Ronnie Pavlov