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

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift 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

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Reminder: iceberg model I M of Burton-Steif: d = 2, A = {− M , . . . , − 1 , 1 , . . . , M } , F = { ij , i : ij < − 1 } . j Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Reminder: iceberg model I M of Burton-Steif: d = 2, A = {− M , . . . , − 1 , 1 , . . . , M } , F = { ij , i : ij < − 1 } . j Only allowed adjacent integers with opposite signs are ± 1. Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Reminder: iceberg model I M of Burton-Steif: d = 2, A = {− M , . . . , − 1 , 1 , . . . , M } , F = { ij , i : ij < − 1 } . j Only allowed adjacent integers with opposite signs are ± 1. I M is strongly irreducible, but we claimed it’s not weakly spatial mixing (has multiple MMEs) Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Reminder: iceberg model I M of Burton-Steif: d = 2, A = {− M , . . . , − 1 , 1 , . . . , M } , F = { ij , i : ij < − 1 } . j Only allowed adjacent integers with opposite signs are ± 1. I M is strongly irreducible, but we claimed it’s not weakly spatial mixing (has multiple MMEs) Suffices to show that for some ǫ > 0 and for any n , µ ( x (0) > 0 | x ( ∂ {− n , . . . , n } d ) = − M ) < 1 2 − ǫ Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Reminder: iceberg model I M of Burton-Steif: d = 2, A = {− M , . . . , − 1 , 1 , . . . , M } , F = { ij , i : ij < − 1 } . j Only allowed adjacent integers with opposite signs are ± 1. I M is strongly irreducible, but we claimed it’s not weakly spatial mixing (has multiple MMEs) Suffices to show that for some ǫ > 0 and for any n , µ ( x (0) > 0 | x ( ∂ {− n , . . . , n } d ) = − M ) < 1 2 − ǫ Then by symmetry, µ ( x (0) < 0 | x ( ∂ {− n , . . . , n } d ) = M ) < 1 2 − ǫ Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Reminder: iceberg model I M of Burton-Steif: d = 2, A = {− M , . . . , − 1 , 1 , . . . , M } , F = { ij , i : ij < − 1 } . j Only allowed adjacent integers with opposite signs are ± 1. I M is strongly irreducible, but we claimed it’s not weakly spatial mixing (has multiple MMEs) Suffices to show that for some ǫ > 0 and for any n , µ ( x (0) > 0 | x ( ∂ {− n , . . . , n } d ) = − M ) < 1 2 − ǫ Then by symmetry, µ ( x (0) < 0 | x ( ∂ {− n , . . . , n } d ) = M ) < 1 2 − ǫ µ ( x (0) > 0 | x ( ∂ {− n , . . . , n } d ) = M ) > 1 2 + ǫ Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Reminder: iceberg model I M of Burton-Steif: d = 2, A = {− M , . . . , − 1 , 1 , . . . , M } , F = { ij , i : ij < − 1 } . j Only allowed adjacent integers with opposite signs are ± 1. I M is strongly irreducible, but we claimed it’s not weakly spatial mixing (has multiple MMEs) Suffices to show that for some ǫ > 0 and for any n , µ ( x (0) > 0 | x ( ∂ {− n , . . . , n } d ) = − M ) < 1 2 − ǫ Then by symmetry, µ ( x (0) < 0 | x ( ∂ {− n , . . . , n } d ) = M ) < 1 2 − ǫ µ ( x (0) > 0 | x ( ∂ {− n , . . . , n } d ) = M ) > 1 2 + ǫ M Then µ + � = µ − ; they give different values to set � [ x (0) = i ] i =1 Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model For some n , suppose that x (0) > 0 and x ( ∂ {− n , . . . , n } d ) = − M Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model For some n , suppose that x (0) > 0 and x ( ∂ {− n , . . . , n } d ) = − M Then there is a maximal connected component of sites containing 0 where x takes positive values Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model For some n , suppose that x (0) > 0 and x ( ∂ {− n , . . . , n } d ) = − M Then there is a maximal connected component of sites containing 0 where x takes positive values Call this component the cluster and its boundary the shoreline for x Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model For some n , suppose that x (0) > 0 and x ( ∂ {− n , . . . , n } d ) = − M Then there is a maximal connected component of sites containing 0 where x takes positive values Call this component the cluster and its boundary the shoreline for x For a fixed shoreline S , define event E S = { x ∈ A {− n ,..., n } d : S is the shoreline for x } Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model For some n , suppose that x (0) > 0 and x ( ∂ {− n , . . . , n } d ) = − M Then there is a maximal connected component of sites containing 0 where x takes positive values Call this component the cluster and its boundary the shoreline for x For a fixed shoreline S , define event E S = { x ∈ A {− n ,..., n } d : S is the shoreline for x } [ x (0) > 0] = � S E S , and the union is disjoint Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model If x ∈ E S , then x positive on C and negative on S := ∂ ( C c ) Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model If x ∈ E S , then x positive on C and negative on S := ∂ ( C c ) In fact then x is − 1 on S and +1 on S Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model If x ∈ E S , then x positive on C and negative on S := ∂ ( C c ) In fact then x is − 1 on S and +1 on S Can “flip” positives in C to negatives, AND change values on S to ANY negative values Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model If x ∈ E S , then x positive on C and negative on S := ∂ ( C c ) In fact then x is − 1 on S and +1 on S Can “flip” positives in C to negatives, AND change values on S to ANY negative values Therefore, µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) < M −| S | Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Therefore, µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) < M −| S | Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Therefore, µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) < M −| S | µ ([ x 0 > 0] | x ( ∂ {− n , . . . , n } d ) = − M ) = S µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) ≤ � S M −| S | � Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Therefore, µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) < M −| S | µ ([ x 0 > 0] | x ( ∂ {− n , . . . , n } d ) = − M ) = S µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) ≤ � S M −| S | � For any n , there are fewer than n 4 n possible S Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Therefore, µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) < M −| S | µ ([ x 0 > 0] | x ( ∂ {− n , . . . , n } d ) = − M ) = S µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) ≤ � S M −| S | � For any n , there are fewer than n 4 n possible S n 4 n M − n = � 4 M µ ([ x 0 > 0] | x ( ∂ {− n , . . . , n } d ) = − M ) < M 2 − 8 M +16 n Entropy and mixing for Z d SFTs Ronnie Pavlov

Non-uniqueness of MMEs for iceberg model Uniqueness of MME for hard square shift Iceberg model Therefore, µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) < M −| S | µ ([ x 0 > 0] | x ( ∂ {− n , . . . , n } d ) = − M ) = S µ ( E S | x ( ∂ {− n , . . . , n } d ) = − M ) ≤ � S M −| S | � For any n , there are fewer than n 4 n possible S n 4 n M − n = � 4 M µ ([ x 0 > 0] | x ( ∂ {− n , . . . , n } d ) = − M ) < M 2 − 8 M +16 n Can be made smaller than 1 2 for large M Entropy and mixing for Z d SFTs Ronnie Pavlov