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 Zd SFTs

Ronnie Pavlov

University of Denver www.math.du.edu/∼rpavlov

1st School on Dynamical Systems and Computation (DySyCo) CMM, Santiago, Chile

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Reminder: iceberg model IM of Burton-Steif: d = 2, A = {−M, . . . , −1, 1, . . . , M}, F = { ij , i

j

: ij < −1}.

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Reminder: iceberg model IM of Burton-Steif: d = 2, A = {−M, . . . , −1, 1, . . . , M}, F = { ij , i

j

: ij < −1}. Only allowed adjacent integers with opposite signs are ±1.

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Reminder: iceberg model IM of Burton-Steif: d = 2, A = {−M, . . . , −1, 1, . . . , M}, F = { ij , i

j

: ij < −1}. Only allowed adjacent integers with opposite signs are ±1. IM is strongly irreducible, but we claimed it’s not weakly spatial mixing (has multiple MMEs)

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Reminder: iceberg model IM of Burton-Steif: d = 2, A = {−M, . . . , −1, 1, . . . , M}, F = { ij , i

j

: ij < −1}. Only allowed adjacent integers with opposite signs are ±1. IM 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

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Reminder: iceberg model IM of Burton-Steif: d = 2, A = {−M, . . . , −1, 1, . . . , M}, F = { ij , i

j

: ij < −1}. Only allowed adjacent integers with opposite signs are ±1. IM 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

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Reminder: iceberg model IM of Burton-Steif: d = 2, A = {−M, . . . , −1, 1, . . . , M}, F = { ij , i

j

: ij < −1}. Only allowed adjacent integers with opposite signs are ±1. IM 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

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Reminder: iceberg model IM of Burton-Steif: d = 2, A = {−M, . . . , −1, 1, . . . , M}, F = { ij , i

j

: ij < −1}. Only allowed adjacent integers with opposite signs are ±1. IM 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

Then µ+ = µ−; they give different values to set

M

• i=1

[x(0) = i]

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Ronnie Pavlov Entropy and mixing for Zd SFTs

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 ES = {x ∈ A{−n,...,n}d : S is the shoreline for x}

Ronnie Pavlov Entropy and mixing for Zd SFTs

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 ES = {x ∈ A{−n,...,n}d : S is the shoreline for x} [x(0) > 0] =

S ES, and the union is disjoint

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

If x ∈ ES, then x positive on C and negative on S := ∂(C c)

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

If x ∈ ES, then x positive on C and negative on S := ∂(C c) In fact then x is −1 on S and +1 on S

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

If x ∈ ES, 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

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

If x ∈ ES, 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(ES | x(∂{−n, . . . , n}d) = −M) < M−|S|

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Therefore, µ(ES | x(∂{−n, . . . , n}d) = −M) < M−|S|

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Therefore, µ(ES | x(∂{−n, . . . , n}d) = −M) < M−|S| µ([x0 > 0] | x(∂{−n, . . . , n}d) = −M) =

• S µ(ES | x(∂{−n, . . . , n}d) = −M) ≤

S M−|S|

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Therefore, µ(ES | x(∂{−n, . . . , n}d) = −M) < M−|S| µ([x0 > 0] | x(∂{−n, . . . , n}d) = −M) =

• S µ(ES | x(∂{−n, . . . , n}d) = −M) ≤

S M−|S|

For any n, there are fewer than n4n possible S

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Therefore, µ(ES | x(∂{−n, . . . , n}d) = −M) < M−|S| µ([x0 > 0] | x(∂{−n, . . . , n}d) = −M) =

• S µ(ES | x(∂{−n, . . . , n}d) = −M) ≤

S M−|S|

For any n, there are fewer than n4n possible S µ([x0 > 0] | x(∂{−n, . . . , n}d) = −M) <

• n

n4nM−n =

4M M2−8M+16

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

Therefore, µ(ES | x(∂{−n, . . . , n}d) = −M) < M−|S| µ([x0 > 0] | x(∂{−n, . . . , n}d) = −M) =

• S µ(ES | x(∂{−n, . . . , n}d) = −M) ≤

S M−|S|

For any n, there are fewer than n4n possible S µ([x0 > 0] | x(∂{−n, . . . , n}d) = −M) <

• n

n4nM−n =

4M M2−8M+16

Can be made smaller than 1

2 for large M

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

So iceberg model has multiple MMEs for large M

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

So iceberg model has multiple MMEs for large M Even though it is as topologically mixing as possible (strong irreducibility), measure-theoretically it is badly nonmixing

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

So iceberg model has multiple MMEs for large M Even though it is as topologically mixing as possible (strong irreducibility), measure-theoretically it is badly nonmixing How does one prove uniqueness/spatial mixing for an SFT?

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Iceberg model

So iceberg model has multiple MMEs for large M Even though it is as topologically mixing as possible (strong irreducibility), measure-theoretically it is badly nonmixing How does one prove uniqueness/spatial mixing for an SFT? Will outline for hard square shift H by nice argument due to van den Berg and Steif

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Lemma: (van den Berg-Steif) If µ and µ′ are uniform Gibbs measures for an SFT X, and if (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0, then µ = µ′.

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Lemma: (van den Berg-Steif) If µ and µ′ are uniform Gibbs measures for an SFT X, and if (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0, then µ = µ′. Proof: Consider any cylinder set [w].

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Lemma: (van den Berg-Steif) If µ and µ′ are uniform Gibbs measures for an SFT X, and if (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0, then µ = µ′. Proof: Consider any cylinder set [w]. For (µ × µ′)-a.e. (x, y) ∈ [w] × [w]c, there is a path surrounding w

• n which x and y agree

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Lemma: (van den Berg-Steif) If µ and µ′ are uniform Gibbs measures for an SFT X, and if (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0, then µ = µ′. Proof: Consider any cylinder set [w]. For (µ × µ′)-a.e. (x, y) ∈ [w] × [w]c, there is a path surrounding w

• n which x and y agree

Define φ which “flips” x and y on the maximal cluster of disagreement intersecting w

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Lemma: (van den Berg-Steif) If µ and µ′ are uniform Gibbs measures for an SFT X, and if (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0, then µ = µ′. Proof: Consider any cylinder set [w]. For (µ × µ′)-a.e. (x, y) ∈ [w] × [w]c, there is a path surrounding w

• n which x and y agree

Define φ which “flips” x and y on the maximal cluster of disagreement intersecting w Because of uniform Gibbs property and the fact that φ “flips” patterns inside equal boundaries, φ preserves µ × µ′

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Lemma: (van den Berg-Steif) If µ and µ′ are uniform Gibbs measures for an SFT X, and if (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0, then µ = µ′. Proof: Consider any cylinder set [w]. For (µ × µ′)-a.e. (x, y) ∈ [w] × [w]c, there is a path surrounding w

• n which x and y agree

Define φ which “flips” x and y on the maximal cluster of disagreement intersecting w Because of uniform Gibbs property and the fact that φ “flips” patterns inside equal boundaries, φ preserves µ × µ′ φ([w] × [w]c) = [w]c × [w], so (µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w])

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

(µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w])

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

(µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w]) Then µ([w])

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

(µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w]) Then µ([w]) = (µ × µ′)([w] × X)

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

(µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w]) Then µ([w]) = (µ × µ′)([w] × X) = (µ × µ′)([w] × [w]) + (µ × µ′)([w] × [w]c)

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

(µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w]) Then µ([w]) = (µ × µ′)([w] × X) = (µ × µ′)([w] × [w]) + (µ × µ′)([w] × [w]c) = (µ × µ′)([w] × [w]) + (µ × µ)([w]c × [w])

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

(µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w]) Then µ([w]) = (µ × µ′)([w] × X) = (µ × µ′)([w] × [w]) + (µ × µ′)([w] × [w]c) = (µ × µ′)([w] × [w]) + (µ × µ)([w]c × [w]) = (µ × µ′)(X × [w])

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

(µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w]) Then µ([w]) = (µ × µ′)([w] × X) = (µ × µ′)([w] × [w]) + (µ × µ′)([w] × [w]c) = (µ × µ′)([w] × [w]) + (µ × µ)([w]c × [w]) = (µ × µ′)(X × [w]) = µ′([w])

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

(µ × µ′)([w] × [w]c) = (µ × µ′)([w]c × [w]) Then µ([w]) = (µ × µ′)([w] × X) = (µ × µ′)([w] × [w]) + (µ × µ′)([w] × [w]c) = (µ × µ′)([w] × [w]) + (µ × µ)([w]c × [w]) = (µ × µ′)(X × [w]) = µ′([w]) Since w was arbitrary, µ = µ′

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Theorem: (van den Berg-Steif) For the hard square shift H and any uniform Gibbs measures µ and µ′, (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0.

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Theorem: (van den Berg-Steif) For the hard square shift H and any uniform Gibbs measures µ and µ′, (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0. Proof: Note that for any δ, δ′ ∈ {0, 1}{±ei }, (µ × µ′)(x(0) = y(0) | x({±ei}) = δ, y({±ei}) = δ′) ≤ 1

2.

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Theorem: (van den Berg-Steif) For the hard square shift H and any uniform Gibbs measures µ and µ′, (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0. Proof: Note that for any δ, δ′ ∈ {0, 1}{±ei }, (µ × µ′)(x(0) = y(0) | x({±ei}) = δ, y({±ei}) = δ′) ≤ 1

2.

This means that for ANY event E in X × X generated by sites OTHER than 0, (µ × µ′)(x(0) = y(0) | E) ≤ 1

2.

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Theorem: (van den Berg-Steif) For the hard square shift H and any uniform Gibbs measures µ and µ′, (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0. Proof: Note that for any δ, δ′ ∈ {0, 1}{±ei }, (µ × µ′)(x(0) = y(0) | x({±ei}) = δ, y({±ei}) = δ′) ≤ 1

2.

This means that for ANY event E in X × X generated by sites OTHER than 0, (µ × µ′)(x(0) = y(0) | E) ≤ 1

2.

(µ × µ′)(x(0) = y(0) | E) can be written as a weighted average of (µ × µ′)(x(0) = y(0) | x({±ei}) = δ, y({±ei}) = δ′)

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Arbitrarily enumerate sites in Zd and define µ × µ′ by conditioning on these sites in order

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Arbitrarily enumerate sites in Zd and define µ × µ′ by conditioning on these sites in order At every step, regardless of what’s been conditioned on already, probability of assigning a disagreement is at most 1

2

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Arbitrarily enumerate sites in Zd and define µ × µ′ by conditioning on these sites in order At every step, regardless of what’s been conditioned on already, probability of assigning a disagreement is at most 1

2

Key fact: If we define the Bernoulli measure ν0.5 which assigns vertices to be “open” with prob. 0.5 and “closed” otherwise, then (µ × µ′)(x, y disagree on the set S) ≤ ν0.5(all sites in S are open)

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Arbitrarily enumerate sites in Zd and define µ × µ′ by conditioning on these sites in order At every step, regardless of what’s been conditioned on already, probability of assigning a disagreement is at most 1

2

Key fact: If we define the Bernoulli measure ν0.5 which assigns vertices to be “open” with prob. 0.5 and “closed” otherwise, then (µ × µ′)(x, y disagree on the set S) ≤ ν0.5(all sites in S are open) Relates problem of uniqueness of MME to percolation theory

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

In fact ν0.5(there is an infinite path of open vertices) = 0, but it’s hard to prove

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

In fact ν0.5(there is an infinite path of open vertices) = 0, but it’s hard to prove Will prove for ν0.2

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

In fact ν0.5(there is an infinite path of open vertices) = 0, but it’s hard to prove Will prove for ν0.2 For any n, there are fewer than 4n paths of n open sites containing 0, each has probability 0.2n

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

In fact ν0.5(there is an infinite path of open vertices) = 0, but it’s hard to prove Will prove for ν0.2 For any n, there are fewer than 4n paths of n open sites containing 0, each has probability 0.2n ν0.2(∃path of n open sites containing 0) ≤ 0.8n

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

In fact ν0.5(there is an infinite path of open vertices) = 0, but it’s hard to prove Will prove for ν0.2 For any n, there are fewer than 4n paths of n open sites containing 0, each has probability 0.2n ν0.2(∃path of n open sites containing 0) ≤ 0.8n ν0.2(there is an infinite path of open vertices) = 0

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

In fact ν0.5(there is an infinite path of open vertices) = 0, but it’s hard to prove Will prove for ν0.2 For any n, there are fewer than 4n paths of n open sites containing 0, each has probability 0.2n ν0.2(∃path of n open sites containing 0) ≤ 0.8n ν0.2(there is an infinite path of open vertices) = 0 Proof for ν0.5 requires much more subtle arguments, but still true, and still probabilities decay exponentially

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

In fact ν0.5(there is an infinite path of open vertices) = 0, but it’s hard to prove Will prove for ν0.2 For any n, there are fewer than 4n paths of n open sites containing 0, each has probability 0.2n ν0.2(∃path of n open sites containing 0) ≤ 0.8n ν0.2(there is an infinite path of open vertices) = 0 Proof for ν0.5 requires much more subtle arguments, but still true, and still probabilities decay exponentially (µ × µ′)(there exists an infinite path P on which x, y disagree) = 0, so by earlier Lemma, µ = µ′

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Uniqueness of MME is proved for H!

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Uniqueness of MME is proved for H! Exponential decay of percolation probabilities implies SSM of H with exponential rate

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Uniqueness of MME is proved for H! Exponential decay of percolation probabilities implies SSM of H with exponential rate As explained earlier, this implies computability of h(H) in polynomial time

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Uniqueness of MME is proved for H! Exponential decay of percolation probabilities implies SSM of H with exponential rate As explained earlier, this implies computability of h(H) in polynomial time More refined techniques imply more general results

Ronnie Pavlov Entropy and mixing for Zd SFTs

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

Hard square shift

Uniqueness of MME is proved for H! Exponential decay of percolation probabilities implies SSM of H with exponential rate As explained earlier, this implies computability of h(H) in polynomial time More refined techniques imply more general results Theorem: (P.) There exists ǫ such that for any nearest neighbor Z2 SFT X with h(X) > log |A| − ǫ, X has a unique MME, and h(X) is computable in polynomial time

Ronnie Pavlov Entropy and mixing for Zd SFTs