[Paper Review] Phase transition in one-dimensional subshifts

Nicolai Haydn|arXiv (Cornell University)|Nov 27, 2011

Mathematical Dynamics and Fractals1 references3 citations

TL;DR

This paper constructs explicit one-dimensional mixing subshifts with positive topological entropy that admit multiple ergodic measures of maximal entropy and multiple mutually singular equilibrium states for Hölder continuous potentials. By embedding multiple full shifts separated by long zero-blocks, the authors demonstrate that even irreducible, mixing systems can fail to have unique equilibrium states, challenging the common assumption of uniqueness under regularity conditions.

ABSTRACT

In this note we give simple examples of a one-dimensional mixing subshift with positive topological entropy which have two distinct measures of maximal entropy. We also give examples of subshifts which have two mutually singular equilibrium states for Hölder continuous functions. We also indicate how the construction can be extended to yield examples with any number of measures of maximal entropy of equilibrium states.

Motivation & Objective

• To demonstrate that one-dimensional mixing subshifts with positive entropy can have multiple ergodic measures of maximal entropy, contradicting the common assumption of uniqueness.
• To construct examples with multiple mutually singular equilibrium states for Hölder continuous potentials, extending beyond the typical uniqueness results.
• To generalize the construction to yield systems with any finite or infinite number of distinct equilibrium states.
• To provide explicit, simple examples that illustrate phase transitions in symbolic dynamics without requiring complex or pathological potentials.

Proposed method

• Define a subshift Σ over an alphabet 𝒜 = {0} ∪ 𝒢 ∪ 𝒴, where 𝒢 and 𝒴 are disjoint sets of green and yellow symbols, each forming full shifts.
• Introduce a transition rule requiring a separating block of zeros of length ≥ τ(|α| + |β|) between words of different colors, with τ > 0 ensuring mixing.
• Construct a potential function g on Σ by extending a Hölder continuous potential f from a single full shift Σ𝒢 to all embedded full shifts Σj, with g set to sup f on zero-words.
• Use the variational principle to show that the pressure P(g) is realized by invariant measures supported on individual embedded full shifts Σj.
• Apply entropy and pressure estimates to show that contributions from multi-colored words are negligible when τ is large enough, ensuring that equilibrium states concentrate on individual color shifts.
• Prove that when P𝒢(f) ≥ sup f + log 2 and τ is sufficiently large, each embedded shift supports a distinct, mutually singular equilibrium state for g.

Experimental results

Research questions

• RQ1 Can a one-dimensional mixing subshift with positive topological entropy have more than one ergodic measure of maximal entropy?
• RQ2 Do Hölder continuous potentials on mixing subshifts necessarily have unique equilibrium states?
• RQ3 Can the number of distinct equilibrium states be controlled and extended to infinity in such systems?
• RQ4 What conditions on the transition rules (e.g., zero-block length τ) ensure that equilibrium states are mutually singular and supported on embedded subshifts?
• RQ5 How does the pressure of the extended potential compare to the pressure on individual embedded subshifts under different τ and ν values?

Key findings

• For τ ≥ log 3 / log ν, the topological entropy of the constructed subshift is log ν, equal to that of each embedded full shift.
• When τ ≥ log 3ν / (P𝒢(f) − sup f) − 1, the system admits two mutually singular equilibrium states for the extended potential g.
• For L ≥ 2 embedded full shifts, if τ ≥ log(5ν) / (P𝒢(f) − sup f) − 1, the system supports L mutually singular equilibrium states.
• With infinitely many embedded full shifts and ν ≥ 3, the system admits infinitely many mutually singular equilibrium states when τ ≥ log 5ν / (P₁(f) − sup f) − 1.
• The pressure P(g) of the extended potential g equals P𝒢(f), confirming that each embedded shift contributes a distinct equilibrium state.
• The construction ensures topological mixing for all τ > 0, showing that non-uniqueness occurs even in systems with strong mixing properties.