arXiv:2607.16994v1

Entropy Density of Uniquely Ergodic Measures for Full Shifts over Amenable Residually Finite Groups

Martha Łącka, Marcel Mroczek

math.DS37B1037A3537A1543A07

Paper metadata

Original submission: July 18, 2026
Claimed MSC: 37B10, 37A35, 37A15, 43A07
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗

Abstract

We study entropy density for full shifts over amenable residually finite groups. Entropy density means that every invariant measure can be approximated in the weak $^*$ topology, together with its entropy, by measures from a distinguished family. In symbolic dynamics it is known, by a result of Weiss, that uniquely ergodic measures are entropy dense among ergodic measures for the shift action of $\mathbb Z$ . We extend this result to full shifts over amenable residually finite groups: invariant measures supported on uniquely ergodic subsystems are entropy dense in the collection of ergodic invariant measures. This applies in particular to full shifts over finitely generated abelian groups. The proof uses Cortez--Petite Følner tilings and a block-replacement construction adapted to finite-index subgroups.

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Generated August 18, 2026

01Statements3 reported findingsCorrect

The approximation theorem and its density consequence are supported. Theorem 27 constructs uniquely ergodic full-shift measures approximating an arbitrary invariant measure for an amenable residually finite group, with matching finite-pattern statistics and entropy. Corollary 28 concludes that uniquely ergodic measures are entropy dense in the invariant-measure simplex in the topology specified by the paper.

Theorem 27Correct

Approximation by uniquely ergodic full-shift measures with entropy control

Pages 17–20 · Theorem 27 and Corollary 28 · arXiv:2607.16994v1

For a countable amenable residually finite group acting on a finite-alphabet full shift, every invariant measure is approximated in the stated weak/d-bar sense by uniquely ergodic measures constructed from periodic tilings. The approximants' entropies converge to the target entropy rather than merely satisfying a one-sided inequality. The finite-index subgroup chain and Følner fundamental domains required by residual finiteness are chosen once in the construction. The result covers nonergodic target measures because the jigsaw block frequencies encode their full finite-pattern law.

Corollary 28Correct

Entropy density of uniquely ergodic measures

Pages 19–20 · Corollary 28 · arXiv:2607.16994v1

Given an invariant measure, Theorem 27 supplies uniquely ergodic measures converging weakly and with entropy converging to its entropy. This is exactly entropy density: every neighbourhood of the measure and entropy interval contains such an approximant. The full shift and finite alphabet hypotheses used for d-bar entropy continuity are retained in the corollary.

Universally good pointsCorrect

Uniform pattern frequencies in the orbit closure

Pages 10–17 · intermediate propositions · arXiv:2607.16994v1

The hierarchical construction makes the frequency of each finite pattern nearly identical in every high-level macroblock. Summability of level errors gives a common limit uniformly over translates and over all configurations in the orbit closure. The standard uniform-frequency criterion then yields unique ergodicity. This stronger uniform statement is what permits the approximation to be represented by a uniquely ergodic measure rather than only by one generic configuration.

02Proofs3 reported findingsCorrect

Universally good points and entropy continuity. Large Følner fundamental domains are tiled by labeled blocks sampled from the target finite-pattern distribution. A hierarchical jigsaw rule places every permitted macroblock with prescribed frequency inside all sufficiently large superblocks. Boundary proportions are summable, so every point in the resulting orbit closure has the same limiting pattern frequencies; this proves unique ergodicity. Coupling each jigsaw configuration with the original block labels changes symbols only near boundaries and on a chosen small exceptional set, giving the d-bar estimate. Entropy continuity in d-bar for finite alphabets then gives convergence of entropy.

Jigsaw constructionCorrect and complete

Universally good points and entropy continuity

Pages 8–20 · Sections 3–5 · arXiv:2607.16994v1

Large Følner fundamental domains are tiled by labeled blocks sampled from the target finite-pattern distribution. A hierarchical jigsaw rule places every permitted macroblock with prescribed frequency inside all sufficiently large superblocks. Boundary proportions are summable, so every point in the resulting orbit closure has the same limiting pattern frequencies; this proves unique ergodicity. Coupling each jigsaw configuration with the original block labels changes symbols only near boundaries and on a chosen small exceptional set, giving the d-bar estimate. Entropy continuity in d-bar for finite alphabets then gives convergence of entropy.

Sections 3–4Correct and complete

Residual tilings and jigsaw frequency control

Pages 8–17 · Sections 3–4 · arXiv:2607.16994v1

Finite-index subgroups provide periodic fundamental-domain tilings, and amenability selects domains with vanishing boundary for every fixed finite test set. Target blocks are distributed among supertiles using rational approximations to their probabilities. The rounding error and tile boundary error are made summable over levels. Induction then proves uniform convergence of all cylinder frequencies and minimal recurrence of the constructed orbit closure.

Section 5Correct and complete

d-bar coupling and entropy convergence

Pages 17–20 · Section 5 · arXiv:2607.16994v1

A coupling pairs each jigsaw supertile with the target block used to label its interior. Disagreement is confined to boundaries, rounding corrections, and the chosen atypical blocks, whose combined upper density tends to zero. Hence the d-bar distance tends to zero. For finite-alphabet amenable shifts, the cited entropy-continuity inequality bounds the entropy difference by a function of that distance, yielding the two-sided convergence required in Theorem 27.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.16994v1
Authors listed: Martha Łącka, Marcel Mroczek
Audit date: August 18, 2026

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Entropy Density of Uniquely Ergodic Measures for Full Shifts over Amenable Residually Finite Groups

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Approximation by uniquely ergodic full-shift measures with entropy control

Entropy density of uniquely ergodic measures

Uniform pattern frequencies in the orbit closure

Universally good points and entropy continuity

Residual tilings and jigsaw frequency control

d-bar coupling and entropy convergence

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