arXiv:2606.28771v2

Conditional entropy realization and approximation by uniquely ergodic measures

Xiaobo Hou, Wanshan Lin, Xueting Tian, Yi Yuan, Xutong Zhao

math.DS37A3537B4037D2037B10

Paper metadata

Original submission: June 27, 2026
Latest revision: August 9, 2026 · version 2
Claimed MSC: 37A35, 37B40, 37D20, 37B10
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 58 pages

Abstract

This paper studies conditional entropy realization and weak* approximation by uniquely ergodic measures with compact support. We prove that, after fixing an admissible potential average and an entropy strictly below the entropy supremum over the corresponding average fiber, every invariant measure satisfying these two exact constraints can be approximated weakly* by uniquely ergodic measures with compact support satisfying the same constraints. Each approximating measure is the unique invariant measure on its minimal support, whose topological entropy equals the prescribed metric entropy. This result holds for three broad classes of systems: topologically expanding maps (including topologically Anosov systems), transitive countable Markov shifts, and symbolic systems with non-uniform structure. The proof uses a nested multi-horseshoe construction, with separate arguments addressing non-invertibility, non-compactness and non-uniformity.

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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Generated August 18, 2026

01Statements3 reported findingsCorrect

The exact entropy-and-average realization theorem, its weak-star density conclusion, the three multi-horseshoe inputs, and the stated conditional-variational and Lyapunov-spectrum consequences are correct under the hypotheses given. The separate arguments for non-invertible, non-compact, and non-uniform systems supply the properties used by the common nested construction.

Theorem ACorrect

Exact constrained realization and density hold in all three classes

Pages 6–8 and 18–55 · Theorem A via Theorems 5.1, 7.4, and 9.2 · arXiv:2606.28771v2

Fix an invariant measure of entropy $h$ and average $a$ , with $h$ strictly below the entropy supremum in the $a$ -fiber. A nearby exact- $a$ measure of entropy above $h$ and a nearby zero-entropy exact- $a$ measure give the initial entropy interpolation; measures with averages on the two sides of $a$ give exact average control. At every later stage, two nested subhorseshoes retain averages on opposite sides of $a$ , have entropy intervals shrinking to $h$ , and have invariant-measure spaces of diameter tending to zero. Their intersection therefore supports one measure $\nu$ with $h_\nu(f)=h$ and $\int\phi\,d\nu=a$ . Taking $\operatorname{supp}(\nu)$ makes the subsystem minimal and uniquely ergodic, and the first-stage distance estimates make $\nu$ arbitrarily close to the prescribed measure. The degenerate case $L_\phi=\{a\}$ is the same construction with the automatic average constraint omitted.

Theorems B, C, and DCorrect

The multi-horseshoe statements provide the claimed entropy and measure control

Pages 13–17, 31–38, and 45–53 · Theorems B, C, and D · arXiv:2606.28771v2

For a topologically expanding map, periodic separated orbit segments supplied by uniform separation are concatenated as pseudo-orbits and uniquely shadowed, producing finite full shifts for an iterate. For a transitive countable Markov shift, finite families of periodic words with controlled Birkhoff averages concatenate freely and stay in a finite alphabet, hence give compact full shifts. For a finite-alphabet subshift with non-uniform structure, edit approachability transfers sufficiently many typical words into a freely concatenable collection; the marker word of minimal period separates phases and the entropy loss is explicitly bounded. In each case the block empirical measures give the two Hausdorff estimates required by the later induction.

Theorems 2.6 and 2.8Correct

The conditional entropy and Lyapunov-spectrum consequences follow

Pages 4–6 and 24–27 · Theorems 2.6 and 2.8 · arXiv:2606.28771v2

For Theorem 2.6, generic points give the lower entropy bound, the saturated-set estimate gives the matching upper bound over the average fiber, and Theorem A realizes every entropy strictly below that supremum by a compactly supported uniquely ergodic measure, so the suprema agree. For Theorem 2.8, multi-average conformality identifies each repeated Lyapunov exponent with the integral of the continuous normalized logarithmic Jacobian on the corresponding invariant bundle. The finite-vector version of the nested construction realizes every vector in the relative interior by a uniquely ergodic measure, which yields both displayed relative-interior and ordinary-interior equalities.

02Proofs4 reported findingsCorrect

The central proofs are correct and complete. The shadowing construction for expanding maps, the finite-alphabet reduction for countable Markov shifts, the edit-and-marker construction for non-uniform subshifts, and the shared decreasing-horseshoe argument collectively establish every substantive conclusion.

Theorem 5.1Correct and complete

The nested-horseshoe proof closes entropy, average, and uniqueness simultaneously

Pages 18–24 · Lemma 5.2 and proof of Theorem 5.1 · arXiv:2606.28771v2

Lemma 5.2 correctly transfers invariant measures between an $f^N$ -invariant base and its $N$ cyclic images; entropy is preserved for $f^N$ and therefore scales by $1/N$ for $f$ . At stage $k$ , variational interpolation in the two sub-full-shifts supplies measures with entropy just above $h$ and averages on opposite sides of $a$ . The multi-horseshoe estimate makes $\operatorname{diam}\mathcal M(f,\Lambda_k)\to0$ , while topological entropies decrease to $h$ . Upper semicontinuity gives the limiting measure entropy at least $h$ , and the topological upper bound gives at most $h$ . The intersection thus has exactly one invariant measure and the required entropy and average.

Theorems B and CCorrect and complete

The two uniform symbolic models have the asserted quantitative controls

Pages 13–17 and 31–38 · proofs of Theorems B and C · arXiv:2606.28771v2

In Theorem B, the shadowing accuracy is chosen below both the positive expansivity constant and the separation scale. Consequently each infinite concatenation has a unique tracer, periodic concatenations give actual periodic points, and the coding maps are conjugacies. In Theorem C, the connector lengths are negligible compared with the typical blocks; all constructed words begin in the same cylinder and can be concatenated freely. The $o(n)$ cylinder variation of every bounded uniformly continuous test function controls all invariant measures on the resulting compact shifts. The entropy cardinality estimates and the recurring return to the chosen cylinder prove the remaining clauses.

Theorems D and 9.2Correct and complete

The non-uniform symbolic construction and its reduction are complete

Pages 43–55 · Theorem 8.9, Lemma 8.11, and proofs of Theorems D and 9.2 · arXiv:2606.28771v2

The cited symbolic approximation theorem supplies entropy-approaching ergodic measures on transitive sofic shifts, while edit approachability and the combinatorial edit bound retain exponentially many distinct free-concatenation words. The count of short-period words guarantees a marker of exact period $T$ ; deleting its cyclic $T$ -blocks makes the $MT$ phases disjoint in the two-sided case and gives bounded-gap specification in the one-sided case. Equations (9.29)–(9.32) control all block empirical measures and propagate the initial positive-entropy perturbation back to the original convex hull. Remark 9.1 then places the subsequent nested construction inside compact finite full shifts, where entropy upper semicontinuity is available.

Climenhaga–Thompson–Yamamoto, non-uniform symbolic horseshoes ↗

Countable-state inputCorrect and complete

The entropy-approachability input is used within its scope

Pages 29–39 · Lemmas 6.3–6.6 and proof of Theorem 7.4 · arXiv:2606.28771v2

Takahasi's theorem applies to every invariant probability measure on a transitive countable Markov shift and supplies ergodic compact-support approximants with convergent entropy. The proof uses it first to replace a possibly infinite-entropy high-entropy fiber measure by finite-entropy compact approximants, and then corrects their averages with periodic measures; the correction coefficient tends to zero. After the first application of Theorem C, all later stages lie in finite full shifts, so the compact-case entropy argument applies without assuming upper semicontinuity on the original non-compact shift.

Takahasi, entropy-approachability for transitive countable Markov shifts ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.28771v2
Authors listed: Xiaobo Hou, Wanshan Lin, Xueting Tian, Yi Yuan, Xutong Zhao
Audit date: August 18, 2026

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Conditional entropy realization and approximation by uniquely ergodic measures

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Exact constrained realization and density hold in all three classes

The multi-horseshoe statements provide the claimed entropy and measure control

The conditional entropy and Lyapunov-spectrum consequences follow

The nested-horseshoe proof closes entropy, average, and uniqueness simultaneously

The two uniform symbolic models have the asserted quantitative controls

The non-uniform symbolic construction and its reduction are complete

The entropy-approachability input is used within its scope

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