arXiv:2608.00398v1

Infinite-Piecewise Expanding Maps: Chaos, Ergodicity and Invariant-Set Complexity

Matheus G. C. Cunha, Douglas D. Novaes, Gabriel Ponce

math.DS37E9937A9926A1828A7828A8034A3637C29

Paper metadata

Original submission: August 1, 2026
Claimed MSC: 37E99, 37A99, 26A18, 28A78, 28A80, 34A36, 37C29
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: Version 01

Abstract

In this paper, we study a class of one-dimensional piecewise maps defined by infinitely many smooth expanding branches. This class arises naturally in the context of non-smooth dynamical systems and includes the first-return maps locally defined near sliding Shilnikov connections. By means of the theory of conformal iterated function systems (CIFS), we investigate several dynamical properties of these maps as well as the topological complexity of their invariant sets. In particular, we show that the dynamics restricted to the invariant set is topologically conjugate to the shift on $\mathbb{N}^{\mathbb{N}}$ . We also establish the existence of a unique conformal measure that is invariant and ergodic under the map.

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Audited against arXiv v1

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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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The dimension bounds, closure decomposition, Cantor-set structure, symbolic conjugacy, and conformal-measure conclusions in Theorem A are correct under hypotheses (H0)–(H3). Two local parameter and index-range corrections do not change any conclusion.

Theorem A(a)–(d)Correct

Geometry of the invariant set and its closure

Pages 3 and 11–13 · Theorem A(a)–(d) and Sections 5.1–5.4 · arXiv:2608.00398v1

The inverse branches form a strongly separated conformal iterated-function system with bounded distortion. Hypothesis (H3) places the pressure divergence threshold at $1/q$ , while the positive open gap makes the pressure negative at $1$ ; hence its unique zero $\widehat t$ lies in $(1/q,1)$ and equals $\dim_H(\Lambda)$ . The recursive sets $Z_k$ add exactly the finite preimages of the singular accumulation set, yielding $\overline\Lambda=\Lambda\mathbin{\dot\cup}Z$ . Countable stability and bi-Lipschitz invariance give the stated dimension and measure formulas, and the gap images establish total disconnectedness while symbolic tail changes establish perfectness.

Theorem A(e)Correct

Conjugacy with the one-sided countable shift

Pages 3–4 and 13–15 · Theorem A(e) and Section 5.5 · arXiv:2608.00398v1

Uniform contraction makes the coding projection from $\mathcal J^{\mathbb N}$ continuous, and disjoint branch intervals make every address unique. For each fixed address prefix, the corresponding cylinder is isolated inside the preceding cylinder, which makes the inverse projection continuous pointwise. Removing the first branch agrees with applying $\varphi$ , so the projection is a homeomorphism intertwining $\varphi|_\Lambda$ and the shift.

Theorem A(f)Correct

Conformal and equivalent invariant measures

Pages 4 and 15–16 · Theorem A(f) and Section 5.6 · arXiv:2608.00398v1

Regularity at $\widehat t=\dim_H(\Lambda)$ supplies the conformal probability measure. Strong separation gives unique coding, and the standard regular-CIFS construction produces the unique shift-invariant probability equivalent to the coding measure; pushing it through the conjugacy gives the stated invariant and ergodic measure equivalent to the conformal measure on $\Lambda$ .

02Proofs4 reported findingsContains incorrect or incomplete proofs

The pressure, dimension, closure, and measure arguments are correct after two local yellow corrections. The printed proof of continuity of the inverse coding map contains a false global implication; the claimed homeomorphism remains true, and a verified pointwise repair is supplied, but the proof segment is incomplete as written.

Section 5.5Incomplete as written

Closeness does not globally force a common symbolic prefix

Pages 14–15 · proof that $\operatorname{proj}^{-1}$ is continuous · arXiv:2608.00398v1

The proof asserts that sufficiently close arbitrary $x,y\in\Lambda$ lie in one common cylinder of a prescribed length. This is false uniformly: first-level intervals may accumulate at $Z_1$ , so points in distinct first-level cylinders can be arbitrarily close. The needed pointwise statement is nevertheless valid. Fix $x$ and a prefix length $k$ . Each interval $F_j$ is isolated, and applying successively the finitely many inverse branches in the address of $x$ shows that the length- $k$ cylinder containing $x$ is relatively open in $\Lambda$ . Thus some neighborhood of this fixed $x$ has the same first $k$ symbols, which proves continuity of $\operatorname{proj}^{-1}$ at $x$ . Repair classification: Verified repair.

Proof of Theorem A(b)Minor formal correction

The minimal-level word has length $\kappa-1$

Page 12 · paragraph following Equation (15) · arXiv:2608.00398v1

From the definition $Z_\kappa=Z_1\cup\bigcup_{j=1}^{\kappa-1}\bigcup_{|\eta|=j}\psi_\eta(Z_1)$ , minimality of $\kappa\geq2$ gives a representing word of length $\kappa-1$ , not $\kappa$ . Replace the two ensuing occurrences of length $\kappa$ by $\kappa-1$ . This yields $x\in\Delta_{\kappa-1}$ , exactly what is required to cover every earlier level; no later statement changes.

Definitions 2 and 3Typo

The conformal parameter range must include values below one

Pages 6 and 15 · regularity and conformal-measure definitions · arXiv:2608.00398v1

The definitions print $\widehat t\geq1$ and $t\geq1$ , but Proposition 5.4 proves and Section 5.6 uses $1/q<\widehat t<1$ . Replace those lower bounds by $\widehat t\geq0$ and $t\geq0$ . The pressure is defined on $[0,\infty)$ and the cited CIFS results apply on that range, so the intended correction is unique and does not alter the argument.

Proposition 5.4 and Appendix ACorrect and complete

Pressure has a unique zero in the claimed interval

Pages 11 and 16–18 · Proposition 5.4, Proposition A.1, and Corollary A.3 · arXiv:2608.00398v1

The two-sided polynomial estimate in (H3) gives convergence of $P_1(t)$ precisely for $t>1/q$ and divergence at the endpoint, with $P_1(t)\to\infty$ as $t\downarrow1/q$ . Bounded distortion transfers this to the pressure. The positive open-gap proportion gives exponential decay of total level length and hence $P_N(1)<1$ for some $N$ . Continuity and strict decrease then give the unique zero $\widehat t\in(1/q,1)$ .

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.00398v1
Authors listed: Matheus G. C. Cunha, Douglas D. Novaes, Gabriel Ponce
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

is proved for every sufficiently small

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is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

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Infinite-Piecewise Expanding Maps: Chaos, Ergodicity and Invariant-Set Complexity

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Geometry of the invariant set and its closure

Conjugacy with the one-sided countable shift

Conformal and equivalent invariant measures

Closeness does not globally force a common symbolic prefix

The minimal-level word has length κ−1\kappa-1κ−1

The conformal parameter range must include values below one

Pressure has a unique zero in the claimed interval

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The minimal-level word has length $\kappa-1$