MathAudit

Paper metadata

Original submission
October 20, 2026
Claimed MSC
37F10, 11J83
arXiv categories
math.DS

Abstract

We study Schmidt's game on the Julia set J(f)J(f) of weak hyperbolic rational maps f:CCtoCC. For each y∈J(f)y\in J(f), let E(f,y)E(f,y) be the set of points in J(f)J(f) whose forward orbits stay a positive distance from yy. We prove that E(f,y)E(f,y) is absolute winning on J(f)J(f) for each y∈J(f)y \in J(f). To do this, we first prove the absolute winning property for the exceptional set associated to the critical points by a potential game argument, and then use uniform expansion to obtain the result for each target point yy.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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.

Current report

Detailed mathematical audit

Generated October 23, 2026
01Statements2 reported findingsNo statement issues found

The absolute-winning theorem for orbit-avoidance sets of expanding-away-from-critical-points, backward-contracting rational maps and its stated consequences are supported.

Theorem ANo issue found

Every one-point exceptional set is absolute winning

PDF pp. 3–4 and 8–22 · arXiv:2610.27796v1

Backward contraction and nice couples give a potential-game strategy that avoids the critical set at all sufficiently small scales. Expansion away from critical points then supplies uniformly controlled inverse branches, allowing the strategy to be transferred from the critical exceptional set to E(f,y)E(f,y) for every y∈J(f)y\in J(f).

Corollaries 1.4 and 1.5No issue found

The strong-winning and nondense-orbit conclusions follow

PDF p. 4 · arXiv:2610.27796v1

Uniform perfectness converts absolute winning to strong winning with a constant depending only on the Julia set. Countable intersections preserve absolute winning, and the nondense-orbit set contains any fixed one-point exceptional set.

02Proofs2 reported findingsNo proof issues found

The potential-game construction, backward-contraction estimates, nice-couple geometry, and expansion transfer cover the stated hypotheses and parameter ranges.

Theorem BNo issue found in the reviewed proof

Uniform expansion transfers avoidance from critical points to an arbitrary target

PDF pp. 8–12 · Section 3 · arXiv:2610.27796v1

Once an orbit remains uniformly separated from the critical set, inverse branches have controlled distortion and shrink at a uniform exponential rate. The game deletions are scheduled at the corresponding scales, and the resulting limit point stays a positive distance from the target.

Theorem ANo issue found in the reviewed proof

The critical exceptional set is won by a summable potential strategy

PDF pp. 12–22 · Section 4 · arXiv:2610.27796v1

Polynomial shrinking bounds the relevant pull-backs, bounded overlap controls the total potential deleted at each turn, and the nested nice couples separate returns to critical neighborhoods. The chosen exponents make the potential sum converge, yielding potential winning and hence absolute winning.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.27796v1
Authors listed
Haoyang Ji
Audit date
October 23, 2026
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