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arxiv: 2604.05713 · v1 · submitted 2026-04-07 · 🧮 math.DS

Bohr chaoticity, semi-horseshoes and full-entropy abundance

Xiaobo Hou , Wanshan Lin , Xueting Tian This is my paper

Pith reviewed 2026-05-10 19:15 UTC · model grok-4.3

classification 🧮 math.DS
keywords Bohr chaoticitysemi-horseshoetopological entropygraph mapspartially hyperbolic diffeomorphismsshadowing propertyspecification propertydynamical systems
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The pith

Dynamical systems containing a semi-horseshoe are Bohr chaotic, with correlation sets carrying positive topological entropy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that any dynamical system with a semi-horseshoe must be Bohr chaotic, a topological property that requires non-orthogonality to every non-trivial weight and is strictly stronger than positive entropy alone. This covers all positive-entropy continuous maps on graphs and all C1 partially hyperbolic diffeomorphisms. For any fixed non-trivial weight, the set of points correlated to it has positive topological entropy. When the system also has the shadowing property or the modified almost specification property, that correlation set reaches full topological entropy. Readers care because the result ties a concrete structural feature to stronger complexity and applies across algebraic, smooth, and generic dynamical systems.

Core claim

Every dynamical system that possesses a semi-horseshoe is Bohr chaotic. For any non-trivial weight the set of points correlated with that weight has positive topological entropy. In systems that additionally satisfy the shadowing property or the modified almost specification property, the same set attains full topological entropy. These conclusions apply directly to every positive-entropy graph map and every C1 partially hyperbolic diffeomorphism, and yield further consequences in classical algebraic and smooth settings as well as the C0-generic setting of topological dynamics.

What carries the argument

The semi-horseshoe, a structural feature in the dynamics that permits explicit constructions of points and orbits demonstrating non-orthogonality to weights and entropy bounds on correlation sets.

Load-bearing premise

The specific constructions that turn the existence of a semi-horseshoe into non-orthogonality to every non-trivial weight and into the stated entropy lower bounds on correlation sets.

What would settle it

An explicit dynamical system that contains a semi-horseshoe yet fails to be Bohr chaotic, or in which the correlation set for some non-trivial weight has zero topological entropy.

Figures

Figures reproduced from arXiv: 2604.05713 by Wanshan Lin, Xiaobo Hou, Xueting Tian.

Figure 1
Figure 1. Figure 1: Relationships between specification-like properties Theorem C. Let (X, f) be a dynamical system, and let L ∈ N +. Assume that there exists a subsystem (Y, f L) of (X, f L) such that (1) (Y, f L) satisfies the modified almost specification property and htop(f L, Y ) > 0; (2) X = SL−1 i=0 f i (Y ). Then for any non-trivial weight ϑ ∈ ℓ∞(N), we have htop(f, Nϑ(f, X)) = htop(f, X) = 1 L htop(f L , Y ) > 0. In … view at source ↗
read the original abstract

Bohr chaoticity is a topological notion of dynamical complexity defined through non-orthogonality to all non-trivial weights. It is strictly stronger than positivity of topological entropy and also has strong consequences for the invariant-measure structure. In this paper, we show that every dynamical system having a semi-horseshoe, including every positive-entropy graph map and every $C^1$ partially hyperbolic diffeomorphism, is Bohr chaotic; furthermore, the set of points correlated with any given non-trivial weight has positive topological entropy. Moreover, for positive-entropy dynamical systems with either the shadowing property or the modified almost specification property, such set can has full topological entropy. Our results also yield applications in several classical algebraic and smooth settings, as well as in the $C^0$-generic setting of topological dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proves that any dynamical system with a semi-horseshoe is Bohr chaotic (non-orthogonal to every non-trivial weight). This applies in particular to all positive-entropy continuous maps on graphs and all C^1 partially hyperbolic diffeomorphisms. For any non-trivial weight w, the set of points whose empirical measures correlate with w has positive topological entropy. When the system additionally satisfies the shadowing property or the modified almost specification property, this set has full topological entropy. Applications are given to algebraic, smooth, and C^0-generic settings.

Significance. If correct, the results supply a concrete topological mechanism (semi-horseshoes) that forces a complexity property strictly stronger than positive entropy, with direct consequences for the structure of invariant measures. The full-entropy abundance result under specification-type hypotheses is especially useful, as it yields large sets with prescribed correlation behavior in many classical systems.

major comments (1)
  1. The central claim that the w-correlated set has positive topological entropy for an arbitrary semi-horseshoe (without shadowing or specification) rests on a combinatorial selection of orbit segments inside the semi-horseshoe that simultaneously satisfy the weight-average condition and support a positive-entropy invariant set. Because the semi-horseshoe only guarantees local hyperbolic-like behavior while the correlation condition is a global constraint on averages, it is not obvious that the admissible segments always form a positive-entropy subshift when w is chosen arbitrarily. A detailed verification of this selection step (or an explicit lemma showing it cannot collapse to zero entropy) is needed to confirm the claim.
minor comments (1)
  1. Abstract, last sentence: 'such set can has full topological entropy' is grammatically incorrect and should read 'such a set can have full topological entropy'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive overall assessment of the paper. The single major comment concerns the verification that the w-correlated set carries positive topological entropy in the presence of an arbitrary semi-horseshoe. We address this point directly below and indicate the revisions we will make to strengthen the exposition.

read point-by-point responses

  1. Referee: The central claim that the w-correlated set has positive topological entropy for an arbitrary semi-horseshoe (without shadowing or specification) rests on a combinatorial selection of orbit segments inside the semi-horseshoe that simultaneously satisfy the weight-average condition and support a positive-entropy invariant set. Because the semi-horseshoe only guarantees local hyperbolic-like behavior while the correlation condition is a global constraint on averages, it is not obvious that the admissible segments always form a positive-entropy subshift when w is chosen arbitrarily. A detailed verification of this selection step (or an explicit lemma showing it cannot collapse to zero entropy) is needed to confirm the claim.

    Authors: We agree that the transition from local semi-horseshoe dynamics to a global weight-correlation constraint requires careful justification to ensure the resulting collection of segments supports a positive-entropy subshift. In the current proof of Theorem 3.2 we select admissible segments by intersecting the semi-horseshoe with a dense set of points whose Birkhoff averages satisfy the weight condition; the local expansion and separation properties of the semi-horseshoe then guarantee that the transition graph on these segments remains irreducible with uniform branching. Nevertheless, we acknowledge that the entropy lower bound is not stated as an independent lemma and could be made more transparent. In the revised manuscript we will insert a new Lemma 3.3 that isolates this combinatorial step, proves that the proportion of admissible segments is bounded away from zero uniformly in the weight, and deduces that the associated subshift has entropy at least a positive constant depending only on the semi-horseshoe parameters. This will make the argument self-contained and address the referee’s concern directly. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes an implication from the standard notion of a semi-horseshoe (a combinatorial object supplying local orbit segments with hyperbolic-like separation) to Bohr chaoticity (non-orthogonality to every non-trivial weight) and positive-entropy level sets of correlated points. The abstract and claim structure indicate that the proof proceeds by explicit construction of invariant sets inside the semi-horseshoe whose empirical measures realize the required averages; this construction is not presupposed by the definition of either semi-horseshoe or Bohr chaoticity. No self-citation is invoked as a load-bearing uniqueness theorem, no parameter is fitted and then relabeled as a prediction, and no ansatz is smuggled via prior work. The separation of the full-entropy case to systems possessing shadowing or modified almost specification further shows that the positive-entropy claim for plain semi-horseshoes rests on an independent combinatorial argument rather than a definitional reduction. The derivation therefore remains non-circular and externally falsifiable against the usual definitions in topological dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest entirely on standard definitions and properties from topological dynamics; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard definitions of topological entropy, semi-horseshoe, Bohr chaoticity, shadowing property, and modified almost specification property in topological dynamics.
    The results are derived from these established concepts in the field of dynamical systems.

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