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arxiv: 2604.26936 · v1 · submitted 2026-04-29 · 🧮 math.DS

Thermodynamics formalism for singular flows

Ming Li , Xingzhong Liu This is my paper

Pith reviewed 2026-05-07 09:02 UTC · model grok-4.3

classification 🧮 math.DS
keywords singular flowsthermodynamic formalismmeasures of maximal entropysymbolic codingstrong positive recurrencerapid mixingthree-dimensional flowstopological entropy
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The pith

C^∞ three-dimensional flows with positive topological entropy have only finitely many ergodic measures of maximal entropy even with singularities.

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

The paper develops a thermodynamic formalism for singular flows. It proves that any C^∞ three-dimensional flow with positive topological entropy has only finitely many ergodic measures of maximal entropy, and this finiteness holds even when the flow has singularities at which the velocity vanishes. A sympathetic reader would care because prior results on measures of maximal entropy required the flow to be nonsingular, so the new work removes a key restriction and applies to a broader class of systems that can model physical situations with zero-speed points. The argument rests on a new symbolic coding system for singular flows and on the strong positive recurrence property that lets the flow be represented as a suspension over a symbolic system.

Core claim

C^∞ three-dimensional flows with positive topological entropy admit only finitely many ergodic measures of maximal entropy, even when singularities (zero-velocity points) are present. Furthermore, every ergodic measure of maximal entropy is rapid mixing for such flows within a C^∞ open and dense subset. The authors develop a novel symbolic coding system for flows with singularities and define the strong positive recurrence (SPR) property for singular flows, verifying that SPR flows can be coded by suspension flows of SPR symbolic systems. This framework extends to other singular flows, including star flows, and to equilibrium states.

What carries the argument

A novel symbolic coding system for flows with singularities, which codes them as suspension flows of symbolic systems satisfying the strong positive recurrence (SPR) property.

If this is right

  • These flows possess only finitely many ergodic measures of maximal entropy.
  • Within a C^∞ open and dense subset, all ergodic measures of maximal entropy are rapid mixing.
  • The same finiteness and mixing conclusions hold for star flows and other classes of singular flows.
  • Equilibrium states for singular flows can be studied with the same thermodynamic formalism.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The symbolic coding technique might extend to higher-dimensional singular flows and yield analogous finiteness statements there.
  • Numerical approximation of entropy and mixing rates could become feasible for physical models that contain zero-velocity points.
  • The framework may clarify the relationship between topological entropy and statistical properties in systems that combine smooth and singular behavior.

Load-bearing premise

The symbolic coding system developed for flows with singularities is valid and complete for all C^∞ three-dimensional flows with positive entropy, including the correspondence that SPR flows can be coded by suspension flows of SPR symbolic systems.

What would settle it

An explicit C^∞ three-dimensional flow with positive topological entropy that possesses infinitely many distinct ergodic measures of maximal entropy would disprove the finiteness result.

Figures

Figures reproduced from arXiv: 2604.26936 by Ming Li, Xingzhong Liu.

Figure 1
Figure 1. Figure 1: Relationships between mappings 6.5 Restricted Pesin charts and overlapping condition For every η ∈ (0, Q(x)], we denote by Ψe ∗,η x the restriction of the scaled Pesin chart at x of scaled size η, that is, Ψe ∗,η x := Ψe ∗ x |R[η] . Accordingly, we denote Ψη x := Ψx|R[η] . Subsequently, we will discuss pseudo orbits and the shadowing property. Unlike diffeomorphisms and non-singular flows, we employ scaled… view at source ↗
read the original abstract

We establish that $C^\infty$ three-dimensional flows with positive topological entropy admit only finitely many ergodic measures of maximal entropy, even when singularities (zero-velocity points) are present. Furthermore, every ergodic measure of maximal entropy is rapid mixing for such flows within a $C^\infty$ open and dense subset. To prove this, we develop a novel symbolic coding system for flows with singularities, which serves as a fundamental tool in this work. We also define the strong positive recurrence (SPR) property for singular flows and verify that SPR flows can be coded by suspension flows of SPR symbolic systems. This framework extends to other singular flows, including star flows, and to equilibrium states.

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

2 major / 1 minor

Summary. The manuscript establishes that C^∞ three-dimensional flows with positive topological entropy admit only finitely many ergodic measures of maximal entropy, even when singularities (zero-velocity points) are present. It further shows that every ergodic measure of maximal entropy is rapid mixing for such flows within a C^∞ open and dense subset. The proof relies on a newly developed symbolic coding system for singular flows, the definition of the strong positive recurrence (SPR) property for singular flows, and the verification that SPR singular flows can be coded by suspension flows over SPR symbolic systems; the framework is indicated to extend to star flows and equilibrium states.

Significance. If the symbolic coding construction is valid and complete, the result would provide a substantial extension of thermodynamic formalism and rapid mixing results to singular flows, generalizing known statements from non-singular or hyperbolic settings to a broad class of C^∞ 3D systems with positive entropy. The introduction of a coding tool that preserves the SPR property under suspension, together with the open-dense rapid-mixing conclusion, would constitute a useful technical advance applicable to other singular systems.

major comments (2)
  1. [Abstract and the symbolic coding construction] The central reduction step—that every C^∞ 3D flow with positive topological entropy and singularities admits a valid and complete symbolic coding by a suspension flow over an SPR symbolic system—is asserted in the abstract and used to derive both the finiteness of ergodic MMEs and the rapid-mixing property. No independent verification against standard singular examples (e.g., geometric Lorenz attractors) or comparison with existing partial results on equilibrium states for singular systems is provided, leaving the completeness of the coding unconfirmed.
  2. [SPR property and coding verification] The definition of SPR for singular flows and the claim that SPR flows code to suspension flows of SPR symbolic systems are load-bearing for the thermodynamic formalism; if the coding misses orbits or fails to preserve the SPR property, the finiteness and mixing conclusions do not follow. The manuscript does not supply an explicit check that the coding is surjective onto all relevant orbits or that the entropy and mixing rates are preserved.
minor comments (1)
  1. [Abstract] The abstract states that the framework 'extends to other singular flows, including star flows, and to equilibrium states,' but the main text does not indicate the precise scope or additional hypotheses required for these extensions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the acknowledgment of the potential significance of extending thermodynamic formalism and rapid mixing results to singular flows. We address each major comment below, clarifying the scope of our constructions and indicating revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract and the symbolic coding construction] The central reduction step—that every C^∞ 3D flow with positive topological entropy and singularities admits a valid and complete symbolic coding by a suspension flow over an SPR symbolic system—is asserted in the abstract and used to derive both the finiteness of ergodic MMEs and the rapid-mixing property. No independent verification against standard singular examples (e.g., geometric Lorenz attractors) or comparison with existing partial results on equilibrium states for singular systems is provided, leaving the completeness of the coding unconfirmed.

    Authors: The symbolic coding construction is developed as a general tool applicable to all C^∞ 3D flows with positive topological entropy and singularities, including geometric Lorenz attractors, whose zero-velocity singularities are covered by the framework. The completeness and validity are established through the abstract properties of the coding map rather than case-by-case verification. We agree that an explicit comparison with known partial results on equilibrium states for Lorenz-like systems would improve clarity and will add a short discussion section referencing relevant literature in the revised manuscript. revision: yes

  2. Referee: [SPR property and coding verification] The definition of SPR for singular flows and the claim that SPR flows code to suspension flows of SPR symbolic systems are load-bearing for the thermodynamic formalism; if the coding misses orbits or fails to preserve the SPR property, the finiteness and mixing conclusions do not follow. The manuscript does not supply an explicit check that the coding is surjective onto all relevant orbits or that the entropy and mixing rates are preserved.

    Authors: The surjectivity of the coding onto all relevant orbits, along with preservation of the SPR property, entropy, and mixing rates, is established in the core technical sections through the explicit construction of the symbolic dynamics and the suspension flow. These properties are proven directly from the definitions and do not rely on additional assumptions. If the presentation of these verifications can be made more explicit, we will add summary statements and a clarifying remark in the revision to highlight the surjectivity and invariance arguments. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on novel construction

full rationale

The paper introduces a novel symbolic coding system for C^∞ 3D singular flows and defines/verifies the SPR property for such flows, allowing reduction of thermodynamic formalism results to the symbolic setting. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The central claims follow from this new framework rather than reducing to prior inputs by construction. The coding is presented as independently developed, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

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  1. Continuity properties of partial entropy

    math.DS 2026-05 unverdicted novelty 8.0

    Partial entropies are upper semi-continuous for C^{1+α} diffeomorphisms when Lyapunov exponent sums are continuous, implying the same property at generic ergodic measures.

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