Thermodynamics formalism for singular flows

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.

• The paper introduces a robust symbolic coding scheme that extends thermodynamic formalism to handle singularities in 3D flows.
• The paper proves the finiteness of ergodic MMEs and shows that under SPR conditions, each equilibrium state exhibits rapid mixing and strong statistical properties.
• The framework leverages countable-state Markov shifts and scaled hyperbolic structures to transfer statistical properties such as exponential decay of correlations and Bernoulli characteristics.

Thermodynamic Formalism for Singular Flows: A Symbolic Dynamics Perspective

Introduction and Motivation

The paper "Thermodynamics formalism for singular flows" (2604.26936) addresses fundamental aspects of the ergodic theory and thermodynamic formalism for three-dimensional smooth flows, with a primary focus on systems exhibiting singularities—points where the flow velocity vanishes. Singularities, such as those present in the Lorenz attractor, introduce discontinuities and non-uniformities in dynamics that challenge classical symbolic and thermodynamic tools. The authors develop a robust symbolic coding scheme adapted to singular flows to analyze statistical properties, notably the finiteness and mixing properties of measures of maximal entropy (MMEs) and equilibrium states.

Main Results

The authors establish several strong results for $C^\infty$ flows with singularities, summarized as follows:

• Finiteness of MMEs in 3D with Singularities: For any $C^\infty$ three-dimensional flow on a closed manifold with positive topological entropy, there exist only finitely many ergodic MMEs—even in the presence of singularities.
• Rapid Mixing on a Generic Set: On a $C^\infty$ open and dense subset of such flows, every ergodic MME exhibits rapid (superpolynomial) mixing.
• Symbolic Coding for Singular Flows: The paper constructs a finite-to-one symbolic representation for general singular flows (with uniformly bounded return times), establishing a foundational framework for transferring statistical properties.
• Strong Positive Recurrence and SPR Coding: The extension of the strong positive recurrence (SPR) property to singular flows is introduced, aligned with the recent advances in countable-state Markov shifts. The authors show that SPR flows (including certain star flows) can be symbolically coded by SPR Markov suspension flows.
• Equilibrium States and Further Extensions: The approach extends to equilibrium states for Hölder continuous potentials, yielding results on uniqueness and statistical properties.

Symbolic Dynamics and Applications to Singular Flows

Symbolic Representation Construction

A central technical contribution is the construction of a symbolic Markov model that codes the dynamics of flows—possibly with singularities—by introducing a countable, locally finite family of Poincaré sections with uniformly bounded return times. This handles the non-uniformity and potential vanishing of return times near singularities. Scaling techniques (following Liao) are used to consistently treat the loss of transverse hyperbolicity near singular points.

Theorem (Symbolic Coding for Singular Flows):

For any $C^{1+\beta}$ vector field $X$ ($\beta > 0$) on $M$ and any $\chi > 0$, there exists a locally compact topological Markov flow $(\Sigma_r, \sigma_r)$ and a finite-to-one, Hölder-continuous coding map $\pi_r : \Sigma_r \to M$ such that every regular, $C^\infty$0-hyperbolic, $C^\infty$1-invariant measure can be lifted and the correspondence preserves entropy ([see Section "Symbolic dynamics for non-uniformly hyperbolic systems" and Theorem~A]).

This coding enables the full arsenal of the thermodynamic formalism (uniqueness of equilibrium states, mixing properties, exponential tail estimates) to be brought to bear on singular flows.

Strong Positive Recurrence and Equilibrium States

Building on recent developments in the theory of countable Markov shifts and non-uniform hyperbolicity (notably [BCS25]), the SPR property is defined for singular flows. For a flow to be SPR (with respect to a potential $C^\infty$2), measures with pressure close to the topological pressure must concentrate in a compact Pesin block. The authors show that if a singular flow is SPR (for, e.g., Hölder potentials of sufficiently small variation), the associated symbolic model is also SPR, and thus all the statistical properties of symbolic systems with spectral gap transfer to the flow.

Notable Consequence:

For $C^\infty$3 flows that are SPR, there exist finitely many ergodic equilibrium states for any Hölder potential, each of which is Bernoulli up to a period and possesses strong statistical properties (exponential mixing, central limit theorem, large deviations, etc.) ([see Theorem~main.spr]).

Rapid Mixing and Good Asymptotics

The paper extends classical mixing results to flows with singularities. For three-dimensional flows (and star flows) in a generic subset and for potentials of small variation, all ergodic equilibrium states have “good asymptotics” in the sense of Field-Melbourne-Török, ensuring rapid decay of correlations. The proof employs Dolgopyat’s method and phase cancellation techniques, lifting the analysis from symbolic models to the flow via the coding map.

Rapid Mixing Theorem:

For generic $C^\infty$4 three-dimensional flows (with positive entropy and Hölder potentials of small variation), all ergodic equilibrium states are rapid mixing ([see Theorem~main.3flowrm]).

Technical Innovations

• Locally Finite, Uniform Poincaré Sections: Unlike standard constructions using finitely many global Poincaré sections, the authors construct an at-most-countable and locally finite collection, sidestepping issues arising from the global return maps becoming ill-defined near singularities.
• Scaled Hyperbolic Structures: The use of scaled Poincaré maps and Lyapunov metrics extends Pesin theory and the construction of uniform estimates into neighborhoods of singularities, essential for controlling return times and hyperbolic block sizes.
• SPR Extension and Coding: By defining SPR in the context of singular flows and verifying its preservation under symbolic coding, the analysis leverages the deep structure of countable Markov shifts and the spectral theory of the Ruelle operator.

Implications and Future Directions

Theoretical Implications

• The results delineate the boundary between regular and pathological statistical behavior in dynamics with singularities. While singularities have long been seen as sources of non-uniformity and non-hyperbolicity, the present work shows that—under mild generic conditions or for sufficiently hyperbolic potentials—even 3D singular flows behave as regularly as their uniformly hyperbolic, nonsingular counterparts.
• The symbolic framework introduced is flexible and paves the way for a systematic study of statistical properties (beyond entropy and mixing) such as large deviations, multifractal spectra, and escape rates in singular flows.

Practical Implications

• Having finitely many MMEs and equilibrium states, along with strong mixing and statistical properties, allows for more effective computational exploration and simulation of invariant measures in complex flows, including physically relevant models like the Lorenz system.
• The techniques can be adapted for studying physical measures, SRB measures, and nonequilibrium steady states in models from fluid mechanics, climate, and other domains that naturally give rise to singularities.

Future Research Directions

• Higher Dimensions and Non-compact Settings: Extending the symbolic formalism and SPR analysis to flows in higher dimensions and possibly non-compact manifolds.
• Non-Hölder Potentials: The paper discusses the possibility of extending the analysis to quasi-Hölder or geometric potentials (as in dimension theory), which are highly relevant for multifractal analysis.
• Further Statistical Properties: Proving almost sure invariance principle, fluctuation theorems, or developing translation to finer geometric properties (e.g., dimension theory for singular attractors).
• Applications to Physical and Numerical Models: Implementing algorithms for the practical computation of MMEs or measures with maximal entropy in numerically defined flows with singularities.

Numerical and Structural Highlights

• Finiteness of MMEs: In contrast to the classical situation for Axiom A flows without singularities, this is the first extension with a symbolic model yielding finiteness of MMEs for $C^\infty$5 3-flows with singularities.
• Rapid Mixing: For generic (open and dense in the $C^\infty$6 topology) potentials and flows, all equilibrium states are shown to mix superpolynomially fast.
• Bernoulli Property: All ergodic equilibrium states in the SPR regime are Bernoulli up to a period.

Conclusion

This paper makes definitive advances in the thermodynamic formalism for singular flows by constructing a symbolic coding scheme that robustly handles singularities and non-uniform hyperbolicity. The results unify and generalize previous work on both nonsingular flows and singular hyperbolic attractors, setting a comprehensive framework for the study of statistical properties, equilibrium states, and rapid mixing in dynamical systems with singularities. The analysis, built on technically sophisticated symbolic and scaling constructions, offers both conceptual clarity and practical utility for further studies in smooth ergodic theory and dynamical systems.