[Paper Review] Unique equilibrium states for geodesic flows over surfaces without focal points

Dong Chen, Lien-Yung Kao+1 more|arXiv (Cornell University)|Aug 2, 2018

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper establishes the uniqueness of equilibrium states for geodesic flows on closed surfaces of genus ≥2 without focal points, under a pressure gap condition. It proves that Hölder continuous potentials and scalar multiples of the geometric potential with coefficient q < 1 have unique equilibrium states that are Bernoulli, fully supported, and equidistribute weighted regular periodic orbits, extending results from Anosov and nonpositively curved systems to the broader class of surfaces without focal points.

ABSTRACT

In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted regular periodic orbits are equidistributed relative to these unique equilibrium states.

Motivation & Objective

• To establish the uniqueness of equilibrium states for geodesic flows on closed surfaces of genus ≥2 without focal points.
• To extend the thermodynamic formalism beyond uniformly hyperbolic (Anosov) systems to non-uniformly hyperbolic settings with singular sets.
• To characterize ergodic properties—specifically Bernoulli, full support, and equidistribution—of these unique equilibrium states.
• To generalize prior results on measures of maximal entropy and equilibrium states from nonpositively curved manifolds to surfaces without focal points.
• To prove the C¹ regularity of the pressure function P(qφᵘ) for q < 1 and show P(qφᵘ) = 0 for q ≥ 1 when the singular set is nonempty.

Proposed method

• Applies the Climenhaga-Thompson framework for uniqueness of equilibrium states, relying on the Bowen property and specification property.
• Uses the pressure gap condition P(Sing, φ) < P(φ) as a key criterion for uniqueness, verified via variational principle and topological entropy estimates.
• Establishes the Bowen property for Hölder potentials and the geometric potential φᵘ via uniform estimates on stable/unstable Jacobi fields and hyperbolicity indices λ, λ_T.
• Employs the flat strip theorem and C² regularity of horocycles in surfaces without focal points to control geometry and derive uniform estimates on orbit segments.
• Applies Pesin’s entropy formula and Ruelle’s inequality to relate Lyapunov exponents and measure-theoretic entropy, enabling entropy-expansion arguments.
• Uses entropy expansiveness and Walters’ theorem to prove C¹ regularity of the pressure function P(qφᵘ) for q < 1.

Experimental results

Research questions

• RQ1 Under what conditions does a geodesic flow on a surface without focal points admit a unique equilibrium state for a given potential?
• RQ2 Do the unique equilibrium states for scalar multiples of the geometric potential φᵘ with q < 1 exhibit Bernoulli and full support properties?
• RQ3 Are weighted regular periodic orbits equidistributed relative to the unique equilibrium state in this setting?
• RQ4 How does the pressure function P(qφᵘ) behave for q < 1 and q ≥ 1, particularly in relation to the singular set?
• RQ5 Can the pressure gap condition P(Sing, φ) < P(φ) be verified for a broad class of potentials, including those vanishing on the singular set?

Key findings

• For any Hölder continuous potential φ or scalar multiple qφᵘ with q < 1, the geodesic flow on a closed surface of genus ≥2 without focal points has a unique equilibrium state μ_φ.
• The unique equilibrium state μ_φ is Bernoulli, implying strong statistical independence and mixing properties.
• The unique equilibrium state μ_φ is fully supported on the unit tangent bundle T¹S, meaning its support is the entire space.
• Weighted regular periodic orbits are equidistributed relative to μ_φ, confirming a deep connection between periodic orbits and equilibrium states.
• The pressure function q ↦ P(qφᵘ) is C¹ for q < 1, and P(qφᵘ) = 0 for q ≥ 1 when the singular set is nonempty.
• The pressure gap condition P(Sing, φ) < P(φ) holds if sup_Sing φ − inf_T¹S φ < h_top(F), which is satisfied for constant potentials and nonnegative potentials vanishing on Sing.