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Construction of equilibrium states revisited

Changguang Dong, Qiujie Qiao

TL;DR

This work develops an explicit geometric construction for $\mathcal{F}$-equilibrium states in a broad class of expanding foliations arising from partially and nonuniformly hyperbolic diffeomorphisms. By introducing unstable entropy and unstable pressure along expanding foliations, the authors prove an abstract criterion under conditions (C1)–(C3) that yields equilibrium states as accumulation points of leafwise, weighted volume pushforwards. The criterion applies to exponential mixing systems, Katok maps, and almost Anosov diffeomorphisms, providing concrete existence results and illuminating how leafwise dynamics determine global thermodynamic behavior. These results extend prior geometric methods beyond Lyapunov stability and offer new avenues for explicit equilibrium-state constructions in nonuniformly hyperbolic settings with transverse growth.

Abstract

In [52], Parmenter and Pollicott establish an abstract criterion that gives a geometric construction of equilibrium states for a class of partially hyperbolic systems. We refine their criterion to cover a much broader class of diffeomorphisms, which include certain diffeomorphisms with exponential mixing property (with respect to volume), Katok maps and ``almost Anosov'' diffeomorphisms. As a special case, we obtain a construction of equilibrium states for ergodic partially hyperbolic affine maps/flows on homogeneous spaces, without any restrictions on the orbit growth along center directions.

Construction of equilibrium states revisited

TL;DR

This work develops an explicit geometric construction for -equilibrium states in a broad class of expanding foliations arising from partially and nonuniformly hyperbolic diffeomorphisms. By introducing unstable entropy and unstable pressure along expanding foliations, the authors prove an abstract criterion under conditions (C1)–(C3) that yields equilibrium states as accumulation points of leafwise, weighted volume pushforwards. The criterion applies to exponential mixing systems, Katok maps, and almost Anosov diffeomorphisms, providing concrete existence results and illuminating how leafwise dynamics determine global thermodynamic behavior. These results extend prior geometric methods beyond Lyapunov stability and offer new avenues for explicit equilibrium-state constructions in nonuniformly hyperbolic settings with transverse growth.

Abstract

In [52], Parmenter and Pollicott establish an abstract criterion that gives a geometric construction of equilibrium states for a class of partially hyperbolic systems. We refine their criterion to cover a much broader class of diffeomorphisms, which include certain diffeomorphisms with exponential mixing property (with respect to volume), Katok maps and ``almost Anosov'' diffeomorphisms. As a special case, we obtain a construction of equilibrium states for ergodic partially hyperbolic affine maps/flows on homogeneous spaces, without any restrictions on the orbit growth along center directions.
Paper Structure (20 sections, 37 theorems, 137 equations)

This paper contains 20 sections, 37 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Some history
  3. Markov partition
  4. Specification property
  5. Geometric construction
  6. Motivations of our work
  7. Main results
  8. Comparison with Other Works
  9. Further problems
  10. Preliminaries
  11. Entropy and pressure
  12. Nonuniform hyperbolicity
  13. Partially hyperbolicity
  14. Unstable Entropy Along Expanding Foliations
  15. A construction of $\mathcal{F}$-equilibrium states
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $M$ be a closed Riemannian manifold and $f: M \rightarrow M$ a diffeomorphism satisfying conditions (C1)- (C3), and $\phi:M \to \mathbb{R}$ a continuous function. For any $x\in M$ and $\delta>0$, consider probability measures $\left(\lambda_{n}\right)_{n=1}^{\infty}$ supported on $W^{u}(x,\delt where $\Phi^{u}(x):=-\log \left| \det\left( Df|E^{u}_x \right) \right|$. Define the time-averaged m

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1: PW Variational Principle for pressure
  • Definition 2.3
  • Definition 2.4
  • ...and 52 more