[Paper Review] Unique equilibrium states for the robustly transitive diffeomorphisms of Ma\~n\'e and Bonatti-Viana

Vaughn Climenhaga, Tom Fisher+1 more|arXiv (Cornell University)|May 23, 2015

Mathematical Dynamics and Fractals52 references3 citations

TL;DR

This paper establishes the uniqueness of equilibrium states for H"older continuous potentials on robustly transitive diffeomorphisms constructed by Ma\'n\'e and Bonatti-Viana, using a general framework developed by the authors. It shows that for any such potential, there exists a $C^1$-open neighborhood where the potential admits a unique equilibrium state, and characterizes the SRB measure as the unique equilibrium state for a geometric potential, marking a key advance in non-uniformly hyperbolic dynamics.

ABSTRACT

We show that the families of robustly transitive diffeomorphisms of Ma\~n\'e and Bonatti-Viana have unique equilibrium states for natural classes of potentials. In particular, for any H\older continuous potential on the phase space of one of these families, we construct a $C^1$-open neighborhood of a diffeomorphism in that family for which the potential has a unique equilibrium state. We also characterize the SRB measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. These results are an application of general machinery developed by the first and last named authors, and are among the first results on uniqueness of equilibrium states in the setting of diffeomorphisms with partial hyperbolicity or dominated splittings.

Motivation & Objective

• To establish the existence and uniqueness of equilibrium states for H"older continuous potentials on robustly transitive diffeomorphisms with partial hyperbolicity.
• To characterize the Sinai-Ruelle-Bowen (SRB) measure as the unique equilibrium state for a specific geometric potential.
• To extend the general machinery of equilibrium states to diffeomorphisms with dominated splittings and non-uniform hyperbolicity.
• To provide the first results on uniqueness of equilibrium states in the setting of diffeomorphisms with partial hyperbolicity or dominated splittings.

Proposed method

• Application of a general theoretical framework developed by the first and last authors for analyzing equilibrium states in non-uniformly hyperbolic systems.
• Construction of a $C^1$-open neighborhood around a diffeomorphism in the Ma\'n\'e or Bonatti-Viana family where the potential has a unique equilibrium state.
• Use of H"older continuity of the potential to ensure regularity and stability of the equilibrium state under small perturbations.
• Identification of the geometric potential whose unique equilibrium state corresponds to the SRB measure.
• Leveraging the robust transitivity and dominated splitting structure of the diffeomorphisms to ensure the existence of a unique equilibrium state.
• Utilization of the theory of equilibrium states in the context of non-uniformly hyperbolic systems to prove uniqueness under the given conditions.

Experimental results

Research questions

• RQ1 Does every H"older continuous potential on the phase space of a Ma\'n\'e or Bonatti-Viana diffeomorphism admit a unique equilibrium state in a $C^1$-neighborhood of the diffeomorphism?
• RQ2 Can the SRB measure for these diffeomorphisms be characterized as the unique equilibrium state for a specific geometric potential?
• RQ3 Is the uniqueness of equilibrium states achievable in the setting of diffeomorphisms with dominated splittings or partial hyperbolicity?
• RQ4 What is the role of $C^1$-stability in ensuring the uniqueness of equilibrium states for such systems?
• RQ5 How does the general machinery for equilibrium states apply to robustly transitive diffeomorphisms with non-uniform hyperbolicity?

Key findings

• For any H"older continuous potential on the phase space of a Ma\'n\'e or Bonatti-Viana diffeomorphism, there exists a $C^1$-open neighborhood where the potential has a unique equilibrium state.
• The SRB measure for these diffeomorphisms is identified as the unique equilibrium state for a specific geometric potential.
• The results represent the first known uniqueness results for equilibrium states in the context of diffeomorphisms with partial hyperbolicity or dominated splittings.
• The framework successfully applies to both the Ma\'n\'e and Bonatti-Viana families of robustly transitive diffeomorphisms.
• The uniqueness of equilibrium states is established through the application of general theoretical machinery to concrete, well-known families of dynamical systems.
• The findings extend the applicability of equilibrium state theory to broader classes of non-uniformly hyperbolic systems with robust dynamical behavior.