On equilibrium states for certain partially hyperbolic endomorphisms with one-dimensional center

TL;DR

This paper investigates the uniqueness and robustness of equilibrium states for certain partially hyperbolic endomorphisms with one-dimensional center, using inverse limit techniques and criteria from recent dynamical systems research.

Contribution

It establishes conditions under which equilibrium states are unique and robust for a class of non-degenerate $C^2$ partially hyperbolic endomorphisms with one-dimensional center.

Findings

Proves uniqueness of equilibrium states under specified conditions.

Demonstrates robustness of equilibrium states against perturbations.

Applies inverse limit techniques to analyze equilibrium states.

Abstract

In this paper, the equilibrium states for a non-degenerate $ C^2 $ partially hyperbolic endomorphism $f$ on a closed Riemannian manifold $M$ with one-dimensional center bundle are investigated. Applying the criterion of Climenhaga-Thompson (\cite{CT21}) and the method of Mongez-Pacifico (\cite{Mongez}), we use the techniques of inverse limit to obtain the uniqueness and robustness of equilibrium states for $f$ and any H\"{o}lder continuous potential satisfying certain conditions about the unstable pressure and stable pressure.