Unique equilibrium states for Viana maps with small potentials

TL;DR

This paper studies the thermodynamic formalism for Viana maps, proving the existence and uniqueness of equilibrium states for certain potentials and their stability under small perturbations.

Contribution

It establishes the existence, uniqueness, and stability of equilibrium states for Viana maps with small oscillation potentials, extending thermodynamic formalism results.

Findings

Unique equilibrium states exist for small oscillation potentials.

Equilibrium states satisfy a large-deviation principle.

Results are stable under small perturbations of the map.

Abstract

We investigate the thermodynamic formalism for Viana maps-skew products obtained by coupling an expanding circle map with a slightly perturbed quadratic family on the fibers. For every H\"older potential $\varphi$ whose oscillation is below an explicit threshold, we show that an equilibrium state not only exists but is unique and satisfies an upper level-2 large-deviation principle. All of these conclusions persist under sufficiently small perturbations of the reference map.