Unique Equilibrium States for Viana Maps for Small Potentials

Abstract: We study the thermodynamic formalism for Viana maps (skew products that couple an expanding circle map with a small perturbation of a quadratic map on the fibers). Working within the Climenhaga–-Thompson framework, we show that for every Hölder potential whose oscillation is below an explicit threshold, there is a unique equilibrium state. The main step is a uniform control of recurrence to the critical region in the fibers, where the derivative degenerates. This yields the pressure gap and the specification estimates needed to apply the method and removes the principal obstruction. These conclusions are robust under sufficiently small perturbations of the reference map.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar