Equilibrium Stability for Open Zooming Systems

TL;DR

This paper proves the stability and uniqueness of equilibrium states in a broad class of open zooming dynamical systems with holes and general potentials, including skew-products, ensuring continuous dependence on dynamics.

Contribution

It establishes equilibrium stability and uniqueness for open zooming systems with general contractions and potentials, extending previous results to more complex systems.

Findings

Equilibrium states depend continuously on the dynamics and potential.

Stability results hold for systems with special holes and general contractions.

Uniqueness of equilibrium state is guaranteed under the studied conditions.

Abstract

We prove that for a wide family of open zooming systems and zooming potentials we have equilibrium stability, i.e., the equilibrium states depend continuously on the dynamics and the potential. We consider the open zooming systems with special holes and quite general contractions and zooming potentials with locally H\"older induced potential, which include the H\"older ones. We also prove stability for skew-products with the base being a zooming system like above. As a consequence of finiteness and stability, we obtain uniqueness of equilibrium state.