Open Zooming Systems: Uniqueness of Equilibrium States

TL;DR

This paper investigates the uniqueness of equilibrium states in open zooming systems, establishing conditions under which uniqueness holds, and demonstrating the density of potentials with unique equilibrium states.

Contribution

It introduces new conditions for uniqueness of equilibrium states in open zooming systems and proves the density of such potentials among continuous potentials.

Findings

Uniqueness of equilibrium states is established for certain classes of zooming potentials.

The set of potentials with unique equilibrium states is dense among continuous potentials.

Results extend previous work on finiteness and stability of equilibrium states.

Abstract

We study open zooming systems and potentials with uniqueness of equilibrium states. The uniqueness is established for a certain class of zooming potentials when the map is topologically exact, including the null one. Also, with equilibrium stability, we prove that there exists a countable and an open sets of continuous potentials with uniqueness which are both dense in the set of continuous potentials with finiteness. The results here are related to the works [14] and [49] where finiteness and stability are studied.