Relative entropy, topological pressure and variational principle for locally compact sofic group actions

TL;DR

This paper extends classical entropy and pressure concepts to locally compact sofic group actions, establishing variational principles and inequalities, and providing conditions for invariant measures.

Contribution

It generalizes classical results by developing relative entropy, pressure, and variational principles for locally compact sofic groups acting on compact spaces.

Findings

Proved an additive inequality relating sofic entropy and relative sofic entropy.

Established the variational principle for topological pressure in this context.

Provided a sufficient condition for a signed measure to be G-invariant.

Abstract

For a locally compact sofic group continuously acting on a compact metric space, we first study the relative sofic entropy and prove an additive inequality relating sofic entropy and relative sofic entropy. Moreover, it is shown that the relative variational principle remains valid in this paper. Secondly, the topological pressure for locally compact sofic group actions is investigated and the variational principle for topological pressure in this sofic context is established. As an application, we show a sufficient condition for a signed measure to be a $G$-invariant measure. These contributions generalize the classical results for countable sofic groups on such spaces.