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

Xianqiang Li, Zhuowei Liu

TL;DR

This work extends core entropy and pressure concepts from countable sofic groups to actions of locally compact sofic groups on compact metric spaces. It introduces two equivalent definitions of relative sofic topological entropy, proves an additive inequality linking absolute and relative entropies, and establishes a relative variational principle for both entropy and measure entropy. The paper then defines sofic topological pressure for locally compact group actions and proves a variational principle that expresses pressure as a supremum of entropies plus integrals of a potential over invariant measures, along with foundational properties of the pressure functional. A practical consequence is a sufficient condition for a signed measure to be $G$-invariant, tying together the thermodynamic formalism with invariant-measure theory in the locally compact, sofic setting. Overall, these results generalize classical findings for countable sofic groups and broaden applicability to continuous group actions in dynamical systems and ergodic theory.

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.

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

TL;DR

This work extends core entropy and pressure concepts from countable sofic groups to actions of locally compact sofic groups on compact metric spaces. It introduces two equivalent definitions of relative sofic topological entropy, proves an additive inequality linking absolute and relative entropies, and establishes a relative variational principle for both entropy and measure entropy. The paper then defines sofic topological pressure for locally compact group actions and proves a variational principle that expresses pressure as a supremum of entropies plus integrals of a potential over invariant measures, along with foundational properties of the pressure functional. A practical consequence is a sufficient condition for a signed measure to be -invariant, tying together the thermodynamic formalism with invariant-measure theory in the locally compact, sofic setting. Overall, these results generalize classical findings for countable sofic groups and broaden applicability to continuous group actions in dynamical systems and ergodic theory.

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 -invariant measure. These contributions generalize the classical results for countable sofic groups on such spaces.

Paper Structure

This paper contains 9 sections, 23 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological dynamical systems and factor maps
  4. Local $G$-spaces and locally compact sofic groups
  5. Relative sofic topological entropy
  6. Relative sofic measure entropy and variational principle
  7. Topological pressure and the variational principle
  8. Variational principle
  9. Properties of sofic topological pressure

Key Result

Proposition 2.3

Let $(M, \alpha)$ be a local $G$-space. Fix a right-Haar measure $\mathrm{Haar}$ on $G$. Then there exists a unique Radon measure $\mathrm{vol}_M$ on $M$ satisfying the following. If $p \in M$, $f_p$ is a chart centered at $p$ and $K \subset \operatorname{dom}(f_p)$ is Borel then We write $\mathrm{vol}$ instead of $\mathrm{vol}_M$, when the choice of $M$ is clear.

Theorems & Definitions (63)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 53 more