Entropy of Toeplitz systems over residually finite groups

TL;DR

This paper investigates the bounds of sofic topological entropy in Toeplitz systems over residually finite groups and proves a theorem on achieving arbitrary entropy levels, advancing understanding of entropy in complex dynamical systems.

Contribution

It introduces bounds for sofic topological entropy in Toeplitz systems and proves the Krieger Theorem in this context, independent of specific sofic approximations.

Findings

Bounded sofic topological entropy for Toeplitz systems.

Proved Krieger Theorem for arbitrary entropy attainment.

Results are formulated independently of natural sofic approximation sequences.

Abstract

The purpose of this work is to bound sofic topological entropy of Toeplitz systems over residually finite groups and to prove the Krieger Theorem about attaining arbitrary entropy by the Toeplitz systems. To achieve these results, we discuss certain properties of the sofic topological entropy in the context of finitely indexed normal subgroups of the group. It will help us to formulate results almost independently of the natural sofic approximation sequences of residually finite groups.