Extracting invariant of conjugacy from independence entropy

TL;DR

This paper introduces an invariant version of independence entropy for symbolic dynamical systems, proving it equals topological entropy for sofic shifts but not for all shift spaces, thus refining entropy invariants.

Contribution

It defines a new invariant of independence entropy and establishes its equality with topological entropy for sofic shifts, highlighting differences in general shift spaces.

Findings

Invariant independence entropy equals topological entropy for sofic shifts.

Counterexample shows equality does not hold for all shift spaces.

Provides a new perspective on entropy invariants in symbolic dynamics.

Abstract

The concept of independence entropy for symbolic dynamical systems was introduced in [LMP13]. This notion of entropy measures the extent to which one can freely insert symbols in positions without violating the constraints defined by the shift space. Independence entropy is not invariant under topological conjugacy. We define an invariant version of the independence entropy by setting sup h_{ind}(X) = sup{h_{ind}(Y )|Y ' X}. This invariant is bounded above by the topological entropy. We prove that equality sup h_{ind}(X) = h(X) holds for all sofic shift spaces over Z, then we give an example showing that equality does not hold for general shift spaces.