Shearer's inequality and Infimum Rule for Shannon entropy and topological entropy

TL;DR

This paper explores Shearer's inequality and the infimum rule in the context of Shannon and topological entropy, establishing key inequalities and formulas for amenable group actions and discussing their applicability.

Contribution

It introduces the infimum rule for Shannon entropy derived from Shearer's inequality and extends the discussion to topological entropy for amenable group actions.

Findings

Shearer's inequality holds for disjoint covers in topological entropy.

The infimum formula accurately computes topological entropy for amenable group actions.

Counterexamples show limitations of Shearer's inequality for non-disjoint covers.

Abstract

We review subbadditivity properties of Shannon entropy, in particular, from the Shearer's inequality we derive the "infimum rule" for actions of amenable groups. We briefly discuss applicability of the "infimum formula" to actions of other groups. Then we pass to topological entropy of a cover. We prove Shearer's inequality for disjoint covers and give counterexamples otherwise. We also prove that, for actions of amenable groups, the supremum over all open covers of the "infimum fomula" gives correct value of topological entropy.