Equivalence of topological entropy definitions for Markov operators via partitions of unity

Determine whether the topological entropy for Markov operators defined using continuous partitions of unity coincides with the topological entropy notions introduced by Downarowicz and Frej (2005).

Background

The paper suggests extending its partition-of-unity-based definitions of metric and topological entropy to Markov operators. The metric version aligns with the axioms of Downarowicz and Frej, and it is not difficult to show the new topological entropy is bounded above by their topological entropy (notably h₂).

The authors explicitly pose the question of whether the partition-of-unity-based topological entropy agrees exactly with the established definitions of topological entropy for Markov operators.