Variational principle for Markov operators

Establish a variational principle for Markov operators that relates the metric entropy (as characterized by axioms in Downarowicz and Frej, 2005) to the topological entropy (the three equivalent notions defined therein), i.e., determine whether a variational equality between these metric and topological notions holds for Markov operators.

Background

Downarowicz and Frej introduced an axiomatic framework for metric entropy of Markov operators and three equivalent notions of topological entropy, proving that these topological entropies coincide and bound the metric entropy from above.

The existence of a full variational principle analogous to the classical topological/metric entropy variational principle for dynamical systems has remained unresolved, and the authors explicitly note this as an open question.