Zero topological entropy is equivalent to unique ergodicity for every dynamical system with the approximate product property, with a minimality version under a stronger condition.
Equilibrium states of intermediate entropies
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abstract
We explore an approach to the conjecture of Katok on intermediate entropies that based on uniqueness of equilibrium states, provided the entropy function is upper semi-continuous. As an application, we prove Katok's conjecture for Ma\~n\'e diffeomorphisms.
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2019 1verdicts
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Unique ergodicity for zero-entropy dynamical systems with the approximate product property
Zero topological entropy is equivalent to unique ergodicity for every dynamical system with the approximate product property, with a minimality version under a stronger condition.