It is well known that, for a dynamical system $T$ on a metric space $(X,d)$, the variational principle connects the definition of metric entropy and topological entropy. In other words, if $$M(X,T) := \{ \mu\,\, \text{probability measure} : \mu= T_*\mu \} $$ is the set of invariant measures for $T$, then

$$h_\text{top}( T)= \sup_{\mu \in M(X,T)} h_{\mu}(T) $$

where $h_\text{top}(T)$ is the topological entropy and $h_{\mu}(T)$ is the metric entropy relative to $\mu$.

I have seen somewhere that, if we denote by $E(X,T) \subset M(X,T)$ the set of invariant ergodic measures for $T$, then

$$h_\text{top}(T)= \sup_{\mu \in E(X,T)} h_{\mu}(T) $$

My questions are: is this true? If it is true, how is it proven?

  • 1
    $\begingroup$ Have you checked in Entropy in Dynamical Systems by Tomasz Downarowicz? I think his Theorem 6.8.1 is what you are after. $\endgroup$ Nov 10, 2021 at 20:29
  • $\begingroup$ thanks! I did not know the book $\endgroup$ Nov 11, 2021 at 13:10

1 Answer 1


It's true for a very simple reason: the entropy of a dynamical system with respect to a (not necessarily ergodic) invariant measure is the average of the entropies of its ergodic components.


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