Refinement of Open Cover has Greater Entropy

Theorem

Let $X$ be a topological Space.

Let $\alpha, \beta$ be open covers of $X$.

Let $\map H \alpha$ and $\map H \beta$ be their entropies.

Suppose that $\beta$ is a refinement of $\alpha$.

Then:

$\map H \alpha \le \map H \beta$

Proof

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Sources

• 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.): Chapter $7$: Topological Entropy