Refinement of Open Cover has Greater Entropy
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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