Let $X=([0,1],f)$ be a dynamical system. If $f$ has positive topological entropy on a proper subset $J$ of $[0,1]$, what can we say about the topological entropy of $f$ on $[0,1]$ (Full space)? Does it remain positive in full space? If yes, why?
2026-03-27 01:46:44.1774576004
What can we say about the topological entropy of a map on full space X?
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The entropy of a subset is always at most the entropy of the whole space (for the same dynamics). This is because you can cover any subset with any cover of the whole space, but not necessarily the other way around. So yes.