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Let $f$ be a homeomorphism on a compact metric space $X$. $K_1\supset K_2\supset\cdots \supset K$ are compact subsets of $X$ such that $f(K_n)=K_n$ and $K=\bigcap K_n$. If $h(f, K_1)<\infty$, do we always have $h(f,K)=\lim h(f,K_n)$?

I can show that this is true for $C^\infty$ diffeomorphisms where the entropy map of invariant measures is upper semi-continuous.

Let $f$ be a homeomorphism on a compact metric space $X$. $K_1\supset K_2\supset\cdots \supset K$ are compact subsets of $X$ such that $f(K_n)=K_n$ and $K=\bigcap K_n$. If $h(f, K_1)<\infty$, do we always have $h(f,K)=\lim h(f,K_n)$?

I can show that this is true for $C^\infty$ diffeomorphisms where the entropy map of invariant measures is upper semi-continuous.

OK. I see the point. This is definitely false for the most general case. For example, take the union of countable hyperbolic toral automorphisms of shrinking size and add one point with Identity map on it.

However, what if $f$ is a diffeomorphism on a compact manifold? I still expect negative answer.

Let $f$ be a homeomorphism on a compact metric space $X$. $K_1\supset K_2\supset\cdots \supset K$ are compact subsets of $X$ such that $f(K_n)=K_n$ and $K=\bigcap K_n$. If $h(f, K_1)<\infty$, do we always have $h(f,K)=\lim h(f,K_n)$?

I can show that this is true for $C^\infty$ diffeomorphisms where the entropy map of invariant measures is semi-upper continuous.

Let $f$ be a homeomorphism on a compact metric space $X$. $K_1\supset K_2\supset\cdots \supset K$ are compact subsets of $X$ such that $f(K_n)=K_n$ and $K=\bigcap K_n$. If $h(f, K_1)<\infty$, do we always have $h(f,K)=\lim h(f,K_n)$?

I can show that this is true for $C^\infty$ diffeomorphisms where the entropy map of invariant measures is upper semi-continuous.