TITLE: Entropy of nested compact invariant sets
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: The strategy of Benoît Kloeckner fails for differentiable maps. Indeed if $K$ is a single point and $K_n$ invariant balls around $K$ it implies that $K$ is an attracting fixed point. Therefore the log of the differential of the diffeo $f$ should be close to zero near $K$ and so does the entropy.
However conter-examples in any finite smoothness ($C^r$ maps with $1\leq r<+\infty$)
were given by Misisurewicz in the early seventies :
Diffeomorphism without any measure with maximal entropy, Bull. Acad. Pol. Sci., Ser. sci. math., astr. et phys.  21 (1973), 903--910