Finite Sets with Fake Observable Cardinality

Abstract

Let $X$ be a compact metric space and let $|A|$ denote the cardinality of a set $A$. We prove that if $f\colon X\to X$ is a homeomorphism and $|X|=\infty$ then for all $δ>0$ there is $A\subset X$ such that $|A|=4$ and for all $k\in Z$ there are $x,y\in f^k(A)$, $x\neq y$, such that $dist(x,y)<δ$. An observer that can only distinguish two points if their distance is grater than $δ$, for sure will say that $A$ has at most 3 points even knowing every iterate of $A$ and that $f$ is a homeomorphism. We show that for hyper-expansive homeomorphisms the same $δ$-observer will not fail about the cardinality of $A$ if we start with $|A|=3$ instead of $4$. Generalizations of this problem are considered via what we call $(m,n)$-expansiveness.