Finite sets with fake observable cardinality
Bull. Korean Math. Soc. 2015 Vol. 52, No. 1, 323-333
https://doi.org/10.4134/BKMS.2015.52.1.323
Printed January 31, 2015
Alfonso Artigue
Universidad de la Rep\'ublica
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 $\delta>0$ 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 $\dist(x,y)<\delta$. An observer that can only distinguish two points if their distance is grater than $\delta$, 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 $\delta$-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.
Keywords : topological dynamics, expansive homeomorphisms
MSC numbers : 54H20, 37B05
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