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FINITE SETS WITH FAKE OBSERVABLE CARDINALITY

  • Artigue, Alfonso
  • Received : 2014.02.18
  • Published : 2015.01.31

Abstract

Let X be a compact metric space and let |A| denote the cardinality of a set A. We prove that if $f:X{\rightarrow}X$ is a homeomorphism and ${\mid}X{\mid}={\infty}$, then for all ${\delta}$ > 0 there is $A{\subset}X$ such that |A| = 4 and for all $k{\in}\mathbb{Z}$ 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 hyperexpansive 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

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Cited by

  1. Levels of Generalized Expansiveness vol.29, pp.3, 2017, https://doi.org/10.1007/s10884-015-9502-6