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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.
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참고문헌 (16)
참고문헌 신청
A. Artigue / 2013 / Hyper-expansive homeomorphisms / Publ. Mat. Urug 14 : 62 ~ 66
A. Artigue / 2013 / N-expansive homeomorphisms on surfaces
C. A. Morales / 2011 / Measure expansive systems / IMPA
C. A. Morales / 2012 / A generalization of expansivity / Discrete Contin. Dyn. Syst 32 (1) : 293 ~ 301
C. A. Morales / 2013 / Expansive Measures / IMPA, 29o Col´oq. Bras. Mat
A. Artigue / 2013 / N-expansive homeomorphisms on surfaces
C. A. Morales / 2011 / Measure expansive systems / IMPA
C. A. Morales / 2012 / A generalization of expansivity / Discrete Contin. Dyn. Syst 32 (1) : 293 ~ 301
C. A. Morales / 2013 / Expansive Measures / IMPA, 29o Col´oq. Bras. Mat
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