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# Limit Sets of Typical Homeomorphisms

Published:2011-04-14
Printed: Jun 2012
• Nilson C. Bernardes,
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ, 21945-970, Brasil
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## Abstract

Given an integer $n \geq 3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f \colon X \to X$, it is true that for $\mu$-almost every point $x$ in $X$ the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero, each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is non-sensitive at each point of $\omega(f,x)$, and the function $a \to \omega(f,a)$ is continuous at $x$.
 Keywords: topological manifolds, homeomorphisms, measures, Baire category, limit sets
 MSC Classifications: 37B20 - Notions of recurrence 54H20 - Topological dynamics [See also 28Dxx, 37Bxx] 28C15 - Set functions and measures on topological spaces (regularity of measures, etc.) 54C35 - Function spaces [See also 46Exx, 58D15] 54E52 - Baire category, Baire spaces
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