Abstract view
Limit Sets of Typical Homeomorphisms


Published:20110414
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, 21945970, Brasil
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
nonsensitive at each point of $\omega(f,x)$, and the function
$a \to \omega(f,a)$ is continuous at $x$.
MSC Classifications: 
37B20, 54H20, 28C15, 54C35, 54E52 show english descriptions
Notions of recurrence Topological dynamics [See also 28Dxx, 37Bxx] Set functions and measures on topological spaces (regularity of measures, etc.) Function spaces [See also 46Exx, 58D15] Baire category, Baire spaces
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
