Limit Sets of Typical Homeomorphisms

Jr., Nilson C. Bernardes

Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 225-232

Résumé

Given an integer $n\,\ge \,3$ , 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 $f\,:\,X\,\to \,X$ , 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 $x$ .

Détail

DOI : 10.4153/CMB-2011-066-2

Mots-clés : 37B20, 54H20, 28C15, 54C35, 54E52, topological manifolds, homeomorphisms, measures, Baire category, limit sets