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Limit Sets of Typical Homeomorphisms

Published online by Cambridge University Press:  20 November 2018

Nilson C. Bernardes Jr.
Nilson C. Bernardes Jr.*
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ, 21945-970, Brasil e-mail: bernardes@im.ufrj.br
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Abstract

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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 $x$ .

Keywords

37B2054H2028C1554C3554E52topological manifoldshomeomorphismsmeasuresBaire categorylimit sets

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012, pp. 225 - 232
Copyright
Copyright © Canadian Mathematical Society 2012

References

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