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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the predictability of discrete dynamical systems


Author: Nilson C. Bernardes Jr.
Journal: Proc. Amer. Math. Soc. 130 (2002), 1983-1992
MSC (2000): Primary 37B25, 37B20, 54H20
Published electronically: November 21, 2001
MathSciNet review: 1896031
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Abstract: Let $X$ be a metric space. A function $f: X \to X$ is said to be non-sensitive at a point $a \in X$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that for any choice of points $a_0 \in B(a;\delta)$, $a_1 \in B(f(a_0);\delta)$, $a_2 \in B(f(a_1);\delta),\ldots$, we have that $d(a_m,f^m(a)) < \epsilon$ for every $m \geq 0$. Let $H(X)$ be the set of all homeomorphisms from $X$ onto $X$ endowed with the topology of uniform convergence. The main goal of the present paper is to prove that for certain spaces $X$, ``most'' functions in $H(X)$ are non-sensitive at ``most'' points of $X$.


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Additional Information

Nilson C. Bernardes Jr.
Affiliation: Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga s/n, 24020-140, Niterói, RJ, Brasil
Email: bernardes@mat.uff.br

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06247-5
PII: S 0002-9939(01)06247-5
Keywords: Homeomorphisms, predictability, recurrence, Baire category, Lebesgue measure
Received by editor(s): August 10, 1999
Received by editor(s) in revised form: January 16, 2001
Published electronically: November 21, 2001
Communicated by: Alan Dow
Article copyright: © Copyright 2001 American Mathematical Society