, Volume 3, Issue 2, pp 149172
On the Dynamics of Homeomorphisms on the Unit Ball of Rn
 Nilson C. BernardesJr.Affiliated withInstituto de Matemática, Universidade Federal Fluminense
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Consider a compact convex subset X of R ^{n} (n ≥ 2) with nonempty interior and let H(X) be the set of all homeomorphisms from X onto X endowed with the supremum metric. We are interested in studying the dynamics of functions in H(X) from the following point of view: Which properties are satisfied by ''most'' functions in H(X), in the sense that the set of all functions in H(X) that do not satisfy the given property is of the first category? We prove that most functions in H(X) have uncountably many periodic points of period m, for each m ≥ 1, but have no attractive cycles. Also, for most functions f ≥ H(X), the set of all periodic points of f has no isolated points, is nowhere dense, has infinitely many connected components, is nowhere closed, is dense in the set of all nonwandering points of f, and has Lebesgue measure zero. Moreover, most functions in H(X) are not sensitive to initial conditions on any subset of X that is somewhere dense, but are sensitive to initial conditions on an uncountable closed connected subset of X. Finally, we prove that most functions in H(X) have infinitely many pairwise disjoint uniform attractors with certain properties, but have no attractors with a dense orbit (hence, no strange attractors).
 Title
 On the Dynamics of Homeomorphisms on the Unit Ball of Rn
 Journal

Positivity
Volume 3, Issue 2 , pp 149172
 Cover Date
 199906
 DOI
 10.1023/A:1009750622797
 Print ISSN
 13851292
 Online ISSN
 15729281
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Homeomorphisms
 orbits
 periodic points
 sensitivity to initial conditions
 attractors
 Authors

 Nilson C. Bernardes Jr. ^{(1)}
 Author Affiliations

 1. Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga s/n, 24020140, Niterói, RJ, Brasil