# On the Dynamics of Homeomorphisms on the Unit Ball of Rn

DOI: 10.1023/A:1009750622797

- Cite this article as:
- Bernardes, N.C. Positivity (1999) 3: 149. doi:10.1023/A:1009750622797

## Abstract

Consider a compact convex subset *X* of **R**^{n} (n ≥ 2) with non-empty 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 non-wandering 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).