Consider a compact convex subset X of Rn (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).
Homeomorphisms orbits periodic points sensitivity to initial conditions attractors