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On a manifold X of dimension at least two, let μ be a nonatomic measure of full support with μ(∂X) = 0. The Oxtoby-Ulam Theorem says that ergodicity of μ is a residual property in the group of homeomorphisms which preserve μ. Daalderop and Fokkink have recently shown that density of periodic points is residual as well. We provide a proof of their result which replaces the dependence upon the Annulus Theorem by a direct construction which assures topologically robust periodic points.
We study generalized recurrence for closed relations on locally compact spaces. This includes continuous maps and real flows. The main tools are Lyapunov functions and their compactifications. Under certain conditions it is shown that the Lyapunov functions determine the topology of the space.
For measures on a Cantor space, the demand that the measure be "good" is a useful homogeneity condition. We examine the question of when a Bernoulli measure on the sequence space for an alphabet of size n is good. Complete answers are given for the n = 2 cases and the rational cases. Partial results are obtained for the general cases.
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