Departmental Colloquium

Title

Expansivity for the action of general groups

Abstract

S. Schwartzman in his Ph.D. thesis in the 1950s observed a deep result in dynamical systems. It states that a homeomorphism of a compact metric space always has two points whose iteration into the future remain closer than an arbitrary small distance. The same occurs when we consider the past. A notable extension to multidimensional dynamics is due to M. Boyle and D. Lind in 1997. It claims that any $\mathbb{Z}^d$-action (that is, $d$-commuting homeomorphisms) admits a half-space and two different points whose iterations under elements of the half-space remain arbitrarly close. While Schwartzman’s result ensures this asymptotic property for both possible half-spaces of the integers, Boyle and Lind’s result guarantees this property for only one of these half-spaces.

In this talk we develop a geometric framework to address asymptoticity and the related property of nonexpansivity in topological dynamics when the acting group is second countable and locally compact. As an application, we show extensions of Schwartzman’s theorem in this context. Also, we get new results when the acting groups is $\mathbb{Z}^d$: any half-space of Rd contains a vector defining a (oriented) nonexpansive direction in the sense of Boyle and Lind.