Santiago Dynamical Systems Seminar: “Generic invariant measures beyond specification”

Abstract: 

Which properties hold for a generic invariant measure of a continuous map T from a compact metric space X to itself? We prove the genericity of rank one measures under a relaxed specification hypothesis (the vague specification or the approximate product property), sharpening Sigmund’s classical genericity theorem. The method applies, in particular, to certain minimal and proximal systems. As a by-product, every measure produced by the Gorodetski–Ilyashenko–Kleptsyn–Nalsky construction of non-uniformly hyperbolic ergodic invariant measures is rank one.

The key new tools are an approximation scheme for the natural extension of T, inspired by the classical approximation of a subshift by a descending sequence of shifts of finite type, and notions of convergence of invariant measures generalising Ornstein’s d-bar metric and Feldman’s f-bar metric. Outside the symbolic setting, these notions of convergence are new and allow several classical results from the theory of d-bar and f-bar to be transferred to general topological dynamical systems.

Based on joint work with M. E. Can, J. Konieczny, M. Kupsa; with D. Buldağ and B. Jacelon; and with M. Łącka and A. Trilles.

Speaker: Dominik Kwietniak (Jagiellonian University)
Venue: Room 2, Faculty of Mathematics, Pontificia Universidad Católica