Seminar 10.31.19 Downarowicz

Title: Asymptotic Pairs Versus Positivity of Entropy

Speaker: Tomasz Downarowicz –¬†Wroclaw University of Technology

Abstract: Consider a dynamical system (X,T) consisting of a compact metric space X and iterates of a self-homeomorphism T of this space. Topological entropy of the system depends on the speed of growth of complexity of orbits. Zero entropy means that this growth is subexponential and positive entropy corresponds to exponential growth. All this seems quite sophisticated and subtle. On the other hand, there is a very simple-minded notion of an asymptotic pair: two points x, y in X are asymptotic pair if their orbits come closer and closer together as time advances. It might seem surprising but this simple concept SUFFICES to distinguish between positive and zero entropy systems. During my talk I will try to familiarize the audience with the main ideas behind this result (which is due to Blanchard-Host-Ruette in one direction and D. and Lacroix in the other). Moreover, in recent time D., Oprocha and Zhang have obtained a similar criterion for positive entropy in actions of any countable abelian group. If time permits. I will briefly say why this case is very different from the Z-case.