Institute of Mathematics


Modul:   MAT760  Ergodic Theory and Dynamical Systems Seminar

Homoclinic points of algebraic dynamical systems

Prof. Dr. Klaus Schmidt talk

Date: 06.05.19  Time: 14.00 - 15.00  Room: ETH HG G 43

An algebraic action of a countable discrete group $\Gamma $ is an action of $\Gamma $ by automorphisms of a compact abelian group $X$. Classical examples are $Z^d$-actions by (commuting) toral automorphisms, Ledrappier’s Example, or analogous examples of shift-actions of $Z^2$ on closed, shift-invariant subgroup of $(R/Z)^{Z^2}$ (which may or may not have positive entropy). For groups bigger than $Z$, algebraic actions offer one of the key playgrounds for the study of positive entropy ergodic actions of such groups. For an algebraic action $\alpha \colon \gamma \to \alpha ^\gamma $ of a group $\Gamma $ on a compact abelian group $X$, a \textit{homoclinic point} is a nonzero point $x\in X$ for which $\alpha ^\gamma x$ converges to $0$ as $\gamma \to\infty $. Such a point is ‘summable’ if $\alpha ^\gamma x —> 0$ sufficiently fast as $\gamma —> \infty $. The existence of a summable homoclinic point implies positive entropy of the action, and for expansive actions the reverse implication holds under quite general conditions. For nonexpansive actions, the existence of summable homoclinic points is much more mysterious, but implies many nice dynamical properties such actions. This talk is based on results and examples by Martin G\"{o}ll, Hanfeng Li, Doug Lind, Evgeny Verbitskiy, and KS.