Entropy, Asymptotic pairs and Pseudo-Orbit Tracing for actions of amenable groups

Chung and Li [Invent. Math. 2015] proved that for every expansive action of a countable polycyclic-by-finite group $\Gamma$ on a compact group $X$ by continuous group automorphisms, positive entropy implies the existence of non-diagonal asymptotic pairs. In the same paper they asked if the this holds in general for an expansive action of a countable amenable group $\Gamma$ on a compact space $X$.

In my talk I plan to explain the notions involved Chung and Li‘s question and discuss a property of dynamical systems called the ``pseudo-orbit tracing property‘‘. R. Bowen introduced the pseudo-orbit tracing property in the 1970‘s for $\mathbb{Z}$-actions while studying Axiom A maps. I will prove that Chung and Li‘s question has an affirmative answer if one also assumes pseudo-orbit tracing, and explain implications for algebraic actions (automorphisms of compact abelian groups).

I will also explain why the answer to Chung and Li‘s question is negative if one doesn‘t assume the pseudo-orbit tracing property, even when the acting group is $\mathbb{Z}$, or when the action is algebraic (but not both).