I will discuss the relation between positive entropy and the existence of non-diagonal homoclinic pairs for expansive algebraic actions of amenable groups. The main tool is the strong topological Markov property. It turns out that these two conditions are equivalent for expansive finitely presented algebraic actions of amenable groups which satisfy the strong Atiyah conjecture and have a bound for the orders of finite subgroups. This is joint work with Sebastian Barbieri and Felipe Garcia-Ramos.

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