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Tuesday, February 4, 2020

**Abstract:** A dynamical system is a pair $(X, G)$ where $X$ is a compact metrizable space and $G$ is a countable group acting on $X$ by homeomorphisms. An endomorphism of $(X, G)$ is a continuous map from $X$ to $X$ which commutes with the action of $G$. A dynamical system is surjunctive if every injective endomorphism is surjunctive, and therefore a homeomorphism. Sofic groups were introduced by Gromov and Weiss as a generalization of both residually finite groups and amenable groups. A celebrated theorem of Gromov (and Weiss) is that if $A$ is a finite set and $G$ is sofic, then $(AG, G)$ is surjunctive. In recent work with Michel Coornaert and Hanfeng Li we generalize the Gromov-Weiss theorem to show that every dynamical system $(X, G)$ with certain suitable properties is surjunctive.