### On closed manifolds admitting an Anosov diffeomorphism but no expanding map

Ergodic Theory and Dynamical Systems Seminar

19th May 2022, 2:00 pm – 3:00 pm

Fry Building, 2.04

Anosov diffeomorphisms and expanding maps are important types of dynamical systems. They combine properties like structural stability and chaos. In his 1967 survey on differentiable dynamical systems, S. Smale raised the question of which closed manifolds can admit these dynamical systems. As a consequence of his theorem on groups of polynomial growth, M. Gromov proved that all expanding maps on closed manifolds are topologically conjugate to affine infra-nilmanifold endomorphisms. A similar statement has been conjectured for Anosov diffeomorpisms. This points us to studying these dynamical systems on infra-nilmanifolds, on which the existence problem becomes an algebraic one on Lie algebras. This seminar will deal with the related question of which infra-nilmanifolds admit an Anosov diffeomorphism but no expanding map and we present an example of minimal dimension.

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