Non-uniformly hyperbolic endomorphisms

Journal article

Martin Andersson, Pablo D. Carrasco, Radu Saghin

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Andersson, M., Carrasco, P. D., & Saghin, R. Non-uniformly hyperbolic endomorphisms. ArXiv:2206.08295.

Andersson, Martin, Pablo D. Carrasco, and Radu Saghin. β€œNon-Uniformly Hyperbolic Endomorphisms.” arXiv:2206.08295 (n.d.).

Andersson, Martin, et al. β€œNon-Uniformly Hyperbolic Endomorphisms.” ArXiv:2206.08295.

We show that in nearly every homotopy class of any non-invertible endomorphism of the two-torus there exists a C1 open set of non-uniformly hyperbolic area preserving maps (one positive and one negative exponent at Lebesgue almost every point), without dominated splitting. Moreover, these maps are continuity points of the (averaged) Lyapunov exponents and, under a mild assumption on their linear part, they are also stably ergodic: any C2 conservative C1 nearby map is ergodic, and in fact metrically isomorphic to a Bernoulli shift.