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Abstract

In this paper, we study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology group has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than the modulus of the largest eigenvalue, then the map is robustly transitive. For this, we introduce Blichfedt’s theorem as a tool for extracting dynamical information from the action of a map in homology. We also treat the case of specially partially hyperbolic endomorphisms, for which we obtain a complete dichotomy: either the map is transitive and conjugated to its linear part, or its unstable foliation must contain an annulus which may either be wandering or periodic.

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... Notice that the work of Martin Andersson and Wagner Ranter [AR24] shows that, if we assume further than the assumptions of Problem 1 that det Df (x) > σ u , for every x ∈ T 2 , then f is locally eventually onto (for all U ⊆ M non-empty open set there is N ∈ N such that f N (U ) = M ), and in particular strongly transitive. Even under this stronger assumption, we do not know the answer to Problem 1. ...
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W. Ranter. About transitivity of surface endomorphisms admitting critical points. PhD Thesis, Instituto de Matemática Pura e Aplicada, 2017.
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  • G Hector
  • U Hirsch
G. Hector and U. Hirsch. Introduction to the Geometry of Foliations, Part A: Foliations on Compact Surfaces, Fundamentals for Arbitrary Codimension, and Holonomy (Aspects of Mathematics, 1), 2nd edn. Vieweg, Braunschweig, 1986.