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Abstract In this work, we deal with a notion of partially hyperbolic endomorphism. We explore topological properties of this definition and we obtain, among other results, obstructions to get center leaf conjugacy with the linear part, for a class of partially hyperbolic endomorphism C 1 -sufficiently close to a hyperbolic linear endomorphism. Indeed such obstructions are related to the number of center directions of a point. We provide examples illustrating these obstructions. We show that for a manifold M with dimension n ⩾ 3, admitting a non-invertible partially hyperbolic endomorphisms, there is a C 1 open and dense subset U of all partially hyperbolic endomorphisms with degree d ⩾ n , such that any f ∈ U is neither c nor u special.
Costa et al. (Wed,) studied this question.