On robustness of orbit spaces for partially hyperbolic endomorphisms

Abstract

In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space (the inverse limit space) M ^f of f is topologically quasi-stable under C ⁰-small perturbations in the following sense: For any covering endomorphism g C ⁰-close to f, there is a continuous map φ from M ^g to $\mathop \prod \limits_{ - \infty }^\infty M$ such that for any {y _i} _i∈Z ∈ φ(M ^g), y _i+1 and f(y _i) differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense: For any pseudo-orbit {x _i} _i∈ℤ, there is a sequence of points {y _i} _i∈ℤ tracing it, in which y _i+1 is obtained from f(y _i) by a motion along the center direction.

Keywords

Partially hyperbolic endomorphism / Orbit space / Quasi-stability / Quasishadowing

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