On robustness of orbit spaces for partially hyperbolic endomorphisms

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) Mf of f is topologically quasi-stable under C0-small perturbations in the following sense: For any covering endomorphism gC0-close to f, there is a continuous map φ from Mg to ∏−∞∞M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop \prod \limits_{ - \infty }^\infty M$$\end{document} such that for any {yi}i∈Z ∈ φ(Mg), yi+1 and f(yi) 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 {xi}i∈ℤ, there is a sequence of points {yi}i∈ℤ tracing it, in which yi+1 is obtained from f(yi) by a motion along the center direction.