On Robustness of Orbit Spaces for Partially Hyperbolic Endomorphisms

In this paper, the robustness of the orbit structure is investigatedfor a partially hyperbolic endomorphism $f$ on a compact manifold$M$. It is first proved that the dynamical structure of its orbitspace (the inverse limit space) $M^f$ of $f$ is topologicallyquasi-stable under $C^0$-small perturbations in the following sense:For any covering endomorphism $g$ $C^0$-close to $f$, there is acontinuous map $varphi$ from $M^g$ to$prodlimits_{-infty}^{infty}M$ such that for any ${y_i}_{iinmathbb{Z}}invarphi(M^g)$, $y_{i+1}$ and $f(y_i)$ differ only by amotion along the center direction. It is then proved that $f$ hasquasi-shadowing property in the following sense: For anypseudo-orbit ${x_i}_{iin mathbb{Z}}$, there is a sequence ofpoints ${y_i}_{iin mathbb{Z}}$ tracing it, in which $y_{i+1}$ isobtained from $f(y_i)$ by a motion along the center direction.