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| On Robustness of Orbit Spaces for Partially Hyperbolic Endomorphisms |
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Citation： |
Lin WANG.On Robustness of Orbit Spaces for Partially Hyperbolic Endomorphisms[J].Chinese Annals of Mathematics B,2016,37(6):899~914 |
| Page view： 1641
Net amount： 1254 |
Authors： |
Lin WANG; |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (No.11371120), the High-level Personnel for
Institutions of Higher Learning in Hebei Province (No.GCC2014052)
and the Natural Science Foundation of Hebei Province
(No.A2013205148). |
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| 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 $\varphi$ from $M^g$ to
$\prod\limits_{-\infty}^{\infty}M$ such that for any $\{y_i\}_{i\in
\mathbb{Z}}\in\varphi(M^g)$, $y_{i+1}$ 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 $\{x_i\}_{i\in \mathbb{Z}}$, there is a sequence of
points $\{y_i\}_{i\in \mathbb{Z}}$ 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,
Quasi-shadowing |
Classification： |
37D30, 37C05, 37C15 |
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