Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

A generalized shadowing lemma

Pages: 627 - 632, Volume 8, Issue 3, July 2002      doi:10.3934/dcds.2002.8.627

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Shaobo Gan - School of Mathematical Science, Peking University, Beijing 100871, China (email)

Abstract: In this paper, we prove a generalized shadowing lemma. Let $f \in$ Diff$(M)$. Assume that $\Lambda$ is a closed invariant set of $f$ and there is a continuous invariant splitting $T\Lambda M = E\oplus F$ on $\Lambda$. For any $\lambda \in (0, 1)$ there exist $L > 0, d_0> 0$ such that for any $d \in (0, d_0]$ and any $\lambda$-quasi-hyperbolic d-pseudoorbit $\{x_i, n_i\}_{i=-\infty}^\infty$, there exists a point $x$ which Ld-shadows $\{x_i, n_i\}_{i=-\infty}^\infty$. Moreover, if $\{x_i, n_i\}_{i=-\infty}^\infty$ is periodic, i.e., there exists an $m > 0$ such that $x_{i+m}= x_i$ and $n_{i+m} = n_i$ for all $i$, then the point $x$ can be chosen to be periodic.

Keywords:  Pseudo-orbit, shadowing property, quasi-hyperbolic.
Mathematics Subject Classification:  37D30.

Available Online: April 2002.