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Strong shadowing property on a weak form hyperbolic set. (Chinese. English summary) Zbl 1199.37042

Summary: In this paper, we give a kind of \(C^1\) map on a weak form hyperbolic set, i.e. Steinlein-hyperbolic set, via the notion of strong shadowing, we obtain a generalized shadowing lemma. Let \(H\) be a Banach space, \(\phi: V\rightarrow H\) a \(C^1\) map on an open subset \(V\) of \(H\), then the \(C^1\) map of \(\phi\) has the strong shadowing property on \(T\). Assuming that \(T\subset V\) is a Steinlein-hyperbolic set of \(\phi\).

MSC:

37D05 Dynamical systems with hyperbolic orbits and sets
37C75 Stability theory for smooth dynamical systems
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