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Erschienen in:
01.10.2021
Quasi-Shadowing for Partially Hyperbolic Flows with a Local Product Structure
Erschienen in: Journal of Dynamical and Control Systems | Ausgabe 1/2023
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Abstract
Let a flow ϕt be partially hyperbolic on Λ. If Λ has a local product structure, then ϕt has the quasi-shadowing property on Λ in the following sense: for any 𝜖 > 0, there exists constant δ > 0 such that for any (δ,1)-pseudo orbit \(\{x_{k}, t_{k}\}_{k\in \mathbb {Z}}\) of ϕt with 1 ≤ tk ≤ 2 for all \(k\in \mathbb {Z}\), there exist a sequence of points \(\{y_{k}\}_{k\in \mathbb {Z}}\) and a reparametrization \(\alpha \in Rep(\mathbb {R},\epsilon )\) such that \(\phi _{\alpha (t)-\alpha ({\Sigma }_{k})}(y_{k})\) trace \(\phi _{t-{\Sigma }_{k}}(x_{k})\) in which yk+ 1 lies in the local center leaf of \(\phi _{\alpha ({\Sigma }_{k+1})-\alpha ({\Sigma }_{k})}(y_{k})\) for k ≥ 0 t ≥ 0 and \(\phi _{\alpha (t)-\alpha (-{\Sigma }_{k})}(y_{k})\) trace \(\phi _{t-(-{\Sigma }_{k})}(x_{k})\) in which yk+ 1 lies in the local center leaf of \(\phi _{\alpha (-{\Sigma }_{k+1})-\alpha (-{\Sigma }_{k})}(y_{k})\) for k < 0, t < 0.