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Positively \(N\)-expansive homeomorphisms and the L-shadowing property. (English) Zbl 1415.37013

Fix a metric space \((X,d)\). A continuous \(f:X\to X\) is positively \(n\)-expansive if there is \(c>0\) such that \(|W_c^s(x)|\le n\) for each \(x\in X\), where \(W_c^s(x)=\{y\in X\ /\ d(f^k(x),f^k(y))\le c \text{ for each } k\ge0\}\). Assume that \(X\) is compact. A homeomorphism \(h:X\to X\) has the L-shadowing property if for each \(\varepsilon>0\) there is \(\delta>0\) such that every \(\delta\)-pseudo orbit \(\langle x_k\rangle_{k\in\mathbb Z}\) (i.e., \(d(h(x_k),x_{k+1})<\delta\) for each \(k\)) that is also a two-sided limit pseudo-orbit (i.e., \(d(h(x_k),x_{k+1})\to0\) as \(|k|\to\infty\)) there is \(z\in X\) that \(\varepsilon\)-shadows (i.e., \(d(h^k(z),x_k)<\varepsilon\) for each \(k\)) and two-sided limit shadows (i.e., \(d(h^k(z),x_k)\to0\) as \(|k|\to\infty\)) the pseudo-orbit \(\langle x_k\rangle\). If a positively \(n\)-expansive homeomorphism on a compact metric space \(X\) either has the L-shadowing property or is transitive and admits the shadowing property then \(X\) is finite.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
54H20 Topological dynamics (MSC2010)
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