Wu, Yuhu; Xue, Xiaoping Shadowing property for induced set-valued dynamical systems of some expansive maps. (English) Zbl 1219.37017 Dyn. Syst. Appl. 19, No. 3-4, 405-414 (2010). Let \(X\) be a compact metrisable space and \(f\) a continuous map \(X\rightarrow X\). A sequence \(x_i\) is an orbit if \(x_{i+1}=f(x_i)\) and a \(\delta\)-pseudo-orbit if \(d(f(x_i),x_{i+1})<\delta\), \(\delta >0\). The \(\delta\)-pseudo-orbit \(x_i\) is \(\varepsilon\)-shadowed by the orbit \(f^i(y)\) if \(d(f^i(y),x_i)<\varepsilon\) for all \(i\). The authors study a shadowing property for induced set-valued dynamical systems of some expansive maps. They show that if \(f\) is a positively expansive open map, then the induced map has the shadowing property. They prove that ball expansive maps also have the shadowing property. Reviewer: Vladimir P. Kostov (Nice) Cited in 6 Documents MSC: 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems Keywords:pseudo-orbit; shadowing property; induced self-valued map; expansive map; ball expansive map PDFBibTeX XMLCite \textit{Y. Wu} and \textit{X. Xue}, Dyn. Syst. Appl. 19, No. 3--4, 405--414 (2010; Zbl 1219.37017)