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Shadowing property for induced set-valued dynamical systems of some expansive maps. (English) Zbl 1219.37017

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.

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
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