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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.
37C15Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
28B20Set-valued set functions and measures; integration of set-valued functions; measurable selections
37F15Expanding maps; hyperbolicity; structural stability