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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 XX. A sequence x i is an orbit if x i+1 =f(x i ) and a δ-pseudo-orbit if d(f(x i ),x i+1 )<δ, δ>0. The δ-pseudo-orbit x i is ε-shadowed by the orbit f i (y) if d(f i (y),x i )<ε 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:
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