DCDS-B
Topological stability in set-valued dynamics
Roger Metzger Carlos Arnoldo Morales Rojas Phillipe Thieullen
Discrete & Continuous Dynamical Systems - B 2017, 22(5): 1965-1975 doi: 10.3934/dcdsb.2017115

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the set-valued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

keywords: Topological stability set-valued map metric space

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