Topological stability in set-valued dynamics (Q520949)

A definition of topological stability for set-valued maps is proposed. Then, the following basic theorems are stated and proved. Theorem 1. Every continuous single-valued map of a metric space, which is topologically stable as a set-valued map, is also topologically stable in the classical sense. Theorem 2. Every topologically stable single-valued map \(f\) of a compact metric space \(X\) satisfies the property that for each \(\varepsilon > 0\) there exists \(\delta> 0\) such that: for every continuous single-valued map \(g:X\to X\) with \(d_H(f,g)< \delta\), there is a continuous map \(h:(\lim_{\leftarrow}g,d^*)\to (\lim_{\leftarrow}f,d^*)\) such that \(D_H(h,\text{Id}_X)< \varepsilon\), \(\sigma_f\circ h=h\circ \sigma_g\). Theorem 3. Every upper semicontinuous positively expansive closed-valued map with the POTP\(_+\) of a compact metric space is topologically stable. Theorem 4. Every topologically stable upper semicontinuous closed-valued map of a compact metric space has the POTP\(_+\); moreover, \(\text{Per}(f)\) is dense in \(\Omega(f)\). All these results are shown to extend the classical single-valued ones.