Topological Stability and Entropy for Certain Set-valued Maps

In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if f is a hyperbolic endomor-phism then for each ε> 0 there exists a C1-neighborhood U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal U}$$\end{document} of f such that the induced set-valued map Ff,U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_{f,{\cal U}}}$$\end{document} has the ε-shadowing property, and moreover, if f is an expanding endomorphism then there exists a C1-neighborhood U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal U}$$\end{document} of f such that the induced set-valued map Ff,U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_{f,{\cal U}}}$$\end{document} has the Lipschitz shadowing property; (2) when a set-valued map F is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable; (3) if f is an expanding endomorphism then for each ε> 0 there exists a C1-neighborhood U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal U}$$\end{document} of f such that h(Ff,U,ε)=h(f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h({F_{f,{\cal U}}},\varepsilon) = h(f)$$\end{document} (4) when F is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of F is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.