Yu ZHANG, Yu Jun ZHU
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 endomorphism then for each $\varepsilon>0$ there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that the induced set-valued map $F_{f,\mathcal{U}}$ has the $\varepsilon$-shadowing property, and moreover, if $f$ is an expanding endomorphism then there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that the induced set-valued map $F_{f,\mathcal{U}}$ 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 $\varepsilon>0$ there exists a $C^1$-neighborhood $\mathcal{U}$ of $f$ such that $h(F_{f,\mathcal{U}}, \varepsilon)=h(f)$; (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.
