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- Title
Topological Stability and Entropy for Certain Set-valued Maps.
- Authors
Zhang, Yu; Zhu, Yu Jun
- Abstract
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 of f such that the induced set-valued map F f , U has the ε-shadowing property, and moreover, if f is an expanding endomorphism then there exists a C1-neighborhood U of f such that the induced set-valued map F f , 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 ε> 0 there exists a C1-neighborhood U of f such that h (F f , U , ε) = 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.
- Subjects
TOPOLOGICAL entropy; SET-valued maps; DIFFERENTIABLE dynamical systems; ENDOMORPHISMS
- Publication
Acta Mathematica Sinica, 2024, Vol 40, Issue 4, p962
- ISSN
1439-8516
- Publication type
Article
- DOI
10.1007/s10114-023-1643-7