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Topological Stability and Entropy for Certain Set-valued Maps

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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 \({\cal U}\) of f such that the induced set-valued map \({F_{f,{\cal U}}}\) has the ε-shadowing property, and moreover, if f is an expanding endomorphism then there exists a C1-neighborhood \({\cal U}\) of f such that the induced set-valued map \({F_{f,{\cal U}}}\) 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 \({\cal U}\) of f such that \(h({F_{f,{\cal U}}},\varepsilon) = h(f)\) (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.

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  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China

    Yu Zhang & Yu Jun Zhu

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  1. Yu Zhang
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  2. Yu Jun Zhu
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Correspondence to Yu Jun Zhu.

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Conflict of Interest The authors declare no conflict of interest.

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Supported by NSFC (Grant Nos. 11771118, 12171400)

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Zhang, Y., Zhu, Y.J. Topological Stability and Entropy for Certain Set-valued Maps. Acta. Math. Sin.-English Ser. 40, 962–984 (2024). https://doi.org/10.1007/s10114-023-1643-7

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