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Topological Stability and Entropy for Certain Set-valued Maps

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Acta Mathematica Sinica,English Series 2024年第4期40卷 962-984页

作者： Yu ZHANG Yu Jun ZHU School of Mathematical Sciences Xiamen UniversityXiamen 361005P.R.China

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 *** is shown that(1)if f is a hyperbolic endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has theε-shadowing property,and moreover,if f is an expanding endomorphism then there exists a C^(1)-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 eachε>0 there exists a C^(1)-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 ***,the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.

关键词： Set-valued map orbit space hyperbolic endomorphism perturbation shadowing expansiveness topological stability entropy

On the Centralizers of Rescaling Separating Differentiable Vector Fields

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Acta Mathematica Sinica,English Series 2024年第11期40卷 2671-2683页

作者： Bo HAN Xiao WEN School of Mathematical Sciences Beihang UniversityBeijing100191P.R.China LMIB Institute of Artificial Intelligence&School of Mathematical SciencesBeihang UniversityBeijing100191P.R.China Beijing Zhongguancun Laboratory Beijing100094P.R.China

In this paper,we introduce a new concept of expansiveness,similar to the separating ***,we consider a compact Riemannian manifold M without boundary and a C^(1)vector field X on M,which generates a flowφ_(t)on *** say that X is rescaling separating on a compact invariant setΛof X if there is a constantδ>0 such that,for any x,y∈Λ,if d(φ_(t)(x),φ_(t)(y))≤δ∥X(φ_(t)(x))∥for all t∈R,then y∈Orb(x).We prove that if X is rescaling separating onΛand every singularity of X inΛis hyperbolic,then any C^(1)vector field Y,whose flow commutes withφ_(t)onΛ,must be collinear to X onΛ.As applications of this result,we show that the centralizer of a rescaling separating C^(1)vector field without nonhyperbolic singularity is *** also proved that there is an open and dense set u⊂χ^(1)(M)such that for any star vector fieldχ∈u,the centralizer of X is collinear to X on the chain recurrent set of X.

关键词： Centralizer expansiveness hyperbolicity

Topological Stability and Pseudo-orbit Tracing Property for Homeomorphisms on Uniform Spaces

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Acta Mathematica Sinica,English Series 2022年第2期38卷 431-442页

作者： Ke Song YAN Fan Ping ZENG School of Information and Statistics Guangxi University of Finance and EconomicsNanning 530003P.R.China

In this paper,we investigate the topological stability and pseudo-orbit tracing property for homeomorphisms on uniform *** introduce the concept of topological stability for homeomorphisms on compact uniform spaces and prove that if a homeomorphism on a compact uniform space is expansive and has pseudo-orbit tracing property,then it is topologically ***,we discuss the topological stability for homeomorphisms on uniform spaces from the view of *** introduce definitions of topologically stable point and shadowable point for homeomorphisms on uniform spaces and show that every shadowable point of an expansive homeomorphism on a compact uniform space is topologically stable.

关键词： Topological stability pseudo-orbit tracing property expansiveness uniform space

The nonexistence of sensitive commutative group actions on graphs

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Science China Mathematics 2007年第8期50卷 1197-1204页

作者： Jie-hua MAI~1 En-hui SHI~(2+) 1 Institute of Mathematics,Shantou University,Shantou 515063,China 2 School of Mathematical Sciences,Suzhou University,Suzhou 215006,China Institute of Mathematics Shantou University Shantou 515063 China School of Mathematical Sciences Suzhou University Suzhou 215006 China

In this paper,we observe a special kind of continuous functions on *** estimating the integrals of these functions,we prove that there are no sensitive commutative group actions on ***,we consider a 1-dimensional continuum composed of a spiral curve and a circle and show that there exist sensitive homeomorphisms on it,which answers negatively a question proposed by Kato in 1993.

关键词： homeomorphism commutative group action graph map sensitivity expansiveness infinite spiral curve 54H20

INVARIANT OF TOPOLOGICAL PRESSURE UNDER SOME SEMI-CONJUGATES

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Applied Mathematics(A Journal of Chinese Universities) 1997年第3期12卷 355-362页

作者： HELIANFA LiHONG SUNWINXIANG DepartmentofMathematics HebeiNormalUniversityShijiazhuang050016. DepartmentofMathematics HebeiNormalUniversityShijiazhuang050016 DepartmentofMathematics PekingUniversityBeijlng100871.

In this paper, we will discuss the construction problems about the invariant sets and invariant measures of continuous maps which map complexes into themselves, using simplicial approximation and Markov chains. In , the author defined a matrix by using r normal subdivision of the n dimensional unit cube, considered it a Markov matrix, and constructed the invariant set and invariant measure. In fact, the matrix he defined is not Markov matrix generally. So we will give and amendment in the last part of this paper. We also construct an invariant set that is the chain recurrent set of the map by means of a non negative matrix which only depends on the map. At last, we will prove the higher dimension Banach variation formula that can simplify the transition matrix.

关键词： Topological pressure inverse limit expansiveness specification