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 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 given.Furthermore,the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.

Forward Expansiveness and Entropies for Subsystems of Z^(k)_(+)-actions

In this paper,forward expansiveness and entropies of"subsystems"2)of Z^(k)_(+)-actions are investigated.Letαbe a Z^(k)_(+)-action on a compact metric space.For each 1≤j≤k-1,denote G^(j)_(+)={V+:=V∩R^(k)_(+):V is a j-dimensional subspace of R^(k)}.We consider the forward expansiveness and entropies forαalong V+∈G^(j)_(+).Adapting the technique of"coding",which was introduced by M.Boyle and D.Lind to investigate expansive subdynamics of Z^(k)-actions,to the Z^(k)_(+)cases,we show that the set E^(j)_(+)(α)of forward expansive j-dimensional V_(+)is open in G^(j)_(+).The topological entropy and measure-theoretic entropy of j-dimensional subsystems ofαare both continuous in E^(j)_(+)(α),and moreover,a variational principle relating them is obtained.For a 1-dimensional ray L∈G^(+)_(1),we relate the 1-dimensional subsystem ofαalong L to an i.i.d.random transformation.Applying the techniques of random dynamical systems we investigate the entropy theory of 1-dimensional subsystems.In particular,we propose the notion of preimage entropy(including topological and measure-theoretical versions)via the preimage structure ofαalong L.We show that the preimage entropy coincides with the classical entropy along any L∈E1+(α)for topological and measure-theoretical versions respectively.Meanwhile,a formula relating the measure-theoretical directional preimage entropy and the folding entropy of the generators is obtained.

Shadowing and Inverse Shadowing for C^1 Endomorphisms

In this paper, we consider the shadowing and the inverse shadowing properties for C^1 endomorphisms. We show that near a hyperbolic set a C^1 endomorphism has the shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also ＂uniform＂ with respect to C^1 perturbation.

Two Notes on Measure-Theoretic Entropy of Random Dynamical Systems

In this paper, Brin-Katok local entropy formula and Katok＇s definition of the measuretheoretic entropy using spanning set are established for the random dynamical system over an invertible ergodic system.

Quasi-shadowing Property on Random Partially Hyperbolic Sets

Abstract In this paper we investigate the quasi-shadowing property for C1 random dynamical sys-terns on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk}＋∞ -∞ on a random partially hyperbolic set there exists a ＂center＂ pseudo orbit {Yk}＋∞ -∞ shadowing it in the sense that yk＋l is obtained from the image of yk by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above ＂center＂ pseudo orbit{yk}＋∞ -∞ can be chosen such that yk＋1 and the image of yk lie in their common center leaf.

On an Entropy of Zk＋-Actions

In this paper,a definition of entropy for Z＋k（k≥2）-actions due to Friedland is studied.Unlike the traditional definition,it may take a nonzero value for actions whose generators have finite（even zero） entropy as single transformations.Some basic properties are investigated and its value for the Z＋k-actions on circles generated by expanding endomorphisms is given.Moreover,an upper bound of this entropy for the Z＋k-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies,which are entropy-like invariants depending on the ＂inverse orbits＂ structure of the system.