Let (X, d) be a bounded metric space and f : X → X be a uniformly continuous surjection. For a given dynamical system (X, f) which may not be compact, we investigate the relation between the asymptotic average shadow...Let (X, d) be a bounded metric space and f : X → X be a uniformly continuous surjection. For a given dynamical system (X, f) which may not be compact, we investigate the relation between the asymptotic average shadowing property(AASP), transitivity and mixing. If f has the AASP, then the following statements hold: (1) f n is chain transitive for every positive integer n; (2) If X is compact and f is an expansive homeomorphism, then f is topologically weakly mixing; (3) If f is equicontinuous, then f is topologically weakly mixing; (4) If X is compact and f is equicontinuous, then f ×f is a minimal homeomorphism. We also show that the one-sided shift map has the AASP and the identity map 1 X does not have the AASP. Furthermore, as its applications, some examples are given.展开更多
1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved....1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]). It is necessary to know how the subsets of X moved. In this direction, we consider the set-valued discrete system associated to f, An+1 = (f|-)(An), n = 0,1,2,... where (f|-) is the natural extension of f to K(X) (the class of all compact subsets of X).展开更多
基金Supported by the NSF of Guangdong Province(10452408801004217)Supported by the Key Scientific and Technological Research Project of Science and Technology Department of Zhanjiang City(2010C3112005)
文摘Let (X, d) be a bounded metric space and f : X → X be a uniformly continuous surjection. For a given dynamical system (X, f) which may not be compact, we investigate the relation between the asymptotic average shadowing property(AASP), transitivity and mixing. If f has the AASP, then the following statements hold: (1) f n is chain transitive for every positive integer n; (2) If X is compact and f is an expansive homeomorphism, then f is topologically weakly mixing; (3) If f is equicontinuous, then f is topologically weakly mixing; (4) If X is compact and f is equicontinuous, then f ×f is a minimal homeomorphism. We also show that the one-sided shift map has the AASP and the identity map 1 X does not have the AASP. Furthermore, as its applications, some examples are given.
文摘1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]). It is necessary to know how the subsets of X moved. In this direction, we consider the set-valued discrete system associated to f, An+1 = (f|-)(An), n = 0,1,2,... where (f|-) is the natural extension of f to K(X) (the class of all compact subsets of X).
