In the present paper we show that a tree map is totally transitive iff it is topologically mixing. Using this result, we prove that the tree maps having a chaotic (or scrambled) subset with full Lebesgue measure is de...In the present paper we show that a tree map is totally transitive iff it is topologically mixing. Using this result, we prove that the tree maps having a chaotic (or scrambled) subset with full Lebesgue measure is dense in the space consisting of all topologically mixing (transitive, respectively) maps.展开更多
In 2007, T Arai and N Chinen proved that every P -chaotic map from a continuum into itself has positive topological entropy and is topologically mixing. In this work, we show that there exists a P -chaotic map defined...In 2007, T Arai and N Chinen proved that every P -chaotic map from a continuum into itself has positive topological entropy and is topologically mixing. In this work, we show that there exists a P -chaotic map defined on a general compact metric space but not on a continuum such that it has zero topological entropy and is not topologically mixing.展开更多
Let X be a metric space. We say that a continuous surjection f:X→X is a topological Anosov map (abbrev. TA map) if f is expansive and has pseudo orbit tracing property with respect to some compatible me...Let X be a metric space. We say that a continuous surjection f:X→X is a topological Anosov map (abbrev. TA map) if f is expansive and has pseudo orbit tracing property with respect to some compatible metric for X . This paper studies the properties of TA maps of non compact metric spaces and gives some conditions for the map to be topologically mixing.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
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
We showed in [G. H], if f is a Cr-topologically mixing map of the circle (r ≥0),then its rotation set will be non-trivial. But this statement is only satisfied for generic(open and dense) subset of all Cr-topological...We showed in [G. H], if f is a Cr-topologically mixing map of the circle (r ≥0),then its rotation set will be non-trivial. But this statement is only satisfied for generic(open and dense) subset of all Cr-topologically mixing maps of the circle (r≥0). Here,we present another proof of the above statement. Moreover, we introduce a topologicallymixing map of the circle with trivial rotation set.展开更多
For a class of mixing transformations of a compact metric space it is proved that each chaoticsubset is'small' but the possibility for any finite subset to display chaotic behavior is 'large'.
Let f:X → X be a positively expansive map and let X be a compact connected metric space. Then f is topologically mixing and f has the pseudo-orbit-tracing property.
文摘In the present paper we show that a tree map is totally transitive iff it is topologically mixing. Using this result, we prove that the tree maps having a chaotic (or scrambled) subset with full Lebesgue measure is dense in the space consisting of all topologically mixing (transitive, respectively) maps.
基金Supported by the Foundation of the Jiangxi Education Department（GJJ11295） Sup- ported by the Natural Science Foundation of Jiangxi Province（20114BAB201006）
文摘In 2007, T Arai and N Chinen proved that every P -chaotic map from a continuum into itself has positive topological entropy and is topologically mixing. In this work, we show that there exists a P -chaotic map defined on a general compact metric space but not on a continuum such that it has zero topological entropy and is not topologically mixing.
文摘Let X be a metric space. We say that a continuous surjection f:X→X is a topological Anosov map (abbrev. TA map) if f is expansive and has pseudo orbit tracing property with respect to some compatible metric for X . This paper studies the properties of TA maps of non compact metric spaces and gives some conditions for the map to be topologically mixing.
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).
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We showed in [G. H], if f is a Cr-topologically mixing map of the circle (r ≥0),then its rotation set will be non-trivial. But this statement is only satisfied for generic(open and dense) subset of all Cr-topologically mixing maps of the circle (r≥0). Here,we present another proof of the above statement. Moreover, we introduce a topologicallymixing map of the circle with trivial rotation set.
文摘For a class of mixing transformations of a compact metric space it is proved that each chaoticsubset is'small' but the possibility for any finite subset to display chaotic behavior is 'large'.
文摘Let f:X → X be a positively expansive map and let X be a compact connected metric space. Then f is topologically mixing and f has the pseudo-orbit-tracing property.
