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.
The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.
In this work,by virtue of the properties of weakly almost periodic points of a dynamical system(X,T) with at least two points,the authors prove that,if the measure center M(T) of T is the whole space,that is,M(T) = X,then the following statements are equivalent:(1)(X,T) is ergodic mixing;(2)(X,T) is topologically double ergodic;(3)(X,T) is weak mixing;(4)(X,T) is extremely scattering;(5)(X,T) is strong scattering;(6)(X×X,T×T) is strong scattering;(7)(X×X,T×T) is extremely scattering;(8) For any subset S of N with upper density 1,there is a c-dense Fσ-chaotic set with respect to S.As an application,the authors show that,for the sub-shift σA of finite type determined by a k×k-(0,1) matrix A,σA is strong mixing if and only if σA is totally transitive.
