On quasi-weakly almost periodic points

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.     

Banach upper density recurrent points of <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-flows

Let X denote a compact metric space with distance d and F:X×R→X or F_t:X→X denote a C⁰-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C⁰-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.     

On Quasi-weakly Almost Periodic Points of Continuous Flows

Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of this paper is to investigate whether there exists an invariant measure generated by the orbit of x such that the support of this measure coincides with the minimal center of attraction of x? In order to solve the problem, two continuous flows are constructed. In one continuous flow,there exist a proper quasi-weakly almost periodic point and an invariant measure generated by its orbit such that the support of this measure coincides with its minimal center of attraction; and in the other,there is a proper quasi-weakly almost periodic point such that the support of any invariant measure generated by its orbit is properly contained in its minimal center of attraction. So the mentioned problem is sufficiently answered in the paper.     

一个A（f）≠M（f）的紧致系统（X，f）

本文证明，存在紧致系统，其几乎周期点集闭包不等于其测度中心．藉此，我们否定地回答了两个未解决的问题．     

TOPOLOGICAL ENTROPY AND IRREGULAR RECURRENCE

This paper is devoted to problems stated by Z. Zhou and F. Li in 2009. They concern relations between almost periodic, weakly almost periodic, and quasi-weakly almost periodic points of a continuous map $f$ and its topological entropy. The negative answer follows by our recent paper. But for continuous maps of the interval and other more general one-dimensional spaces we give more results; in some cases the answer is positive.     

Topological conjugation and asymptotic stability in impulsive semidynamical systems

We prove several results concerning topological conjugation of two impulsive semidynamical systems. In particular, we prove that the homeomorphism which defines the topological conjugation takes impulsive points to impulsive points; it also preserves limit sets, prolongational limit sets and properties as the minimality of positive impulsive orbits as well as stability and invariance with respect to the impulsive system. We also present the concepts of attraction and asymptotic stability in this setting and prove some related results.     

Tilings,substitution systems and dynamical systems generated by them

The object of this work is to study the properties of dynamical systems defined by tilings. A connection to symbolic dynamical systems defined by one- and two-dimensional substitution systems is shown. This is used in particular to show the existence of a tiling system such that its corresponding dynamical system is minimal and topological weakly mixing. We remark that for one-dimensional tilings the dynamical system always contains periodic points.     

Almost periodically forced circle flows

We study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li-Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2,R)-valued, continuous or discrete cocycle.Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multi-frequency-driven dynamical irregularities and complexities in non-monotone dynamical systems.     

Topological pressure of continuous flows without fixed points

In this paper, we give the equivalent definitions of topological pressure for flows by using spanning sets, weakly spanning sets, strongly separated sets and tracing sets, respectively. We get an inequality between the topological pressures of Lipschitz conjugate flows, and prove that the topological pressure of expansive flows with tracing property can be described by its periodic orbits.     

A note on quasi-weakly almost periodic point

Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597–606(2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in[Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493–502(2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.