真拟弱几乎周期点和拟正则点

T为紧致度量空间X上的连续映射,M（X）为X上所有Borel概率测度.设x∈X,记Mx（T）为概率测度序列{1n∑n 1i=0δTi（x）}在M（X）中的极限点的集合,其中δx表示支撑集是{x}的点测度.记W（T）和QW（T）分别为T的弱几乎周期点和拟弱几乎周期点集.本文证明,如果（X,T）非平凡且满足specifcation性质,则存在x,y∈QW（T）／W（T）（称为真拟弱几乎周期点）,分别满足μ∈Mx（T）,x∈Supp（μ）和ν∈My（T）,y∈/Supp（ν）,回答了周作领等提出的公开问题.Mx（T）在弱拓扑中是紧致连通集,所以,要么是单点集,要么是不可数集.如果x∈QW（T）／W（T）,则Mx（T）是不可数集.一个自然的问题是,怎么刻画M x（T）是单点集的点x（这时x称为拟正则点）.本文给出M x（T）是单点集的充要条件.     

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.     

External rays to periodic points

We prove that for every polynomial-like holomorphic mapP, ifaεK (filled-in Julia set) and the componentK _aofK containinga is either a point ora is accessible along a continuous curve from the complement ofK andK _ais eventually periodic, thena is accessible along an external ray. Ifa is a repelling or parabolic periodic point, then the set of arguments of the external rays converging toa is a nonempty closed “rotation set”, finite (ifK _ais not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in ℂ. In the Appendix we discuss constructions via cutting and glueing, fromP to its external map with a “hedgehog”, and backward. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis, sponsored by the Minerva Foundation (Germany). Supported by the Polish KBN Grants 210469101 “Iteracje i Fraktale” and 210909101 “Uklady Dynamiczne”.     

Homotopical theory of periodic points of periodic homeomorphisms on closed surfaces

In this work we show that the Wecken theorem for periodic points holds for periodic homeomorphisms on closed surfaces, which therefore completes the periodic point theory in such a special case. Using it we derive the set of homotopy minimal periods for such homeomorphisms. Moreover we show that the results hold for homotopically periodic self-maps of closed surfaces. This let us to re-formulate our results as a statement on properties of elements of finite order in the group of outer automorphisms of the fundamental group of a surface with non-positive Euler characteristic.     

Turning points for adiabatically perturbed periodic equations

We make a study of uniform asymptotic solutions of some general adiabatic differential equations on the intervals containing a single turning point or a pair of turning points. We reduce this to the study of models which can be explicitly solved. We apply these results to the case of an adiabatically perturbed differential equation with periodic coefficients.     

Entropy and periodic points for transitive maps

The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the -star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.

     

The structure of graph maps without periodic points

In this paper we show that any graph map without periodic points has only one minimal set. We describe a class of graph maps without periodic points. Our main result is to give a structure theorem of graph maps without periodic points, which states that any graph map without periodic points must be topologically conjugate to one of the described class. In addition, we give some applications of the structure theorem.     

Boundedness results for periodic points on algebraic varieties

We give some conditions under which the periods of a self map of an algebraic variety are bounded.     

Diffeomorphisms of the plane with stable periodic points

We consider self-diffeomorphisms of the plane with a hyperbolic fixed point and a nontransversal homoclinic point. We show that a neighborhood of the homoclinic point may contain countably many stable periodic sets whose characteristic exponents are bounded away from zero.     

On fixed and periodic points of quasi-contractive mappings

Some theorems on fixed, common fixed and periodic points for quasi-contractive mappings have been obtained.