逐点伪轨跟踪性质及其应用

本文给出紧致度量空间逐点伪轨跟踪性质的定义,该定义是伪轨跟踪性质定义的推广.作为应用,证明如下结论:(i)若f具有逐点伪轨跟踪性质,且对任意k∈Z ,fk为链转换的,那么对任意k∈Z ,fk为开集转换;(ii)若f具有逐点伪轨跟踪性质,且对任意n∈Z ,fn为链转换的,则f具有初始敏感依赖性质;(iii)若f为开集混合的,且具有逐点伪轨跟踪性质,那么f具有性质P;(iv)设f:(X,d)→(X,d)是同胚映射,那么f具有逐点伪轨跟踪性质当且仅当移位映射σf具有逐点伪轨跟踪性质.     

伪轨跟踪与完全正熵

本文主要给出了紧致度量空间上的连续满射f具有完全正熵的一个必要条件及当f具有伪轨跟踪性质时,f有完全正熵的一些等价条件．     

逆极限空间的伪轨跟踪性

证明了对于由{xi,φi,fi}∞i=l生成的逆极限系统(X∞,f∞),如果每个fi具有伪轨跟踪性,则诱导映射f∞也具有伪轨跟踪性.并构造了一个例子说明它的逆命题不成立.还证明了零维紧致度量群的自同构拓扑共轭于一族有限型子转移生成的逆极限系统.     

平均伪轨的部分跟踪

引进和研究了q-平均跟踪性质.如果对任意的ε0,存在δ0,使得映射f的每一条δ-平均伪轨都能够被X中的某点沿着下密度大于q的时间集ε-跟踪,则称映射f具有q-平均跟踪性质.证明了:一个映射f有平均跟踪性质当且仅当对任意的q∈[0,1),f有q-平均跟踪性质.指出:对某些p∈[0,1),存在具有p-平均跟踪性质,但没有平均跟踪性质的映射.此外,还讨论了q-平均跟踪的其他一些动力性质及其应用.     

伪轨跟踪与一致正熵

本文对于满足伪轨跟踪性质的紧致度量空间上的连续满射f，给出了f具有一致正熵的几个充要条件和一个必要条件．     

提升和投射具有伪轨跟踪性质或可扩性的连续流

本文证明了紧度量空间上的连续流经提升或投射后，其伪轨跟踪性质及可扩性是不变的；做为应用给出了不定向闭曲面上具有伪轨跟踪性质的Ｃｒ流的特征。     

时变参数动力系统的两种跟踪性质

在时变参数动力系统中引入链传递,伪轨跟踪以及渐进伪轨跟踪的概念,并通过这些概念讨论时变参数动力系统的伪轨跟踪和渐进伪轨跟踪的性质.证明了扩张的时变参数动力系统满足伪轨跟踪性质蕴含其满足渐近伪轨跟踪性质;论证了时变参数动力系统的积系统满足伪轨跟踪和渐进伪轨跟踪性质的充要条件是其每一个分系统也满足相应的性质.最后构造出了一个时变参数动力系统的例子:(∑∞X,F),证明了(∑∞X,F)是拓扑传递的,并且满足渐近伪轨跟踪性质.     

平均跟踪与伪轨跟踪

证明了紧致度量空间中有伪轨跟踪的distal同胚不具有平均跟踪性，并给出了有平均跟踪性的同胚是极小的一个充分条件。     

逆极限空间的伪轨跟踪性

证明了对于由｛Ｘｉ,ｉ,ｆｉ｝ｉ＝１生成的逆极限系统（Ｘ∞,ｆ∞）,如果每个ｆｉ具有伪轨跟踪性,则诱 导映射ｆ∞也具有伪轨跟踪性．并构造了一个例子说明它的逆命题不成立．还证明了零维紧致度量群 的自同构拓扑共轭于一族有限型子转移生成的逆极限系统．     

逐点伪轨跟踪性质与混沌

本文研究了逐点伪轨跟踪性质与拓扑混合等混沌性态的关系,给出了$f$具有逐点伪轨跟踪性质时,$f$具有一致正熵和完全正熵的一些等价条件.     

Recurrence and the asymptotic pseudo-orbit tracing property

In this paper, asymptotic pseudo-orbits and the asymptotic pseudo-orbit tracing property are considered for continuous maps from a compact metric space into itself. We investigate transitive properties and recurrent properties for a map with the asymptotic pseudo-orbit tracing property. In particular, a condition for which a homeomorphism is a non-wandering homeomorphism is obtained.     

Topological Stability and Pseudo-orbit Tracing Property for Homeomorphisms on Uniform Spaces

Acta Mathematica Sinica, English Series - In this paper, we investigate the topological stability and pseudo-orbit tracing property for homeomorphisms on uniform spaces. We introduce the concept of...     

Implications of pseudo-orbit tracing property for continuous maps on compacta

We look at the dynamics of continuous self-maps of compact metric spaces possessing the pseudo-orbit tracing property (i.e., the shadowing property). Among other things we prove the following: (i) the set of minimal points is dense in the non-wandering set Ω(f), (ii) if f has either a non-minimal recurrent point or a sensitive minimal subsystem, then f has positive topological entropy, (iii) if X is infinite and f is transitive, then f is either an odometer or a syndetically sensitive non-minimal map with positive topological entropy, (iv) if f has zero topological entropy, then Ω(f) is totally disconnected and f restricted to Ω(f) is an equicontinuous homeomorphism.     

On homeomorphisms with the two-sided limit shadowing property

We prove that the two-sided limit shadowing property is among the strongest known notions of pseudo-orbit tracing. It implies shadowing, average shadowing, asymptotic average shadowing and specification properties. We also introduce a weaker notion that is called two-sided limit shadowing with a gap and prove that it implies shadowing and transitivity. We show that those two properties allow to characterize topological transitivity and mixing in a class of expansive homeomorphisms and hence they characterize transitive (mixing) shifts of finite type.     

Conditions of C0 flows on closed surfaces having the pseudo-orbit tracing property

Let M be a closed surface, orientable or non-orientable, and letf be a C° flow onM of which all singular points are isolated. Thenf has the pseudo-orbit tracing property if and only if (i) for anyx ∈ M, both the ωlimit set ω(x) and the α-limit set α(x) ofx contain only one orbit; (ii) for any regular pointx off, if ω(x) is not quasi-attracting, then α(x) is quasi-exclusive; (iii) every saddle point off is strict, and at most 4-forked. Project supported by the National Natural Science Foundation of China.     

On Robustness of Orbit Spaces for Partially Hyperbolic Endomorphisms

In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space(the inverse limit space) M~f of f is topologically quasi-stable under C~0-small perturbations in the following sense: For any covering endomorphism g C~0-close to f, there is a continuous map φ from M~g to Multiply form -∞ to ∞ M such that for any {y_i }_(i∈Z) ∈φ(M~g), y_(i+1) and f(y_i) differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense: For any pseudo-orbit {x_i }_(i∈Z),there is a sequence of points {y_i }_(i∈Z) tracing it, in which y_(i+1) is obtained from f(y_i) by a motion along the center direction.     

On partial shadowing of complete pseudo-orbits

In recent years many new definitions of shadowing, using a notion of ergodic (or average) pseudo-orbit were introduced. While, under the assumption of chain mixing, average pseudo-orbit usually can be equivalently replaced by pseudo-orbit in these definitions of shadowing, it is not completely clear when these shadowing properties (i.e. approximation of a pseudo-orbit by a real orbit on a sufficiently large set of indices) can or cannot occur. In this paper we analyze necessary and sufficient conditions for shadowing over a set with positive density (or syndetic). While we do not provide full characterization, a few relations to standard notions from topological dynamics are obtained.     

具有渐近平均跟踪性质的系统

简记渐近平均跟踪性质为AASP.对于紧致度量空间上的连续映射f,证明了:(1)f有AASP当且仅当其逆极限空间上的移位映射有AASP;(2)若f有AASP且是等度连续的,则f是极小同胚.此外,讨论了AASP的拓扑共轭不变性.