Pointwise Pseudo-Orbit Tracing Property and Its Application

In this paper, the pointwise pseudo-orbit tracing property is defined on a compact metric space, and it is a generalization of pseudo-orbit tracing property. As applications, we prove the following results: (i) If / has pointwise pseudo-orbit tracing property, for any k ∈ Z+, and fk is chain transitive, then for any k ∈ Z+, fk has open set transitive ; (ii) If f has pointwise pseudo-orbit tracing property, and for any n ∈ Z+,fn is chain transitive, then f has sensitive dependence on initial conditions; (iii) If f is open set mixing and has pointwise pseudo-orbit tracing property, then f has property P; (iv) Let f : (X, d) →(X, d) be a homeomophism, then f is pointwise pseudo-orbit tracing property if and only if the shift map σf is pointwise pseudo-orbit tracing property.

Distal flows with pseudo-orbit tracing property

In ref. [1] Smale pointed out that the existence of minimal sets is an important problem. The main problem is to determine what kind of spaces are minimal with respect to some flows on it. A comprehensive report of this problem was given in ref. [2]. The distal flows were studied in refs. [3, 4]. Recently, Komuro proved that the

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

In this paper,we investigate the topological stability and pseudo-orbit tracing property for homeomorphisms on uniform spaces.We introduce the concept of topological stability for homeomorphisms on compact uniform spaces and prove that if a homeomorphism on a compact uniform space is expansive and has pseudo-orbit tracing property,then it is topologically stable.Moreover,we discuss the topological stability for homeomorphisms on uniform spaces from the view of localization.We introduce definitions of topologically stable point and shadowable point for homeomorphisms on uniform spaces and show that every shadowable point of an expansive homeomorphism on a compact uniform space is topologically stable.

Necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property

IN this letter we discuss the necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property. According to ref. [1], on a closed surface, every minimal set of a C^r(r≥2) flow is trivial, but it is possible for a C^0 flow to contain non-trivial minimal sets. Thus C^0 flows on closed surfaces are more complicated than C^r(r≥2) flows.

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

Let M be a closed surface,orientable or non-orientable,and let f be a C0 flow on M of which all singular points are isolated.Then f has the pseudo-orbit tracing property if and only if (i) for any x∈M,both the ω-limit set ω(x) and the α-limit set α(x) of x contain only one orbit; (ii) for any regular point x of f,if ω(x) is not quasi-attracting,then α(x) is quasi-exclusive; (iii) every saddle point of f is strict,and at most 4-forked.

An equivalent condition for Anosov maps with zero entropies

Sakai defined the Anosov maps on compact metric spaces. Sun proved that Anosovmaps have the orbit-topological stability, Markov partitions and ζ-funtions. Aformulation for the topological entropy of an Anosov map wes given by Sun in reference. In this note we will also study the topological entropy of an Anosov map, but we willpay attention to the relation between the entropy and the number of periodic points. Thefollowing consequenee will be proved.

Diffeomorphisms with C^1-stably Average Shadowing

Let M be a closed smooth manifold M, and let f ： M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set A of f. We show that if f has the C^1-stably average shadowing property on A, then A admits a dominated splitting.

A kind of symbolic dynamical systems describing non-chaotic maps

1 A kind of non-chaotic dynamical systems on the symbolic space Let A0=A1=A2=…={0,1},and X=multiply from i=0 to ∞ Ai. For every integer k≥2, define metrics dkand dk’ on X by dk（a,b）=max{k-i·|ai-bi|: i=0, 1, 2,…}, dk’（a, b）=sum from i=0 to ∞ k-i·|ai-bi| forany a=（a0, a0, a2,…） and any b=（b0, b1, b2,…）∈X. It is easy to see that all dk anddk’ induce the same topological structure. Thus we now may only consider d≡d2. Thespace （X, d） is usually called a symbolic space. For simplicity, write X for （X, d）. It

STRUCTURAL STABILITY OF SEMIDIFFERENTIABLE DYNAMICAL SYSTEMS

The properties of expanding and contracting invariant sets ot an endomorphism are dis-cussed. The strong hyperbolicity and strong Axiom A property of endomorphisms have beendefined, and an R-stability theorem was proved.