Remetrization and a New Type of Recurrence

In topological dynamics, various types of recurrence has been defined.Each type of recurrence is a generalization of periodicity ([3], [4]) and is closely related ([1], [H]) to some order relation ([9], [12]).The aim of this paper is to define a new type of recurrence and to establish its basic properties.Our new type of recurrence is a modification of Conley's chain recurrence ([6]), in fact, it is defined mathrmb.mathrmvweakening the requirements in Conley's definition.On the other hand, our new type of recurrence still implies recurrence in the sense of Auslander.Throughout this paper, let X be a locally compact separable metrizable space and let $T:Rtimes Xrightarrow X$ , (t, x)rightarrow Tₜx=T(t, x) be a dynamical system on X .The positive semitrajectory through x is denoted by gamma⁺(x) .The first prolongation and the first prolongational limit set of mathrmx are defined by D₁(x)=yin X| there exist sequences xₙ₁ⁱⁿᶠᵗʸsubset X , tₙ₁ⁱⁿᶠᵗʸsubset R⁺ such that xₙrightarrow x , T_tₙxₙrightarrow y and J₁(x)=capD₁(Tₜx)|tgeq 0 , respectively.The basic dynamical concepts are defined in terms of a compatible metric $d-$ however, in case of com- pact metrizable spaces, they do not depend on the choice of this particular metric.This is no longer true if the compactness assumption is dropped ([3]).Our new type of recurrence is defined in terms of a compatible metric as well.But, even though in the compact case, our new type of recurrence is not independent on the choice of this particular metric.1.Auslander recurrence Define a quasi order (i.e. a reflexive and transitive relation) $dashv$ on X by letting $ydashv x$ if and only if yingamma⁺(x) .The transitive closure of $dashv$ i.e. the intersection of all closed quasi orders containing $dashv$ is denoted by precA (a quasi order $.$ $prec$ is closed if (x, y)in Xtimes X|xprec y is a closed subset in the product topology).Using the notion of higher prolongations and higher prolongational limit sets, precA can be constructed explicitely via transfinit induction ([1]).For any $xin X$ , define D(x)=yin X|yprecAx , J(x)=capD(Tₜx)|tgeq 0 .It is well-known that D₁(x)subset D(x) , J₁(x)subset J(x) , D(x)=gamma⁺(x)cup J(x) and that for every t , $sin R$ , $yin J(x)$