Non-wandering point

Let us recall the concept of the non-wandering point. It is such a point xeX that for any t0 > 0 (arbitrarily high) and e > 0 (arbitrarily low) there exist such t > t0 and yeX that... 

Theorem 9. Let some system have rj2 slow relaxations. Then we can find a non-wandering point x eX that does not lie in a>T. 

Therefore we shall focus on non-wandering points. Even from the name, one may anticipate a certain recurrence . 

Since the set of wandering points is open, its complement, which is the set of non-wandering points, is closed. We will denote it by Afi. Let us show that it is not empty under our assumptions. First of all, notice that the set of (j-limit points of any semi-trajectory is non-empty. This follows from the compactness of G,... 

The central sub-class of non-wandering points are points which are stable in the sense of Poisson. The main feature of a Poisson-stable point is not only the recurrence of its neighborhood but the recurrence of the trajectory itself. The definition of Poisson-stable points below is different in some ways but equivalent to the definition given in Chap. 1. 

Let us return to the set A4i of non-wandering points. We have established that it is non-empty, closed and invariant (consists of whole trajectories). The set Ail may be regarded as the phase space of a dynamical system, and therefore one may repeat the procedure and construct the set A 2 consisting of non-wandering points in A i. Clearly, Ai2 Q Ai. Just like Ai, the set AA2... 

Theorem 7.7. (Closing lemma, Pugh) Let xq be a non-wandering point of a smooth flow. Then, arbitrarily close in -topology, there exists a smooth flow which has a periodic orbit passing through the point xq... 

In this case, given an arbitrary sequence Tn -> co, one can find a sequence n 00, such that U returns to itself infinitely many times. One may easily see that if a point xq is non-wandering, then x(t, xq) G for all t (—oo, +oo), and any point on the trajectory is non-wandering too. 

Equilibrium states and periodic orbits are non-wandering trajectories. In the former case, any neighborhood of an equilibrium state will contain it forever in the case of a periodic orbit, any of its points returns infinitely many times to an initial neighborhood simply because of periodicity. 

Since each point on a P-trajectory is non-wandering, this result is also valid for points stable in the sense of Poisson. The closing lenuna implies the following meaningful corollary a rough system with a P-trajectory possesses infinitely many periodic orbits. 

Another example is a family of two-dimensional C -smooth diffeomor-phisms whose non-wandering set does not change until the boundary of Morse-Smale diffeomorphisms is reached. The situation is illustrated in Fig. 8.2.3. The two fixed points 0 and O2 have positive multipliers, and Wq contacts Wq along a heteroclinic trajectory, and so do Wq and. This example... 

The closure of an unclosed Poisson-stable trajectory whose return times are unbounded for some e > 0 is called a quasiminimal set. A quasiminimal set contains, besides Poisson-stable trajectories which are dense everywhere in it, some other invariant and closed subsets. These may be equilibrium states, periodic orbits, non-resonant invariant tori, other minimal sets, homoclinic and heteroclinic orbits, etc., among which a P-trajectory is wandering. This gives a clue to why the recurrent times of the non-trivial unclosed P-trajectory are unbounded. Furthermore, this also points out that Poisson-stable trajectories of a quasiminimal set, due to their unpredictable behavior in time, are of... 

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