Recurrent point

of a dynamical system

A point of a dynamical system (also denoted by , see [2]) in a metric space that satisfies the following condition: For any there exists a such that all points of the trajectory are contained in an -neighbourhood of any arc of time length of this trajectory (in other words, with any , the -neighbourhood of the set

contains all of the trajectory ). In this case is called a recurrent trajectory.

Birkhoff's theorem: If the space is complete (e.g. ), then: 1) for a point to be recurrent it is necessary and sufficient that the closure of its trajectory be a compact minimal set; and 2) for the existence of a recurrent point it is sufficient that there be a point that is stable according to Lagrange (see Lagrange stability).

A recurrent point is stable according to Poisson (see Poisson stability), and if the space is complete, also stable according to Lagrange (see Lagrange stability). An almost-periodic (in particular, a fixed or periodic) point of a dynamical system is recurrent. In general, any point of a strictly ergodic dynamical system (in a complete space) is recurrent, but the restriction of a dynamical system to the closure of a recurrent trajectory (a minimal set) need not be a strictly ergodic dynamical system (Markov's example, see [2]).

References

[1]	G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927)
[2]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)

Comments

An almost-periodic point of a dynamical system on a metric space is a point with the following property: For every the set

is relatively dense in , i.e., there exists a length such that every interval in with length contains a point of . (Thus, one might say that the function is almost-periodic; cf. Almost-period.)

Another important notion is that of an almost-recurrent point: A point such that for every the set

is relatively dense in , where is the open -ball around . (This definition can easily be generalized to the case of a flow or a cascade on an arbitrary topological space.) The relationship of these notions with that of a recurrent point, Birkhoff's theorem and a number of other implications can be visualized as follows:

Figure: r080190a

Here the implication indicated by the dotted arrow holds only in a complete space, and denotes the closure of the trajectory of . The property "Sx is Lyapunov stable rel(ative) Sx" means that the family of functions from into is equicontinuous on (see Lyapunov stability). For a refinement of this diagram (including the notions of a pseudo-recurrent point and a uniformly Poisson-stable trajectory) see [a3].

In the literature on topological dynamics (in particular, in the literature directly or indirectly influenced by [a2]) another terminology is in use:'

<tbody> </tbody> Above, [2], [a3] [a2] [a1] almost periodic — — recurrent — — almost recurrent almost periodic uniformly recurrent Poisson-stable recurrent recurrent (non-) wandering (not) regionally recurrent —

(To be precise, in [a1] "recurrent" means positive Poisson stable, i.e., belongs only to the -limit set of its own trajectory.)

References

[a1]	H. Furstenberg, "Recurrence in ergodic theory and combinatorial number theory" , Princeton Univ. Press (1981)
[a2]	W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955)
[a3]	K.S. [K.S. Sibirskii] Sibirsky, "Introduction to topological dynamics" , Noordhoff (1975) (Translated from Russian)