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:'
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|
(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) |
Recurrent point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Recurrent_point&oldid=15436