Limit set

In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system.
Contents
Types
 fixed points
 periodic orbits
 limit cycles
 attractors.
In general limits sets can be very complicated as in the case of strange attractors, but for 2dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all possible limit sets as a union of fixed points and periodic orbits.
Definition for iterated functions
Let X be a metric space, and let be a continuous function. The ωlimit set of , denoted by ω(x,f), is the set of cluster points of the forward orbit of the iterated function f. Hence, if and only if there is a strictly increasing sequence of natural numbers such that as . Another way to express this is
The points in the limit set are nonwandering (but may not be recurrent points).
If f is a homeomorphism (that is, a bicontinuous bijection), then the αlimit set is defined in a similar fashion, but for the backward orbit; i.e. α(x,f) = ω(x,f ^{− 1}).
Both sets are finvariant, and if X is compact, they are compact and nonempty.
Definition for flows
Given a real dynamical system (T, X, φ) with flow , a point x and an orbit γ through x, we call a point y an ωlimit point of γ if there exists a sequence in R so that
 .
Analogously we call y an αlimit point if there exists a sequence in R so that
 .
The set of all ωlimit points (αlimit points) for a given orbit γ is called ωlimit set (αlimit set) for γ and denoted lim_{ω} γ (lim_{α} γ).
If the ωlimit set (αlimit set) is disjunct from the orbit γ, that is lim_{ω} γ ∩ γ = ∅ (lim_{α} γ ∩ γ = ∅) , we call lim_{ω} γ (lim_{α} γ) a ωlimit cycle (αlimit cycle).
Alternatively the limit sets can be defined as
and
Examples
 For any periodic orbit γ of a dynamical system, lim_{ω} γ = lim_{α} γ = γ
 For any fixed point x_{0} of a dynamical system, lim_{ω} x_{0} = lim_{α} x_{0} = x_{0}
Properties
 lim_{ω} γ and lim_{α} γ are closed
 if X is compact then lim_{ω} γ and lim_{α} γ are nonempty, compact and connected
 lim_{ω} γ and lim_{α} γ are φinvariant, that is φ(R × lim_{ω} γ) = lim_{ω} γ and φ(R × lim_{α} γ) = lim_{α} γ
See also
 Julia set
 Stable set
 Limit cycle
 Periodic point
 Nonwandering set
 Kleinian group
This article incorporates material from Omegalimit set on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
Categories: Limit sets
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