# 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.

## Types

In general limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional 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

$\omega (x,f)=\bigcap _{n\in \mathbb {N} }{\overline {\{f^{k}(x):k>n\}}},$ where ${\overline {S}}$ denotes the closure of set $S$ . The closure is here needed, since we have not assumed that the underlying metric space of interest to be a complete metric space. The points in the limit set are non-wandering (but may not be recurrent points). This may also be formulated as the outer limit (limsup) of a sequence of sets, such that

$\omega (x,f)=\bigcap _{n=1}^{\infty }{\overline {\bigcup _{k=n}^{\infty }\{f^{k}(x)\}}}.$ ## Definition for flows

Given a real dynamical system (T, X, φ) with flow $\varphi :\mathbb {R} \times X\to X$ , a point x and an orbit γ through x, we call a point y an ω-limit point of γ if there exists a sequence $(t_{n})_{n\in \mathbb {N} }$ in R so that

$\lim _{n\to \infty }t_{n}=\infty$ $\lim _{n\to \infty }\varphi (t_{n},x)=y$ .

Analogously we call y an α-limit point if there exists a sequence $(t_{n})_{n\in \mathbb {N} }$ in R so that

$\lim _{n\to \infty }t_{n}=-\infty$ $\lim _{n\to \infty }\varphi (t_{n},x)=y$ .

The set of all ω-limit points (α-limit points) for a given orbit γ is called ω-limit set (α-limit set) for γ and denoted limω γ (limα γ).

If the ω-limit set (α-limit set) is disjoint from the orbit γ, that is limω γ ∩ γ = ∅ (limα γ ∩ γ = ∅), we call limω γ (limα γ) a ω-limit cycle (α-limit cycle).

Alternatively the limit sets can be defined as

$\lim _{\omega }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t>s\}}}$ and

$\lim _{\alpha }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t ### Properties

• limω γ and limα γ are closed
• if X is compact then limω γ and limα γ are nonempty, compact and connected
• limω γ and limα γ are φ-invariant, that is φ(R × limω γ) = limω γ and φ(R × limα γ) = limα γ