In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in the study of dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s, 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

• fixed point
  Fixed point (mathematics)
  In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
  
  s
• periodic orbits
• limit cycles
• attractor
  Attractor
  An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
  
  s.

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

Poincaré–Bendixson theorem

In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane.-Theorem:...

 provides a simple characterization of all possible limit sets as a union of fixed points and periodic orbits.

Definition for iterated functions

Let be a metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

, and let be a continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

. The -limit set of , denoted by , is the set of cluster points of the forward orbit of the iterated function

Iterated function

In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...

 . Hence, if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 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 non-wandering (but may not be recurrent point

Recurrent point

In mathematics, a recurrent point for function f is a point that is in the limit set of the iterated function f. Any neighborhood containing the recurrent point will also contain iterates of it as well.-Definition:...

s).

If is a homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

 (that is, a bicontinuous bijection), then the -limit set is defined in a similar fashion, but for the backward orbit; i.e. .

Both sets are -invariant, and if is compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

, they are compact and nonempty.

Definition for flows

Given a real dynamical system (T, X, φ) with flow

Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...

 , a point x and an orbit

Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...

 γ through x, we call a point y an ω-limit point of γ if there exists a sequence in R so that.

Analogously we call y an α-limit point if there exists a sequence in R so that.

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 lim_ω γ (lim_α γ).

If the ω-limit set (α-limit 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 ω-limit cycle (α-limit 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
  Fixed point
  "Fixed point" has many meanings in science, most of them mathematical.* Fixed point * Fixed-point combinator* Fixed-point arithmetic, a manner of doing arithmetic on computers* Benchmark , fixed points used by geodesists...
  
    of a dynamical system, lim_ω = lim_α =

Properties

• lim_ω γ and lim_α γ are closed
  Closed set
  In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
  
  
• if X is compact then lim_ω γ and lim_α γ are nonempty, compact and connected
  Connected space
  In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
  
  
• lim_ω γ and lim_α γ are φ-invariant, that is φ(R × lim_ω γ) = lim_ω γ and φ(R × lim_α γ) = lim_α γ

See also

• Julia set
  Julia set
  In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
  
  
• Stable set
  Stable manifold
  In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor...
  
  
• Limit cycle
• Periodic point
  Periodic point
  In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.- Iterated functions :...
  
  
• Non-wandering set
• Kleinian group
  Kleinian group
  In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...