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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in the study of
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
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. A system that has reached its limiting set is said to be at equilibrium.


Types

* fixed points * periodic orbits * limit cycles * attractors In general, limits sets can be very complicated as in the case of
strange attractor In the mathematics, mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor va ...
s, but for 2-dimensional dynamical systems the
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, cylinder, or two-sphere. Theorem Given a differentiable real dynamical system defined on an op ...
provides a simple characterization of all nonempty, compact \omega-limit sets that contain at most finitely many fixed points as a fixed point, a periodic orbit, or a union of fixed points and homoclinic or heteroclinic orbits connecting those fixed points.


Definition for iterated functions

Let X be a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, and let f:X\rightarrow X be a
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
. The \omega-limit set of x\in X, denoted by \omega(x,f), is the set of cluster points of the forward orbit \_ of the
iterated function In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some ...
f. Hence, y\in \omega(x,f)
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
there is a strictly increasing sequence of natural numbers \_ such that f^(x)\rightarrow y as k\rightarrow\infty. Another way to express this is :\omega(x,f) = \bigcap_ \overline, where \overline denotes the ''closure'' of set S. 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_^\infty \overline. If f is a
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
(that is, a bicontinuous bijection), then the \alpha-limit set is defined in a similar fashion, but for the backward orbit; ''i.e.'' \alpha(x,f)=\omega(x,f^). Both sets are f-invariant, and if X is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, they are compact and nonempty.


Definition for flows

Given a real dynamical system (T,X,\varphi) with flow \varphi:\mathbb\times X\to X, a point x, we call a point y an \omega-limit point of ''x'' if there exists a sequence (t_n)_ in \mathbb so that :\lim_ t_n = \infty :\lim_ \varphi(t_n, x) = y . For an
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
\gamma of (T,X,\varphi), we say that y is an \omega-limit point of \gamma, if it is an \omega-limit point of some point on the orbit. Analogously we call ''y'' an \alpha-limit point of ''x'' if there exists a sequence (t_n)_ in \mathbb so that :\lim_ t_n = -\infty :\lim_ \varphi(t_n, x) = y . For an
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
\gamma of (T,X,\varphi), we say that ''y'' is an \alpha-limit point of \gamma, if it is an \alpha-limit point of some point on the orbit. The set of all \omega-limit points (\alpha-limit points) for a given orbit \gamma is called \omega-limit set (\alpha-limit set) for \gamma and denoted \lim_ \gamma (\lim_ \gamma). If the \omega-limit set (\alpha-limit set) is disjoint from the orbit \gamma, that is \lim_ \gamma \cap \gamma =\varnothing (\lim_ \gamma \cap \gamma =\varnothing), we call \lim_ \gamma (\lim_ \gamma) a ω-limit cycle (
α-limit cycle In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself ...
). Alternatively the limit sets can be defined as :\lim_\omega \gamma := \bigcap_\overline and :\lim_\alpha \gamma := \bigcap_\overline.


Examples

* For any periodic orbit \gamma of a dynamical system, \lim_ \gamma =\lim_ \gamma =\gamma * For any fixed point x_0 of a dynamical system, \lim_ x_0 =\lim_ x_0 =x_0


Properties

* \lim_ \gamma and \lim_ \gamma are closed * if X is compact then \lim_ \gamma and \lim_ \gamma are
nonempty In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, whi ...
,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
and connected * \lim_ \gamma and \lim_ \gamma are \varphi-invariant, that is \varphi(\mathbb\times\lim_ \gamma)=\lim_ \gamma and \varphi(\mathbb\times\lim_ \gamma)=\lim_ \gamma


See also

*
Julia set In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of ...
* Stable set * Limit cycle * Periodic point * Non-wandering set *
Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...


References


Further reading

* {{PlanetMath attribution, id=4316, title=Omega-limit set