# Category: Limit sets

Accumulation point
In mathematics, a limit point, accumulation point, or cluster point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with res
Julia set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou
Isolating neighborhood
In the theory of dynamical systems, an isolating neighborhood is a compact set in the phase space of an invertible dynamical system with the property that any orbit contained entirely in the set belon
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 certa
No-wandering-domain theorem
In mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985. The theorem states that a rational map f : Ĉ → Ĉ with deg(f) ≥ 2 does not have a wa
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 t
Pugh's closing lemma
In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows: Let be a diffeomorphism of a
Filled Julia set
The filled-in Julia set of a polynomial is a Julia set and its interior, non-escaping set
Attractor
In the 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 c
Recurrent point
In mathematics, a recurrent point for a function f is a point that is in its own limit set by f. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of i
Siegel disc
Siegel disc is a connected component in the Fatou set where the dynamics is analytically conjugate to an irrational rotation.
Periodic points of complex quadratic mappings
This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that inv
Hyperbolic equilibrium point
In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimens
Herman ring
In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou component where the rational function is conformally conjugate to an irrational rotation of the standard annulus.
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 emb
Wandering set
In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the syst
Hyperbolic set
In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundles, o
Limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals i
Classification of Fatou components
In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equat
Strange nonchaotic attractor
In mathematics, a strange nonchaotic attractor (SNA) is a form of attractor which, while converging to a limit, is strange, because it is not piecewise differentiable, and also non-chaotic, in that it