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81 relations: Abel equation, Aitken's delta-squared process, Algorithm, American Mathematical Monthly, Artin–Mazur zeta function, Associative property, Émile Picard, Banach fixed-point theorem, Beta function (physics), Brouwer fixed-point theorem, Carleman matrix, Chebyshev polynomials, Closed-form expression, Composition operator, Computer science, Cycle detection, Denotational semantics, Dynamical system, Eigenfunction, Ernst Schröder, Exponentiation, Fixed point (mathematics), Fixed-point iteration, Fixed-point theorem, Flow (mathematics), Fractal, Fractional calculus, Free group, Function (mathematics), Function composition, Functional (mathematics), Functional derivative, Functional square root, Geometric progression, Group action, Homeomorphism, Identity function, Infinite compositions of analytic functions, Invariant measure, Irrational rotation, Iterated function system, Iteration, Iterative method, Lambda calculus, Limit point, Limit set, List of chaotic maps, Logistic map, Mandelbrot set, Markov chain, ..., Mathematics, Monoid, Neighbourhood, Orbit (dynamics), Periodic point, Rational difference equation, Recurrence relation, Recursion (computer science), Renormalization group, Rotation number, Schröder's equation, Sequence, Series acceleration, Set (mathematics), Sharkovskii's theorem, Shift operator, Shift space, Stable manifold, Steffensen's method, Stochastic matrix, Subshift of finite type, Summation, Taylor series, Tent map, Tetration, Topological conjugacy, Topological group, Transfer operator, Transformation semigroup, Trigonometric functions, Wandering set. Expand index (31 more) »
Abel equation
The Abel equation, named after Niels Henrik Abel, is a type of functional equation which can be written in the form or, equivalently, and controls the iteration of.
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Aitken's delta-squared process
In numerical analysis, Aitken's delta-squared process or Aitken Extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence.
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Algorithm
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.
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American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.
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Artin–Mazur zeta function
In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals.
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Associative property
In mathematics, the associative property is a property of some binary operations.
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Émile Picard
Prof Charles Émile Picard FRS(For) FRSE (24 July 1856 – 11 December 1941) was a French mathematician.
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Banach fixed-point theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.
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Beta function (physics)
In theoretical physics, specifically quantum field theory, a beta function, β(g), encodes the dependence of a coupling parameter, g, on the energy scale, μ, of a given physical process described by quantum field theory.
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Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer.
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Carleman matrix
In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication.
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Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.
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Closed-form expression
In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.
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Composition operator
In mathematics, the composition operator C_\phi with symbol \phi is a linear operator defined by the rule where f \circ\phi denotes function composition.
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Computer science
Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.
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Cycle detection
In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values.
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Denotational semantics
In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called denotations) that describe the meanings of expressions from the languages.
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Dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.
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Eigenfunction
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.
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Ernst Schröder
Friedrich Wilhelm Karl Ernst Schröder (25 November 1841 in Mannheim, Baden, Germany – 16 June 1902 in Karlsruhe, Germany) was a German mathematician mainly known for his work on algebraic logic.
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Exponentiation
Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.
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Fixed point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.
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Fixed-point iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions.
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Fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x).
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Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid.
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Fractal
In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.
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Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator and of the integration operator and developing a calculus for such operators generalizing the classical one.
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Free group
In mathematics, the free group FS over a given set S consists of all expressions (a.k.a. words, or terms) that can be built from members of S, considering two expressions different unless their equality follows from the group axioms (e.g. st.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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Function composition
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.
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Functional (mathematics)
In mathematics, the term functional (as a noun) has at least two meanings.
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Functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional to a change in a function on which the functional depends.
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Functional square root
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition.
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Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.
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Identity function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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Infinite compositions of analytic functions
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions.
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Invariant measure
In mathematics, an invariant measure is a measure that is preserved by some function.
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Irrational rotation
In the mathematical theory of dynamical systems, an irrational rotation is a map where θ is an irrational number.
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Iterated function system
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar.
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Iteration
Iteration is the act of repeating a process, to generate a (possibly unbounded) sequence of outcomes, with the aim of approaching a desired goal, target or result.
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Iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.
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Lambda calculus
Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.
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Limit point
In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.
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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.
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List of chaotic maps
In mathematics, a chaotic map is a map (.
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Logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations.
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Mandelbrot set
The Mandelbrot set is the set of complex numbers c for which the function f_c(z).
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Markov chain
A Markov chain is "a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event".
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
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Neighbourhood
A neighbourhood (British English), or neighborhood (American English; see spelling differences), is a geographically localised community within a larger city, town, suburb or rural area.
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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.
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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.
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Rational difference equation
A rational difference equation is a nonlinear difference equation of the form where the initial conditions x_, x_,\dots, x_ are such that the denominator never vanishes for any.
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Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.
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Recursion (computer science)
Recursion in computer science is a method of solving a problem where the solution depends on solutions to smaller instances of the same problem (as opposed to iteration).
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Renormalization group
In theoretical physics, the renormalization group (RG) refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales.
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Rotation number
In mathematics, the rotation number is an invariant of homeomorphisms of the circle.
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Schröder's equation
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function, find the function such that: Schröder's equation is an eigenvalue equation for the composition operator, which sends a function to.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Series acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series.
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Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
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Sharkovskii's theorem
In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii who published it in 1964, is a result about discrete dynamical systems.
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Shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation.
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Shift space
In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system.
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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.
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Steffensen's method
In numerical analysis, Steffensen's method is a root-finding technique similar to Newton's method, named after Johan Frederik Steffensen.
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Stochastic matrix
In mathematics, a stochastic matrix (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix) is a square matrix used to describe the transitions of a Markov chain.
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Subshift of finite type
In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory.
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Summation
In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.
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Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
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Tent map
In mathematics, the tent map with parameter μ is the real-valued function fμ defined by the name being due to the tent-like shape of the graph of fμ.
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Tetration
In mathematics, tetration (or hyper-4) is the next hyperoperation after exponentiation, and is defined as iterated exponentiation.
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Topological conjugacy
In mathematics, two functions are said to be topologically conjugate to one another if there exists a homeomorphism that will conjugate the one into the other.
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Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.
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Transfer operator
In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals.
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Transformation semigroup
In algebra, a transformation semigroup (or composition semigroup) is a collection of functions from a set to itself that is closed under function composition.
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Trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.
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Wandering set
In those branches of mathematics called dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing in such systems.
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Redirects here:
Fractional iteration, Function Iteration, Function iteration, Iterated map, Iteration orbit, Iterative function, Iterative functional-differential equation, Picard sequence.
References
[1] https://en.wikipedia.org/wiki/Iterated_function
