![An iterated direct similarity yields spirals](https://upload.wikimedia.org/wikipedia/commons/0/06/An_iterated_direct_similarity_yields_spirals.svg)
In
mathematics, an iterated function is a function (that is, a function from some
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
to itself) which is obtained by
composing another function with itself a certain number of times. The process of repeatedly applying the same function is called
iteration
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right:
:
with the circle‑shaped symbol of function composition.
Iterated functions are objects of study in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
,
fractals
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
,
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s, mathematics and
renormalization group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
physics.
Definition
The formal definition of an iterated function on a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''X'' follows.
Let be a set and be a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
.
Defining as the ''n''-th iterate of (a notation introduced by
Hans Heinrich Bürmann
Hans Heinrich Bürmann (died 21 June 1817, in Mannheim) was a German mathematician and teacher. He ran an "academy of commerce" in Mannheim since 1795 where he used to teach mathematics. He also served as a censor in Mannheim. He was nominated He ...
and
John Frederick William Herschel
Sir John Frederick William Herschel, 1st Baronet (; 7 March 1792 – 11 May 1871) was an English polymath active as a mathematician, astronomer, chemist, inventor, experimental photographer who invented the blueprint and did botanical wo ...
), where ''n'' is a non-negative integer, by:
and
where is the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on and denotes
function composition. That is,
:,
always
associative.
Because the notation may refer to both iteration (composition) of the function or
exponentiation of the function (the latter is commonly used in
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...
), some mathematicians choose to use to denote the compositional meaning, writing for the -th iterate of the function , as in, for example, meaning . For the same purpose, was used by
Benjamin Peirce
Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philoso ...
[while is taken for the th derivative] whereas
Alfred Pringsheim
Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician and patron of the arts. He was born in Ohlau, Prussian Silesia (now Oława, Poland) and died in Zürich, Switzerland.
Family and academic career
Pringsheim came ...
and
Jules Molk
Jules Molk (8 December 1857 in Strasbourg, France – 7 May 1914 in Nancy) was a French mathematician who worked on elliptic functions.
The French Academy of Sciences awarded him the Prix Binoux for 1913.
He was appointed to the chair of applie ...
suggested instead.
Abelian property and iteration sequences
In general, the following identity holds for all non-negative integers and ,
:
This is structurally identical to the property of
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...
that , i.e. the special case .
In general, for arbitrary general (negative, non-integer, etc.) indices and , this relation is called the translation functional equation, cf.
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 that sen ...
and
Abel equation
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
:f(h(x)) = h(x + 1)
or
:\alpha(f(x)) = \alpha(x)+1.
The forms are equivalent when is invertible. or control the iteration of .
Equivalence
The seco ...
. On a logarithmic scale, this reduces to the nesting property of
Chebyshev polynomials, , since .
The relation also holds, analogous to the property of exponentiation that .
The sequence of functions is called a Picard sequence, named after
Charles Émile Picard
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was ...
.
For a given in , the
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of values is called the
orbit
In celestial mechanics, an orbit 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 object or position in space such as ...
of .
If for some integer , the orbit is called a periodic orbit. The smallest such value of for a given is called the period of the orbit. The point itself is called a
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
Given a ...
. The
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.
For any function that maps a finite set to itself, and any initial value in , the sequence of itera ...
problem in computer science is the
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
ic problem of finding the first periodic point in an orbit, and the period of the orbit.
Fixed points
If for some in (that is, the period of the orbit of is ), then is called a
fixed point of the iterated sequence. The set of fixed points is often denoted as . There exist a number of
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'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors cla ...
s that guarantee the existence of fixed points in various situations, including the
Banach fixed point theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...
and the
Brouwer fixed point theorem.
There are several techniques for
convergence acceleration In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the ...
of the sequences produced by
fixed point iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.
More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed-point iterat ...
. For example, the
Aitken method applied to an iterated fixed point is known as
Steffensen's method, and produces quadratic convergence.
Limiting behaviour
Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an
attractive fixed point. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an
unstable fixed point.
When the points of the orbit converge to one or more limits, the set of
accumulation points of the orbit is known as the
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 ca ...
or the ω-limit set.
The ideas of attraction and repulsion generalize similarly; one may categorize iterates into
stable sets and
unstable sets, according to the behaviour of small
neighborhoods under iteration. (Also see
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 ...
.)
Other limiting behaviours are possible; for example,
wandering points are points that move away, and never come back even close to where they started.
Invariant measure
If one considers the evolution of a density distribution, rather than that of individual point dynamics, then the limiting behavior is given by the
invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, ...
. It can be visualized as the behavior of a point-cloud or dust-cloud under repeated iteration. The invariant measure is an eigenstate of the Ruelle-Frobenius-Perron operator or
transfer operator
Transfer may refer to:
Arts and media
* ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović
* ''Transfer'' (1966 film), a short film
* ''Transfer'' (journal), in management studies
...
, corresponding to an eigenvalue of 1. Smaller eigenvalues correspond to unstable, decaying states.
In general, because repeated iteration corresponds to a shift, the transfer operator, and its adjoint, the
Koopman operator can both be interpreted as
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 . In time series analysis, the shift operator is called the lag operator.
Shift ...
s action on a
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. In fact, shift spaces and '' symbolic dynamical systems'' are often considered synon ...
. The theory of
subshifts of finite type provides general insight into many iterated functions, especially those leading to chaos.
Fractional iterates and flows, and negative iterates
![TrivFctRootExm svg](https://upload.wikimedia.org/wikipedia/commons/6/6e/TrivFctRootExm_svg.svg)
The notion must be used with care when the equation has multiple solutions, which is normally the case, as in
Babbage's equation of the functional roots of the identity map. For example, for and , both and are solutions; so the expression doesn't denote a unique function, just as numbers have multiple algebraic roots. The issue is quite similar to the expression "
0/0" in arithmetic. A trivial root of ''f'' can always be obtained if ''f''s domain can be extended sufficiently, cf. picture. The roots chosen are normally the ones belonging to the orbit under study.
Fractional iteration of a function can be defined: for instance, a
half iterate of a function is a function such that . This function can be written using the index notation as . Similarly, is the function defined such that , while may be defined as equal to , and so forth, all based on the principle, mentioned earlier, that . This idea can be generalized so that the iteration count becomes a continuous parameter, a sort of continuous "time" of a continuous
orbit
In celestial mechanics, an orbit 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 object or position in space such as ...
.
In such cases, one refers to the system as a
flow. (cf. Section on
conjugacy
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
below.)
Negative iterates correspond to function inverses and their compositions. For example, is the normal inverse of , while is the inverse composed with itself, i.e. . Fractional negative iterates are defined analogously to fractional positive ones; for example, is defined such that , or, equivalently, such that .
Some formulas for fractional iteration
One of several methods of finding a series formula for fractional iteration, making use of a fixed point, is as follows.
# First determine a fixed point for the function such that .
# Define for all ''n'' belonging to the reals. This, in some ways, is the most natural extra condition to place upon the fractional iterates.
# Expand around the fixed point ''a'' as a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
,
# Expand out
# Substitute in for , for any ''k'',
# Make use of the
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
to simplify terms,
There is a special case when ,
This can be carried on indefinitely, although inefficiently, as the latter terms become increasingly complicated. A more systematic procedure is outlined in the following section on Conjugacy.
Example 1
For example, setting gives the fixed point , so the above formula terminates to just
which is trivial to check.
Example 2
Find the value of
where this is done ''n'' times (and possibly the interpolated values when ''n'' is not an integer). We have . A fixed point is .
So set and expanded around the fixed point value of 2 is then an infinite series,
which, taking just the first three terms, is correct to the first decimal place when ''n'' is positive–cf.
Tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as rep ...
: . (Using the other fixed point causes the series to diverge.)
For , the series computes the inverse function .
Example 3
With the function , expand around the fixed point 1 to get the series
which is simply the Taylor series of ''x''
(''b''''n'' ) expanded around 1.
Conjugacy
If and are two iterated functions, and there exists a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
such that , then and are said to be
topologically conjugate In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated func ...
.
Clearly, topological conjugacy is preserved under iteration, as . Thus, if one can solve for one iterated function system, one also has solutions for all topologically conjugate systems. For example, the
tent map
A tent () is a shelter consisting of sheets of fabric or other material draped over, attached to a frame of poles or a supporting rope. While smaller tents may be free-standing or attached to the ground, large tents are usually anchored using g ...
is topologically conjugate to the
logistic map. As a special case, taking , one has the iteration of as
:, for any function .
Making the substitution yields
:, a form known as the
Abel equation
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
:f(h(x)) = h(x + 1)
or
:\alpha(f(x)) = \alpha(x)+1.
The forms are equivalent when is invertible. or control the iteration of .
Equivalence
The seco ...
.
Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at = 0, (0) = 0, one may often solve
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 that sen ...
for a function Ψ, which makes locally conjugate to a mere dilation, , that is
:.
Thus, its iteration orbit, or flow, under suitable provisions (e.g., ), amounts to the conjugate of the orbit of the monomial,
:,
where in this expression serves as a plain exponent: ''functional iteration has been reduced to multiplication!'' Here, however, the exponent no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit: the
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
of the Picard sequence (cf.
transformation semigroup In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations ( functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transfo ...
) has generalized to a full
continuous group
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
.
![Sine_iterations](https://upload.wikimedia.org/wikipedia/commons/4/4d/Sine_iterations.svg)
This method (perturbative determination of the principal
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. As an equation, th ...
Ψ, cf.
Carleman matrix In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alon ...
) is equivalent to the algorithm of the preceding section, albeit, in practice, more powerful and systematic.
Markov chains
If the function is linear and can be described by a
stochastic matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ...
, that is, a matrix whose rows or columns sum to one, then the iterated system is known as a
Markov chain.
Examples
There are
many chaotic maps.
Well-known iterated functions include the
Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
and
iterated function systems
In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981.
IFS fractal ...
.
Ernst Schröder,
in 1870, worked out special cases of the
logistic map, such as the chaotic case , so that , hence .
A nonchaotic case Schröder also illustrated with his method, , yielded , and hence .
If is the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of a group element on a set, then the iterated function corresponds to a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
.
Most functions do not have explicit general
closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
s for the ''n''-th iterate. The table below lists some
that do. Note that all these expressions are valid even for non-integer and negative ''n'', as well as non-negative integer ''n''.
Note: these two special cases of are the only cases that have a closed-form solution. Choosing ''b'' = 2 = –''a'' and ''b'' = 4 = –''a'', respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table.
Some of these examples are related among themselves by simple conjugacies. A few further examples, essentially amounting to simple conjugacies of Schröder's examples can be found in ref.
Means of study
Iterated functions can be studied with the
Artin–Mazur zeta function and with
transfer operator
Transfer may refer to:
Arts and media
* ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović
* ''Transfer'' (1966 film), a short film
* ''Transfer'' (journal), in management studies
...
s.
In computer science
In
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
, iterated functions occur as a special case of
recursive functions, which in turn anchor the study of such broad topics as
lambda calculus, or narrower ones, such as the
denotational semantics of computer programs.
Definitions in terms of iterated functions
Two important
functionals can be defined in terms of iterated functions. These are
summation:
:
and the equivalent product:
:
Functional derivative
The
functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
of an iterated function is given by the recursive formula:
:
Lie's data transport equation
Iterated functions crop up in the series expansion of combined functions, such as .
Given the
iteration velocity, or
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.
It i ...
,
:
for the
th iterate of the function , we have
[ ]
:
For example, for rigid advection, if , then . Consequently, , action by a plain
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 . In time series analysis, the shift operator is called the lag operator.
Shift ...
.
Conversely, one may specify given an arbitrary , through the generic
Abel equation
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
:f(h(x)) = h(x + 1)
or
:\alpha(f(x)) = \alpha(x)+1.
The forms are equivalent when is invertible. or control the iteration of .
Equivalence
The seco ...
discussed above,
:
where
:
This is evident by noting that
:
For continuous iteration index , then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group,
:
The initial flow velocity suffices to determine the entire flow, given this exponential realization which automatically provides the general solution to the ''translation functional equation'',
[Aczel, J. (2006), ''Lectures on Functional
Equations and Their Applications'' (Dover Books on Mathematics, 2006), Ch. 6, .]
:
See also
*
Irrational rotation
In the mathematical theory of dynamical systems, an irrational rotation is a map
: T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1
where ''θ'' is an irrational number. Under the identification of a circle wit ...
*
Iterated function system
*
Iterative method
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pr ...
*
Rotation number
In mathematics, the rotation number is an invariant of homeomorphisms of the circle.
History
It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theore ...
*
Sarkovskii's theorem
*
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 D
:D f(x) = \frac f(x)\,,
and of the integration ...
*
Recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
*
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 that sen ...
*
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. In other words, a functional square root of a function is a function satisfying ...
*
Abel function
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
:f(h(x)) = h(x + 1)
or
:\alpha(f(x)) = \alpha(x)+1.
The forms are equivalent when is invertible. or control the iteration of .
Equivalence
The se ...
*
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 ...
*
Flow (mathematics)
*
Tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as rep ...
*
Functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
Notes
References
External Links
{{cite web , url=https://www.researchgate.net/publication/362010262 , author-link=John Gill (climber) , first=John , last=Gill , title=A Primer on the Elementary Theory of Infinite Compositions of Complex Functions , publisher=Colorado State University , date=January 2017
Dynamical systems
Fractals
Sequences and series
Fixed points (mathematics)
Functions and mappings
Functional equations