SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Iterated function</title>

</head>

Iterated function

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

In <a href="Mathematics" title="Mathematics">mathematics</a>, an iterated function is a function that is obtained by <a href="Function_composition" title="Function composition">composing</a> another function with itself two or several times. The process of repeatedly applying the same function is called <a href="Iteration" title="Iteration">iteration</a>. In this process, starting from some initial object, the result of applying a given function is fed again into the function as input, and this process is repeated.

For example, on the image on the right:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=F(K),\ M=F\circ F(K)=F^{2}(K).}</annotation>

</semantics>

</math><img src="./5e02677c480bf75ab856d97830f4d335d713da3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.494ex; height:3.176ex;" alt="{\displaystyle L=F(K),\ M=F\circ F(K)=F^{2}(K).}" loading="lazy"></dd></dl>

Iterated functions are studied in <a href="Computer_science" title="Computer science">computer science</a>, <a href="Fractals" class="mw-redirect" title="Fractals">fractals</a>, <a href="Dynamical_system" title="Dynamical system">dynamical systems</a>, mathematics and <a href="Renormalization_group" title="Renormalization group">renormalization group</a> physics.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The formal definition of an iterated function on a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> X follows.

Let X be a set and f: X → X be a <a href="Function_(mathematics)" title="Function (mathematics)">function</a>.

Defining f n as the n-th iterate of f, where n is a non-negative integer, by:

<msup>

</mrow>

</msup>

<mover>

</mrow>

</mrow>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{0}~{\stackrel {\mathrm {def} }{=}}~\operatorname {id} _{X}}</annotation>

</semantics>

</math>

and

<msup>

</mrow>

</msup>

<mover>

</mrow>

</mrow>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n+1}~{\stackrel {\mathrm {def} }{=}}~f\circ f^{n},}</annotation>

</semantics>

</math>

where idX is the <a href="Identity_function" title="Identity function">identity function</a> on X and (f <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \circ }</annotation>

</semantics>

</math><img src="./99add39d2b681e2de7ff62422c32704a05c7ec31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" loading="lazy"> g)(x) = f (g(x)) denotes <a href="Function_composition" title="Function composition">function composition</a>. This notation has been traced to and <a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">John Frederick William Herschel</a> in 1813.<a href="#cite_note-Herschel_1813-1">[1]</a><a href="#cite_note-Herschel_1820-2">[2]</a><a href="#cite_note-Peano_1903-3">[3]</a><a href="#cite_note-Cajori_1929-4">[4]</a> Herschel credited <a href="Hans_Heinrich_B%C3%BCrmann" title="Hans Heinrich Bürmann">Hans Heinrich Bürmann</a> for it, but without giving a specific reference to the work of Bürmann, which remains undiscovered.<a href="#cite_note-5">[5]</a>

Because the notation f n may refer to both iteration (composition) of the function f or <a href="Exponentiation#Iterated_functions" title="Exponentiation">exponentiation of the function</a> f (the latter is commonly used in <a href="Trigonometric_functions" title="Trigonometric functions">trigonometry</a>), some mathematicians choose to use ∘ to denote the compositional meaning, writing f∘n(x) for the n-th iterate of the function f(x), as in, for example, f∘3(x) meaning f(f(f(x))). For the same purpose, f [n](x) was used by <a href="Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a><a href="#cite_note-Peirce_1852-6">[6]</a><a href="#cite_note-Cajori_1929-4">[4]</a><a href="#cite_note-7">[nb 1]</a> whereas <a href="Alfred_Pringsheim" title="Alfred Pringsheim">Alfred Pringsheim</a> and <a href="Jules_Molk" title="Jules Molk">Jules Molk</a> suggested nf(x) instead.<a href="#cite_note-Pringsheim-Molk_1907-8">[7]</a><a href="#cite_note-Cajori_1929-4">[4]</a><a href="#cite_note-NB_Rucker-9">[nb 2]</a>

<div class="mw-heading mw-heading2"><h2 id="Abelian_property_and_iteration_sequences">Abelian property and iteration sequences</h2></div>

In general, the following identity holds for all non-negative integers m and n,

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{m}\circ f^{n}=f^{n}\circ f^{m}=f^{m+n}~.}</annotation>

</semantics>

</math><img src="./15a113a332e038e08a79f0fee3534a8bdc81fa25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.143ex; height:2.843ex;" alt="{\displaystyle f^{m}\circ f^{n}=f^{n}\circ f^{m}=f^{m+n}~.}" loading="lazy"></dd></dl>

This is structurally identical to the property of <a href="Exponentiation" title="Exponentiation">exponentiation</a> that aman = am + n.

In general, for arbitrary general (negative, non-integer, etc.) indices m and n, this relation is called the translation functional equation, cf. <a href="Schr%C3%B6der's_equation" title="Schröder's equation">Schröder's equation</a> and <a href="Abel_equation" title="Abel equation">Abel equation</a>. On a logarithmic scale, this reduces to the nesting property of <a href="Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a>, Tm(Tn(x)) = Tm n(x), since Tn(x) = cos(n arccos(x)).

The relation (f m)n(x) = (f n)m(x) = f mn(x) also holds, analogous to the property of exponentiation that (am)n = (an)m = amn.

The sequence of functions f n is called a Picard sequence,<a href="#cite_note-10">[8]</a><a href="#cite_note-11">[9]</a> named after <a href="Charles_%C3%89mile_Picard" class="mw-redirect" title="Charles Émile Picard">Charles Émile Picard</a>.

For a given x in X, the <a href="Sequence" title="Sequence">sequence</a> of values fn(x) is called the <a href="Orbit_(dynamics)" title="Orbit (dynamics)">orbit</a> of x.

If f n (x) = f n+m (x) for some integer m > 0, the orbit is called a periodic orbit. The smallest such value of m for a given x is called the period of the orbit. The point x itself is called a <a href="Periodic_point" title="Periodic point">periodic point</a>. The <a href="Cycle_detection" title="Cycle detection">cycle detection</a> problem in computer science is the <a href="Algorithm" title="Algorithm">algorithmic</a> problem of finding the first periodic point in an orbit, and the period of the orbit.

<div class="mw-heading mw-heading2"><h2 id="Fixed_points">Fixed points</h2></div>

If x = f(x) for some x in X (that is, the period of the orbit of x is 1), then x is called a <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a> of the iterated sequence. The set of fixed points is often denoted as Fix(f). There exist a number of <a href="Fixed-point_theorem" title="Fixed-point theorem">fixed-point theorems</a> that guarantee the existence of fixed points in various situations, including the <a href="Banach_fixed_point_theorem" class="mw-redirect" title="Banach fixed point theorem">Banach fixed point theorem</a> and the <a href="Brouwer_fixed_point_theorem" class="mw-redirect" title="Brouwer fixed point theorem">Brouwer fixed point theorem</a>.

There are several techniques for <a href="Convergence_acceleration" class="mw-redirect" title="Convergence acceleration">convergence acceleration</a> of the sequences produced by <a href="Fixed_point_iteration" class="mw-redirect" title="Fixed point iteration">fixed point iteration</a>.<a href="#cite_note-12">[10]</a> For example, the <a href="Aitken_method" class="mw-redirect" title="Aitken method">Aitken method</a> applied to an iterated fixed point is known as <a href="Steffensen's_method" title="Steffensen's method">Steffensen's method</a>, and produces quadratic convergence.

<div class="mw-heading mw-heading2"><h2 id="Limiting_behaviour">Limiting behaviour</h2></div>

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 <a href="Attractive_fixed_point" class="mw-redirect" title="Attractive fixed point">attractive fixed point</a>. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an <a href="Unstable_fixed_point" class="mw-redirect" title="Unstable fixed point">unstable fixed point</a>.<a href="#cite_note-13">[11]</a>

When the points of the orbit converge to one or more limits, the set of <a href="Accumulation_point" title="Accumulation point">accumulation points</a> of the orbit is known as the <a href="Limit_set" title="Limit set">limit set</a> or the ω-limit set.

The ideas of attraction and repulsion generalize similarly; one may categorize iterates into <a href="Stable_manifold" title="Stable manifold">stable sets</a> and <a href="Unstable_set" class="mw-redirect" title="Unstable set">unstable sets</a>, according to the behavior of small <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighborhoods</a> under iteration. Also see <a href="Infinite_compositions_of_analytic_functions" title="Infinite compositions of analytic functions">infinite compositions of analytic functions</a>.

Other limiting behaviors are possible; for example, <a href="Wandering_point" class="mw-redirect" title="Wandering point">wandering points</a> are points that move away, and never come back even close to where they started.

<div class="mw-heading mw-heading2"><h2 id="Invariant_measure">Invariant measure</h2></div>

If one considers the evolution of a density distribution, rather than that of individual point dynamics, then the limiting behavior is given by the <a href="Invariant_measure" title="Invariant measure">invariant measure</a>. 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 <a href="Transfer_operator" title="Transfer operator">transfer operator</a>, 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 <a href="Koopman_operator" class="mw-redirect" title="Koopman operator">Koopman operator</a> can both be interpreted as <a href="Shift_operator" title="Shift operator">shift operators</a> action on a <a href="Shift_space" title="Shift space">shift space</a>. The theory of <a href="Subshifts_of_finite_type" class="mw-redirect" title="Subshifts of finite type">subshifts of finite type</a> provides general insight into many iterated functions, especially those leading to chaos.

<div class="mw-heading mw-heading2"><h2 id="Fractional_iterates_and_flows,_and_negative_iterates">Fractional iterates and flows, and negative iterates</h2></div>

The notion f1/n must be used with care when the equation gn(x) = f(x) has multiple solutions, which is normally the case, as in <a href="Functional_square_root" title="Functional square root">Babbage's equation</a> of the functional roots of the identity map. For example, for n = 2 and f(x) = 4x − 6, both g(x) = 6 − 2x and g(x) = 2x − 2 are solutions; so the expression f 1/2(x) does not denote a unique function, just as numbers have multiple algebraic roots. 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 <a href="Functional_square_root" title="Functional square root">half iterate</a> of a function f is a function g such that g(g(x)) = f(x).<a href="#cite_note-14">[12]</a> This function g(x) can be written using the index notation as f 1/2(x) . Similarly, f 1/3(x) is the function defined such that f1/3(f1/3(f1/3(x))) = f(x), while f2/3(x) may be defined as equal to f 1/3(f1/3(x)), and so forth, all based on the principle, mentioned earlier, that f m ○ f n = f m + n. This idea can be generalized so that the iteration count n becomes a continuous parameter, a sort of continuous "time" of a continuous <a href="Orbit_(dynamics)" title="Orbit (dynamics)">orbit</a>.<a href="#cite_note-15">[13]</a><a href="#cite_note-16">[14]</a>

In such cases, one refers to the system as a <a href="Flow_(mathematics)" title="Flow (mathematics)">flow</a> (cf. section on <a href="#Conjugacy">conjugacy</a> below.)

If a function is bijective (and so possesses an inverse function), then negative iterates correspond to function inverses and their compositions. For example, f −1(x) is the normal inverse of f, while f −2(x) is the inverse composed with itself, i.e. f −2(x) = f −1(f −1(x)). Fractional negative iterates are defined analogously to fractional positive ones; for example, f −1/2(x) is defined such that f −1/2(f −1/2(x)) = f −1(x), or, equivalently, such that f −1/2(f 1/2(x)) = f 0(x) = x.

<div class="mw-heading mw-heading3"><h3 id="Some_formulas_for_fractional_iteration">Some formulas for fractional iteration</h3></div>

One of several methods of finding a series formula for fractional iteration, making use of a fixed point, is as follows.<a href="#cite_note-17">[15]</a>

<ol><li>First determine a fixed point for the function such that f(a) = a.</li>

<li>Define f n(a) = a for all n belonging to the reals. This, in some ways, is the most natural extra condition to place upon the fractional iterates.</li>

<li>Expand fn(x) around the fixed point a as a <a href="Taylor_series" title="Taylor series">Taylor series</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{n}(x)=f^{n}(a)+(x-a)\left.{\frac {d}{dx}}f^{n}(x)\right|_{x=a}+{\frac {(x-a)^{2}}{2}}\left.{\frac {d^{2}}{dx^{2}}}f^{n}(x)\right|_{x=a}+\cdots }">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)=f^{n}(a)+(x-a)\left.{\frac {d}{dx}}f^{n}(x)\right|_{x=a}+{\frac {(x-a)^{2}}{2}}\left.{\frac {d^{2}}{dx^{2}}}f^{n}(x)\right|_{x=a}+\cdots }</annotation>

</semantics>

</math></li>

<li>Expand out <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{n}(x)=f^{n}(a)+(x-a)f'(a)f'(f(a))f'(f^{2}(a))\cdots f'(f^{n-1}(a))+\cdots }">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)=f^{n}(a)+(x-a)f'(a)f'(f(a))f'(f^{2}(a))\cdots f'(f^{n-1}(a))+\cdots }</annotation>

</semantics>

</math></li>

<li>Substitute in for fk(a) = a, for any k, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{n}(x)=a+(x-a)f'(a)^{n}+{\frac {(x-a)^{2}}{2}}(f''(a)f'(a)^{n-1})\left(1+f'(a)+\cdots +f'(a)^{n-1}\right)+\cdots }">

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)=a+(x-a)f'(a)^{n}+{\frac {(x-a)^{2}}{2}}(f''(a)f'(a)^{n-1})\left(1+f'(a)+\cdots +f'(a)^{n-1}\right)+\cdots }</annotation>

</semantics>

</math></li>

<li>Make use of the <a href="Geometric_progression" title="Geometric progression">geometric progression</a> to simplify terms, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{n}(x)=a+(x-a)f'(a)^{n}+{\frac {(x-a)^{2}}{2}}(f''(a)f'(a)^{n-1}){\frac {f'(a)^{n}-1}{f'(a)-1}}+\cdots }">

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)=a+(x-a)f'(a)^{n}+{\frac {(x-a)^{2}}{2}}(f''(a)f'(a)^{n-1}){\frac {f'(a)^{n}-1}{f'(a)-1}}+\cdots }</annotation>

</semantics>

</math> There is a special case when f '(a) = 1, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{n}(x)=x+{\frac {(x-a)^{2}}{2}}(nf''(a))+{\frac {(x-a)^{3}}{6}}\left({\frac {3}{2}}n(n-1)f''(a)^{2}+nf'''(a)\right)+\cdots }">

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)=x+{\frac {(x-a)^{2}}{2}}(nf''(a))+{\frac {(x-a)^{3}}{6}}\left({\frac {3}{2}}n(n-1)f''(a)^{2}+nf'''(a)\right)+\cdots }</annotation>

</semantics>

</math></li></ol>

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.

<div class="mw-heading mw-heading4"><h4 id="Example_1">Example 1</h4></div>

For example, setting f(x) = Cx + D gives the fixed point a = D/(1 − C), so the above formula terminates to just

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)={\frac {D}{1-C}}+\left(x-{\frac {D}{1-C}}\right)C^{n}=C^{n}x+{\frac {1-C^{n}}{1-C}}D~,}</annotation>

</semantics>

</math>

which is trivial to check.

<div class="mw-heading mw-heading4"><h4 id="Example_2">Example 2</h4></div>

Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdots }}}}">

<msup>

<msqrt>

</msqrt>

</mrow>

<msup>

<msqrt>

</msqrt>

</mrow>

<msup>

<msqrt>

</msqrt>

</mrow>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdots }}}}</annotation>

</semantics>

</math><img src="./381704cc48abebf8defee1933e806f2816071b10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.161ex; height:4.343ex;" alt="{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdots }}}}" loading="lazy"> where this is done n times (and possibly the interpolated values when n is not an integer). We have f(x) = √2x. A fixed point is a = f(2) = 2.

So set x = 1 and f n (1) expanded around the fixed point value of 2 is then an infinite series,

<msup>

<msqrt>

</msqrt>

</mrow>

<msup>

<msqrt>

</msqrt>

</mrow>

<msup>

<msqrt>

</msqrt>

</mrow>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdots }}}=f^{n}(1)=2-(\ln 2)^{n}+{\frac {(\ln 2)^{n+1}((\ln 2)^{n}-1)}{4(\ln 2-1)}}-\cdots }</annotation>

</semantics>

</math>

which, taking just the first three terms, is correct to the first decimal place when n is positive. Also see <a href="Tetration" title="Tetration">Tetration</a>: f n(1) = n√2. Using the other fixed point a = f(4) = 4 causes the series to diverge.

For n = −1, the series computes the inverse function <style data-mw-deduplicate="TemplateStyles:r1214402035">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}

/* end https://en.wikipedia.org/ */

</style>⁠2+ln x/ln 2⁠.

<div class="mw-heading mw-heading4"><h4 id="Example_3">Example 3</h4></div>

With the function f(x) = xb, expand around the fixed point 1 to get the series

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)=1+b^{n}(x-1)+{\frac {1}{2}}b^{n}(b^{n}-1)(x-1)^{2}+{\frac {1}{3!}}b^{n}(b^{n}-1)(b^{n}-2)(x-1)^{3}+\cdots ~,}</annotation>

</semantics>

</math>

which is simply the Taylor series of x(bn ) expanded around 1.

<div class="mw-heading mw-heading2"><h2 id="Conjugacy">Conjugacy</h2></div>

If f and g are two iterated functions, and there exists a <a href="Homeomorphism" title="Homeomorphism">homeomorphism</a> h such that g = h−1 ○ f ○ h , then f and g are said to be <a href="Topological_conjugacy" title="Topological conjugacy">topologically conjugate</a>.

Clearly, topological conjugacy is preserved under iteration, as gn = h−1  ○ f n ○ h. Thus, if one can solve for one iterated function system, one also has solutions for all topologically conjugate systems. For example, the <a href="Tent_map" title="Tent map">tent map</a> is topologically conjugate to the <a href="Logistic_map" title="Logistic map">logistic map</a>. As a special case, taking f(x) = x + 1, one has the iteration of g(x) = h−1(h(x) + 1) as

<dl><dd>gn(x) = h−1(h(x) + n),   for any function h.</dd></dl>

Making the substitution x = h−1(y) = ϕ(y) yields

<dl><dd>g(ϕ(y)) = ϕ(y+1),   a form known as the <a href="Abel_equation" title="Abel equation">Abel equation</a>.</dd></dl>

Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at x = 0, f(0) = 0, one may often solve<a href="#cite_note-18">[16]</a> <a href="Schr%C3%B6der's_equation" title="Schröder's equation">Schröder's equation</a> for a function Ψ, which makes f(x) locally conjugate to a mere dilation, g(x) = f '(0) x, that is

Thus, its iteration orbit, or flow, under suitable provisions (e.g., f '(0) ≠ 1), amounts to the conjugate of the orbit of the monomial,

where n in this expression serves as a plain exponent: functional iteration has been reduced to multiplication! Here, however, the exponent n no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit:<a href="#cite_note-19">[17]</a> the <a href="Monoid" title="Monoid">monoid</a> of the Picard sequence (cf. <a href="Transformation_semigroup" title="Transformation semigroup">transformation semigroup</a>) has generalized to a full <a href="Continuous_group" class="mw-redirect" title="Continuous group">continuous group</a>.<a href="#cite_note-20">[18]</a>

This method (perturbative determination of the principal <a href="Eigenfunction" title="Eigenfunction">eigenfunction</a> Ψ, cf. <a href="Carleman_matrix" title="Carleman matrix">Carleman matrix</a>) is equivalent to the algorithm of the preceding section, albeit, in practice, more powerful and systematic.

<div class="mw-heading mw-heading2"><h2 id="Markov_chains">Markov chains</h2></div>

If the function is linear and can be described by a <a href="Stochastic_matrix" title="Stochastic matrix">stochastic matrix</a>, that is, a matrix whose rows or columns sum to one, then the iterated system is known as a <a href="Markov_chain" title="Markov chain">Markov chain</a>.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

There are <a href="List_of_chaotic_maps" title="List of chaotic maps">many chaotic maps</a>. Well-known iterated functions include the <a href="Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> and <a href="Iterated_function_systems" class="mw-redirect" title="Iterated function systems">iterated function systems</a>.

<a href="Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a>,<a href="#cite_note-schr-22">[20]</a> in 1870, worked out special cases of the <a href="Logistic_map" title="Logistic map">logistic map</a>, such as the chaotic case f(x) = 4x(1 − x), so that Ψ(x) = arcsin(√x)2, hence f n(x) = sin(2n arcsin(√x))2.

A nonchaotic case Schröder also illustrated with his method, f(x) = 2x(1 − x), yielded Ψ(x) = −⁠1/2⁠ ln(1 − 2x), and hence fn(x) = −⁠1/2⁠((1 − 2x)2n − 1).

If f is the <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of a group element on a set, then the iterated function corresponds to a <a href="Free_group" title="Free group">free group</a>.

Most functions do not have explicit general <a href="Closed-form_expression" title="Closed-form expression">closed-form expressions</a> for the n-th iterate. The table below lists some<a href="#cite_note-schr-22">[20]</a> that do. Note that all these expressions are valid even for non-integer and negative n, as well as non-negative integer n.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">

</th>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)}</annotation>

</semantics>

</math><img src="./ea882858ec426f59544d07d2999853b19feed9d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.678ex; height:2.843ex;" alt="{\displaystyle f^{n}(x)}" loading="lazy">

</th></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+b}</annotation>

</semantics>

</math><img src="./d492724473c8f6822061d34b3e3edec3818c0969.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.168ex; height:2.343ex;" alt="{\displaystyle x+b}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+nb}</annotation>

</semantics>

</math><img src="./470fff89a334aa2448bf188794d988a8006f1084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.562ex; height:2.343ex;" alt="{\displaystyle x+nb}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax+b\ (a\neq 1)}</annotation>

</semantics>

</math><img src="./e677531e2d5f870709e18488007941578d141c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.278ex; height:2.843ex;" alt="{\displaystyle ax+b\ (a\neq 1)}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{n}x+{\frac {a^{n}-1}{a-1}}b}</annotation>

</semantics>

</math><img src="./20ef1781b9c3d514c78963fcb91afdb9547fde9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.903ex; height:5.343ex;" alt="{\displaystyle a^{n}x+{\frac {a^{n}-1}{a-1}}b}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax^{b}\ (b\neq 1)}</annotation>

</semantics>

</math><img src="./5b1f6255779cde9ce13b186986620e3484c2896f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.146ex; height:3.176ex;" alt="{\displaystyle ax^{b}\ (b\neq 1)}" loading="lazy">

</td>

<msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{\frac {b^{n}-1}{b-1}}x^{b^{n}}}</annotation>

</semantics>

</math><img src="./2efeac22997c4641b421a9c4c5581acaabd5b131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.742ex; height:4.009ex;" alt="{\displaystyle a^{\frac {b^{n}-1}{b-1}}x^{b^{n}}}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+{\frac {b^{2}-2b}{4a}}}</annotation>

</semantics>

</math><img src="./36876c358082081559f6c43dc2ac58f94d2fe0fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.51ex; height:5.676ex;" alt="{\displaystyle ax^{2}+bx+{\frac {b^{2}-2b}{4a}}}" loading="lazy"> (see note)

</td>

<mfrac>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2\alpha ^{2^{n}}-b}{2a}}}</annotation>

</semantics>

</math><img src="./9a4acc712d3449749f585827ae3c4c3671854500.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.343ex; height:5.676ex;" alt="{\displaystyle {\frac {2\alpha ^{2^{n}}-b}{2a}}}" loading="lazy">

where:

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {2ax+b}{2}}}</annotation>

</semantics>

</math><img src="./bd8ea1d312419cad59ab7f1cd4c79799f3db8c48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.982ex; height:5.343ex;" alt="{\displaystyle \alpha ={\frac {2ax+b}{2}}}" loading="lazy"></li></ul>

</td></tr>

<tr>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+{\frac {b^{2}-2b-8}{4a}}}</annotation>

</semantics>

</math><img src="./3445ebdc3460e166f28176b3e6b82d5e64ae5a5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.513ex; height:5.676ex;" alt="{\displaystyle ax^{2}+bx+{\frac {b^{2}-2b-8}{4a}}}" loading="lazy"> (see note)

</td>

<mfrac>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2\alpha ^{2^{n}}+2\alpha ^{-2^{n}}-b}{2a}}}</annotation>

</semantics>

</math><img src="./aea4afd970ce3d4c5448bfaa3fa8d324bc4db07e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.131ex; height:5.676ex;" alt="{\displaystyle {\frac {2\alpha ^{2^{n}}+2\alpha ^{-2^{n}}-b}{2a}}}" loading="lazy">

where:

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {2ax+b\pm {\sqrt {(2ax+b)^{2}-16}}}{4}}}</annotation>

</semantics>

</math><img src="./747dcf5b7db66fb63c863d015729b4b9fbcf0735.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.735ex; height:6.176ex;" alt="{\displaystyle \alpha ={\frac {2ax+b\pm {\sqrt {(2ax+b)^{2}-16}}}{4}}}" loading="lazy"></li></ul>

</td></tr>

<tr>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {ax+b}{cx+d}}}</annotation>

</semantics>

</math><img src="./2e4f53b0ac0f7d25be36aaf6334a2a4e938a5a24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.234ex; height:5.676ex;" alt="{\displaystyle {\frac {ax+b}{cx+d}}}" loading="lazy">   (<a href="Fractional_linear_transformation" class="mw-redirect" title="Fractional linear transformation">fractional linear transformation</a>)<a href="#cite_note-23">[21]</a>

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {a}{c}}+{\frac {bc-ad}{c}}\left[{\frac {(cx-a+\alpha )\alpha ^{n-1}-(cx-a+\beta )\beta ^{n-1}}{(cx-a+\alpha )\alpha ^{n}-(cx-a+\beta )\beta ^{n}}}\right]}</annotation>

</semantics>

</math><img src="./80f439f7ab492078f725ac4f6c3b237ea4eb035a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.202ex; height:7.509ex;" alt="{\displaystyle {\frac {a}{c}}+{\frac {bc-ad}{c}}\left[{\frac {(cx-a+\alpha )\alpha ^{n-1}-(cx-a+\beta )\beta ^{n-1}}{(cx-a+\alpha )\alpha ^{n}-(cx-a+\beta )\beta ^{n}}}\right]}" loading="lazy">

where:

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {a+d+{\sqrt {(a-d)^{2}+4bc}}}{2}}}</annotation>

</semantics>

</math><img src="./3888d38051408c930e15d96813a113feadf27f87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.029ex; height:6.176ex;" alt="{\displaystyle \alpha ={\frac {a+d+{\sqrt {(a-d)^{2}+4bc}}}{2}}}" loading="lazy"></li>

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta ={\frac {a+d-{\sqrt {(a-d)^{2}+4bc}}}{2}}}</annotation>

</semantics>

</math><img src="./dc686a0c5570f0e18bec9026e4a57ea5be14d9df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.874ex; height:6.176ex;" alt="{\displaystyle \beta ={\frac {a+d-{\sqrt {(a-d)^{2}+4bc}}}{2}}}" loading="lazy"></li></ul>

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}{\Big (}h{\bigl (}g(x){\bigr )}{\Big )}}</annotation>

</semantics>

</math><img src="./d08de4b20c180d0bbbbde886bb54821df8ad0d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.95ex; height:4.843ex;" alt="{\displaystyle g^{-1}{\Big (}h{\bigl (}g(x){\bigr )}{\Big )}}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}{\Bigl (}h^{n}{\bigl (}g(x){\bigr )}{\Bigr )}}</annotation>

</semantics>

</math><img src="./46b3d568c5f6cc91c0b9aaa9cd48e894822b4a83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.169ex; height:4.843ex;" alt="{\displaystyle g^{-1}{\Bigl (}h^{n}{\bigl (}g(x){\bigr )}{\Bigr )}}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}{\bigl (}g(x)+b{\bigr )}}</annotation>

</semantics>

</math><img src="./dcc0a0ffea91d25dd46e9050e3b9ce9a46af7817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.674ex; height:3.343ex;" alt="{\displaystyle g^{-1}{\bigl (}g(x)+b{\bigr )}}" loading="lazy">   (generic <a href="Abel_equation" title="Abel equation">Abel equation</a>)

</td>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}{\bigl (}g(x)+nb{\bigr )}}</annotation>

</semantics>

</math><img src="./e2266fe6c35a684fbee5db2b8518a2975f1b5c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.068ex; height:3.343ex;" alt="{\displaystyle g^{-1}{\bigl (}g(x)+nb{\bigr )}}" loading="lazy">

</td></tr>

<tr>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}+b}}}</annotation>

</semantics>

</math><img src="./adfccb234b204723eaee2bb8f0efcd468ecb958b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.546ex; height:3.509ex;" alt="{\displaystyle {\sqrt {x^{2}+b}}}" loading="lazy">

</td>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}+bn}}}</annotation>

</semantics>

</math><img src="./9effd7b07b0320e55c3e6f9afb7ff5fb5a3b7ecb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.94ex; height:3.509ex;" alt="{\displaystyle {\sqrt {x^{2}+bn}}}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}{\Bigl (}a\ g(x)+b{\Bigr )}\ (a\neq 1\vee b=0)}</annotation>

</semantics>

</math><img src="./4acb61f62126f0ad96b1c6e10ada2431435d964a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.852ex; height:4.843ex;" alt="{\displaystyle g^{-1}{\Bigl (}a\ g(x)+b{\Bigr )}\ (a\neq 1\vee b=0)}" loading="lazy">

</td>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{-1}{\Bigl (}a^{n}g(x)+{\frac {a^{n}-1}{a-1}}b{\Bigr )}}</annotation>

</semantics>

</math><img src="./8fa13e9097ae67c2677e5f4c5c3af979d04c7a6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.055ex; height:5.343ex;" alt="{\displaystyle g^{-1}{\Bigl (}a^{n}g(x)+{\frac {a^{n}-1}{a-1}}b{\Bigr )}}" loading="lazy">

</td></tr>

<tr>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {ax^{2}+b}}}</annotation>

</semantics>

</math><img src="./edff8540e2887cf681a921113f49e4966040f683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.775ex; height:3.509ex;" alt="{\displaystyle {\sqrt {ax^{2}+b}}}" loading="lazy">

</td>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {a^{n}x^{2}+{\frac {a^{n}-1}{a-1}}b}}}</annotation>

</semantics>

</math><img src="./8ca8daddfee8dde4bcbb918915ee9d47374955fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.281ex; height:6.343ex;" alt="{\displaystyle {\sqrt {a^{n}x^{2}+{\frac {a^{n}-1}{a-1}}b}}}" loading="lazy">

</td></tr>

<tr>

<msub>

</mrow>

</msub>

<mi>arccos</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{m}(x)=\cos(m\arccos x)}</annotation>

</semantics>

</math><img src="./6396c64701a7880af15a775dbb8ed44e0802e135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.552ex; height:2.843ex;" alt="{\displaystyle T_{m}(x)=\cos(m\arccos x)}" loading="lazy"> (<a href="Chebyshev_polynomials#Trigonometric_definition" title="Chebyshev polynomials">Chebyshev polynomial</a> for integer m)

</td>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mi>arccos</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{mn}=\cos(m^{n}\arccos x)}</annotation>

</semantics>

</math><img src="./a560ab8494e90d4fd48d9722a7c387408ccf2bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.618ex; height:2.843ex;" alt="{\displaystyle T_{mn}=\cos(m^{n}\arccos x)}" loading="lazy">

</td></tr></tbody></table>

Note: these two special cases of ax2 + bx + c 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.

<div class="mw-heading mw-heading2"><h2 id="Means_of_study">Means of study</h2></div>

Iterated functions can be studied with the <a href="Artin%E2%80%93Mazur_zeta_function" title="Artin–Mazur zeta function">Artin–Mazur zeta function</a> and with <a href="Transfer_operator" title="Transfer operator">transfer operators</a>.

<div class="mw-heading mw-heading2"><h2 id="In_computer_science">In computer science</h2></div>

In <a href="Computer_science" title="Computer science">computer science</a>, iterated functions occur as a special case of <a href="Recursion_(computer_science)" title="Recursion (computer science)">recursive functions</a>, which in turn anchor the study of such broad topics as <a href="Lambda_calculus" title="Lambda calculus">lambda calculus</a>, or narrower ones, such as the <a href="Denotational_semantics" title="Denotational semantics">denotational semantics</a> of computer programs.

<div class="mw-heading mw-heading2"><h2 id="Definitions_in_terms_of_iterated_functions">Definitions in terms of iterated functions</h2></div>

Two important <a href="Functional_(mathematics)" title="Functional (mathematics)">functionals</a> can be defined in terms of iterated functions. These are <a href="Summation" title="Summation">summation</a>:

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{b+1,\sum _{i=a}^{b}g(i)\right\}\equiv \left(\{i,x\}\rightarrow \{i+1,x+g(i)\}\right)^{b-a+1}\{a,0\}}</annotation>

</semantics>

</math><img src="./736e4d0c499f23e06c49da1317b5bfe8c4964817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.263ex; height:7.509ex;" alt="{\displaystyle \left\{b+1,\sum _{i=a}^{b}g(i)\right\}\equiv \left(\{i,x\}\rightarrow \{i+1,x+g(i)\}\right)^{b-a+1}\{a,0\}}" loading="lazy"></dd></dl>

and the equivalent product:

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{b+1,\prod _{i=a}^{b}g(i)\right\}\equiv \left(\{i,x\}\rightarrow \{i+1,xg(i)\}\right)^{b-a+1}\{a,1\}}</annotation>

</semantics>

</math><img src="./0e84265d2b313515dd9d9f34ed743ed909234a47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.037ex; height:7.509ex;" alt="{\displaystyle \left\{b+1,\prod _{i=a}^{b}g(i)\right\}\equiv \left(\{i,x\}\rightarrow \{i+1,xg(i)\}\right)^{b-a+1}\{a,1\}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Functional_derivative">Functional derivative</h2></div>

The <a href="Functional_derivative" title="Functional derivative">functional derivative</a> of an iterated function is given by the recursive formula:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\delta f^{N}(x)}{\delta f(y)}}=f'(f^{N-1}(x)){\frac {\delta f^{N-1}(x)}{\delta f(y)}}+\delta (f^{N-1}(x)-y)}</annotation>

</semantics>

</math><img src="./8b2c2f80661a94f479f5a9eff3640049d01ea097.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.282ex; height:6.676ex;" alt="{\displaystyle {\frac {\delta f^{N}(x)}{\delta f(y)}}=f'(f^{N-1}(x)){\frac {\delta f^{N-1}(x)}{\delta f(y)}}+\delta (f^{N-1}(x)-y)}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Lie's_data_transport_equation">Lie's data transport equation</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Shift_operator#Functions_of_a_real_variable" title="Shift operator">Shift operator § Functions of a real variable</a></div>

Iterated functions crop up in the series expansion of combined functions, such as g(f(x)).

Given the <a href="Koenigs_function#Structure_of_univalent_semigroups" title="Koenigs function">iteration velocity</a>, or <a href="Beta_function_(physics)" title="Beta function (physics)">beta function (physics)</a>,

<msub>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(x)=\left.{\frac {\partial f^{n}(x)}{\partial n}}\right|_{n=0}}</annotation>

</semantics>

</math><img src="./7018cf64432fe0f1a8bb9897334a19554dd39f65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.163ex; height:6.509ex;" alt="{\displaystyle v(x)=\left.{\frac {\partial f^{n}(x)}{\partial n}}\right|_{n=0}}" loading="lazy"></dd></dl>

for the nth iterate of the function f, we have<a href="#cite_note-24">[22]</a>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(f(x))=\exp \left[v(x){\frac {\partial }{\partial x}}\right]g(x).}</annotation>

</semantics>

</math><img src="./44cc4696d80602916c4349d9fa5247f3d9bd7ec4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.101ex; height:6.176ex;" alt="{\displaystyle g(f(x))=\exp \left[v(x){\frac {\partial }{\partial x}}\right]g(x).}" loading="lazy"></dd></dl>

For example, for rigid advection, if f(x) = x + t, then v(x) = t. Consequently, g(x + t) = exp(t ∂/∂x) g(x), action by a plain <a href="Shift_operator" title="Shift operator">shift operator</a>.

Conversely, one may specify f(x) given an arbitrary v(x), through the generic <a href="Abel_equation" title="Abel equation">Abel equation</a> discussed above,

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=h^{-1}(h(x)+1),}</annotation>

</semantics>

</math><img src="./81549857ffad322944df5112397624cf78268029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.125ex; height:3.176ex;" alt="{\displaystyle f(x)=h^{-1}(h(x)+1),}" loading="lazy"></dd></dl>

where

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(x)=\int {\frac {1}{v(x)}}\,dx.}</annotation>

</semantics>

</math><img src="./167a3bfa968c1c6ed754b07f8b194b2639237290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.839ex; height:6.009ex;" alt="{\displaystyle h(x)=\int {\frac {1}{v(x)}}\,dx.}" loading="lazy"></dd></dl>

This is evident by noting that

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}(x)=h^{-1}(h(x)+n)~.}</annotation>

</semantics>

</math><img src="./374126dcc67043dbc070a35f9c52bc426d900cd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.198ex; height:3.176ex;" alt="{\displaystyle f^{n}(x)=h^{-1}(h(x)+n)~.}" loading="lazy"></dd></dl>

For continuous iteration index t, then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group,

<msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{t~{\frac {\partial ~~}{\partial h(x)}}}g(x)=g(h^{-1}(h(x)+t))=g(f_{t}(x)).}</annotation>

</semantics>

</math><img src="./1af2cf8a716be24a5de6dcf0e8cfb87c167fa4cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.346ex; height:4.509ex;" alt="{\displaystyle e^{t~{\frac {\partial ~~}{\partial h(x)}}}g(x)=g(h^{-1}(h(x)+t))=g(f_{t}(x)).}" loading="lazy"></dd></dl>

The initial flow velocity v suffices to determine the entire flow, given this exponential realization which automatically provides the general solution to the translation functional equation,<a href="#cite_note-acz-25">[23]</a>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{t}(f_{\tau }(x))=f_{t+\tau }(x)~.}</annotation>

</semantics>

</math><img src="./b94c828e013bb1b00f9a0464c7c5d8a183edc691.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.693ex; height:2.843ex;" alt="{\displaystyle f_{t}(f_{\tau }(x))=f_{t+\tau }(x)~.}" loading="lazy"></dd></dl>

/* start https://en.wikipedia.org/ */

.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}

/* end https://en.wikipedia.org/ */

</style><div class="div-col" style="column-width: 15em;">

<ul><li><a href="Irrational_rotation" title="Irrational rotation">Irrational rotation</a></li>

<li><a href="Iterated_function_system" title="Iterated function system">Iterated function system</a></li>

<li><a href="Iterative_method" title="Iterative method">Iterative method</a></li>

<li><a href="Rotation_number" title="Rotation number">Rotation number</a></li>

<li><a href="Sarkovskii's_theorem" class="mw-redirect" title="Sarkovskii's theorem">Sarkovskii's theorem</a></li>

<li><a href="Fractional_calculus" title="Fractional calculus">Fractional calculus</a></li>

<li><a href="Recurrence_relation" title="Recurrence relation">Recurrence relation</a></li>

<li><a href="Schr%C3%B6der's_equation" title="Schröder's equation">Schröder's equation</a></li>

<li><a href="Functional_square_root" title="Functional square root">Functional square root</a></li>

<li><a href="Abel_function" class="mw-redirect" title="Abel function">Abel function</a></li>

<li><a href="B%C3%B6ttcher's_equation" title="Böttcher's equation">Böttcher's equation</a></li>

<li><a href="Infinite_compositions_of_analytic_functions" title="Infinite compositions of analytic functions">Infinite compositions of analytic functions</a></li>

<li><a href="Flow_(mathematics)" title="Flow (mathematics)">Flow (mathematics)</a></li>

<li><a href="Tetration" title="Tetration">Tetration</a></li>

<li><a href="Functional_equation" title="Functional equation">Functional equation</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

<li id="cite_note-7"><a href="#cite_ref-7">^</a> while f (n) is taken for the <a href="Derivative#Lagrange's_notation" title="Derivative">nth derivative</a>

</li>

<li id="cite_note-NB_Rucker-9"><a href="#cite_ref-NB_Rucker_9-0">^</a> <a href="Alfred_Pringsheim" title="Alfred Pringsheim">Alfred Pringsheim</a>'s and <a href="Jules_Molk" title="Jules Molk">Jules Molk</a>'s (1907) notation nf(x) to denote <a href="Function_composition" title="Function composition">function compositions</a> must not be confused with <a href="Rudolf_von_Bitter_Rucker" class="mw-redirect" title="Rudolf von Bitter Rucker">Rudolf von Bitter Rucker</a>'s (1982) <a href="Rudy_Rucker_notation" class="mw-redirect" title="Rudy Rucker notation">notation</a> nx, introduced by Hans Maurer (1901) and <a href="Reuben_Louis_Goodstein" class="mw-redirect" title="Reuben Louis Goodstein">Reuben Louis Goodstein</a> (1947) for <a href="Tetration" title="Tetration">tetration</a>, or with <a href="David_Patterson_Ellerman" class="mw-redirect" title="David Patterson Ellerman">David Patterson Ellerman</a>'s (1995) nx pre-superscript notation for <a href="Nth_root" title="Nth root">roots</a>.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFHerschel1813" class="citation journal cs1"><a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">Herschel, John Frederick William</a> (1813) [1812-11-12]. <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1813.0005">"On a Remarkable Application of Cotes's Theorem"</a>. <a href="Philosophical_Transactions_of_the_Royal_Society_of_London" class="mw-redirect" title="Philosophical Transactions of the Royal Society of London">Philosophical Transactions of the Royal Society of London</a>. 103 (Part 1). London: <a href="Royal_Society_of_London" class="mw-redirect" title="Royal Society of London">Royal Society of London</a>, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall: 8–26 [10]. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1813.0005">10.1098/rstl.1813.0005</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/107384">107384</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118124706">118124706</a>.</cite>

</li>

<li id="cite_note-Herschel_1820-2"><a href="#cite_ref-Herschel_1820_2-0">^</a> <cite id="CITEREFHerschel1820" class="citation book cs1"><a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">Herschel, John Frederick William</a> (1820). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PWcSAAAAIAAJ&pg=PA5">"Part III. Section I. Examples of the Direct Method of Differences"</a>. A Collection of Examples of the Applications of the Calculus of Finite Differences. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 [5–6]. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200804031020/https://books.google.de/books?hl=de&id=PWcSAAAAIAAJ&jtp=5">Archived</a> from the original on 2020-08-04. Retrieved 2020-08-04.</cite> <a rel="nofollow" class="external autonumber" href="https://archive.org/details/acollectionexam00lacrgoog">[1]</a> (NB. Inhere, Herschel refers to his <a href="#CITEREFHerschel1813">1813 work</a> and mentions <a href="Hans_Heinrich_B%C3%BCrmann" title="Hans Heinrich Bürmann">Hans Heinrich Bürmann</a>'s older work.)

</li>

<li id="cite_note-Peano_1903-3"><a href="#cite_ref-Peano_1903_3-0">^</a> <cite id="CITEREFPeano1903" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1903). Formulaire mathématique (in French). Vol. IV. p. 229.</cite>

</li>

<li id="cite_note-Cajori_1929-4">^ <a href="#cite_ref-Cajori_1929_4-0">a</a> <a href="#cite_ref-Cajori_1929_4-1">b</a> <a href="#cite_ref-Cajori_1929_4-2">c</a> <cite id="CITEREFCajori1952" class="citation book cs1"><a href="Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1952) [March 1929]. "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bT5suOONXlgC">A History of Mathematical Notations</a>. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA: <a href="Open_court_publishing_company" class="mw-redirect" title="Open court publishing company">Open court publishing company</a>. pp. 108, 176–179, 336, 346. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-60206-714-1</bdi>. Retrieved 2016-01-18. <q>[…] §473. Iterated logarithms […] We note here the symbolism used by <a href="Alfred_Pringsheim" title="Alfred Pringsheim">Pringsheim</a> and <a href="Jules_Molk" title="Jules Molk">Molk</a> in their joint Encyclopédie article: "2logb a = logb (logb a), …, k+1logb a = logb (klogb a)."<a href="#CITEREFPringsheimMolk1907">[a]</a> […] §533. <a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">John Herschel</a>'s notation for inverse functions, sin−1 x, tan−1 x, etc., was published by him in the <a href="Philosophical_Transactions_of_London" class="mw-redirect" title="Philosophical Transactions of London">Philosophical Transactions of London</a>, for the year 1813. He says (<a href="#CITEREFHerschel1813">p. 10</a>): "This notation cos.−1 e must not be understood to signify 1/cos. e, but what is usually written thus, arc (cos.=e)." He admits that some authors use cos.m A for (cos. A)m, but he justifies his own notation by pointing out that since d2 x, Δ3 x, Σ2 x mean dd x, ΔΔΔ x, ΣΣ x, we ought to write sin.2 x for sin. sin. x, log.3 x for log. log. log. x. Just as we write d−n V=∫n V, we may write similarly sin.−1 x=arc (sin.=x), log.−1 x.=cx. Some years later Herschel explained that in 1813 he used fn(x), f−n(x), sin.−1 x, etc., "as he then supposed for the first time. The work of a German Analyst, <a href="Hans_Heinrich_B%C3%BCrmann" title="Hans Heinrich Bürmann">Burmann</a>, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan−1, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."<a href="#CITEREFHerschel1820">[b]</a> […] §535. Persistence of rival notations for inverse function.— […] The use of Herschel's notation underwent a slight change in <a href="Benjamin_Peirce" title="Benjamin Peirce">Benjamin Peirce</a>'s books, to remove the chief objection to them; Peirce wrote: "cos[−1] x," "log[−1] x."<a href="#CITEREFPeirce1852">[c]</a> […] §537. Powers of trigonometric functions.—Three principal notations have been used to denote, say, the square of sin x, namely, (sin x)2, sin x2, sin2 x. The prevailing notation at present is sin2 x, though the first is least likely to be misinterpreted. In the case of sin2 x two interpretations suggest themselves; first, sin x ⋅ sin x; second,<a href="#CITEREFPeano1903">[d]</a> sin (sin x). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log2 x, where log x ⋅ log x and log (log x) are of frequent occurrence in analysis. […] The notation sinn x for (sin x)n has been widely used and is now the prevailing one. […]</q></cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help) (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFGulickFord2024" class="citation book cs1">Gulick, Denny; Ford, Jeff (2024). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aVQIEQAAQBAJ&pg=PA2">Encounters with Chaos and Fractals</a> (3rd ed.). CRC Press. p. 2. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781003835776</bdi>.</cite>

</li>

<li id="cite_note-Peirce_1852-6"><a href="#cite_ref-Peirce_1852_6-0">^</a> <cite id="CITEREFPeirce1852" class="citation book cs1"><a href="Benjamin_Peirce" title="Benjamin Peirce">Peirce, Benjamin</a> (1852). Curves, Functions and Forces. Vol. I (new ed.). Boston, USA. p. 203.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link)

</li>

<li id="cite_note-Pringsheim-Molk_1907-8"><a href="#cite_ref-Pringsheim-Molk_1907_8-0">^</a> <cite id="CITEREFPringsheimMolk1907" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Alfred_Pringsheim" title="Alfred Pringsheim">Pringsheim, Alfred</a>; <a href="Jules_Molk" title="Jules Molk">Molk, Jules</a> (1907). Encyclopédie des sciences mathématiques pures et appliquées (in French). Vol. I. p. 195. Part I.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFKuczma1968" class="citation book cs1"><a href="Marek_Kuczma" title="Marek Kuczma">Kuczma, Marek</a> (1968). Functional equations in a single variable. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFKuczma1990" class="citation book cs1">Kuczma, M., Choczewski B., and Ger, R. (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/iterativefunctio0000kucz">Iterative Functional Equations</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-35561-3</bdi>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFCarlesonGamelin1993" class="citation book cs1">Carleson, L.; Gamelin, T. D. W. (1993). <a rel="nofollow" class="external text" href="https://archive.org/details/complexdynamics0000carl">Complex dynamics</a>. Universitext: Tracts in Mathematics. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-97942-5</bdi>.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> Istratescu, Vasile (1981). Fixed Point Theory, An Introduction, D. Reidel, Holland. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-277-1224-7</bdi>.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/66538">"Finding f such that f(f(x))=g(x) given g"</a>. MathOverflow.</cite>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFAldrovandiFreitas1998" class="citation journal cs1">Aldrovandi, R.; Freitas, L. P. (1998). "Continuous Iteration of Dynamical Maps". J. Math. Phys. 39 (10): 5324. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/9712026">physics/9712026</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998JMP....39.5324A">1998JMP....39.5324A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.532574">10.1063/1.532574</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11449%2F65519">11449/65519</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119675869">119675869</a>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFBerkolaikoRabinovichHavlin1998" class="citation journal cs1">Berkolaiko, G.; Rabinovich, S.; Havlin, S. (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjmaa.1998.5986">"Analysis of Carleman Representation of Analytical Recursions"</a>. J. Math. Anal. Appl. 224: 81–90. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjmaa.1998.5986">10.1006/jmaa.1998.5986</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://tetration.org/index.php/Fractional_Iteration">"Tetration.org"</a>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> Kimura, Tosihusa (1971). "On the Iteration of Analytic Functions", <a rel="nofollow" class="external text" href="http://www.math.sci.kobe-u.ac.jp/~fe/">Funkcialaj Ekvacioj</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120426011125/http://www.math.sci.kobe-u.ac.jp/~fe/">Archived</a> 2012-04-26 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> 14, 197-238.

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFCurtrightZachos2009" class="citation journal cs1"><a href="Thomas_Curtright" title="Thomas Curtright">Curtright, T. L.</a>; <a href="Cosmas_Zachos" title="Cosmas Zachos">Zachos, C. K.</a> (2009). "Evolution Profiles and Functional Equations". Journal of Physics A. 42 (48): 485208. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0909.2424">0909.2424</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009JPhA...42V5208C">2009JPhA...42V5208C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1751-8113%2F42%2F48%2F485208">10.1088/1751-8113/42/48/485208</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115173476">115173476</a>.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> For explicit instance, example 2 above amounts to just f n(x) = Ψ−1((ln 2)n Ψ(x)), for any n, not necessarily integer, where Ψ is the solution of the relevant <a href="Schr%C3%B6der's_equation" title="Schröder's equation">Schröder's equation</a>, Ψ(√2x) = ln 2 Ψ(x). This solution is also the infinite m limit of (f m(x) − 2)/(ln 2)m.

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> Curtright, T. L. <a rel="nofollow" class="external text" href="http://www.physics.miami.edu/~curtright/Schroeder.html">Evolution surfaces and Schröder functional methods.</a>

</li>

<li id="cite_note-schr-22">^ <a href="#cite_ref-schr_22-0">a</a> <a href="#cite_ref-schr_22-1">b</a> <cite id="CITEREFSchröder1870" class="citation journal cs1"><a href="Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Schröder, Ernst</a> (1870). "Ueber iterirte Functionen". Math. Ann. 3 (2): 296–322. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01443992">10.1007/BF01443992</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:116998358">116998358</a>.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> Brand, Louis, "A sequence defined by a difference equation," <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a> 62, September 1955, 489–492. <a rel="nofollow" class="external text" href="https://www.jstor.org/discover/10.2307/2307362">online</a>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFBerksonPorta1978" class="citation journal cs1">Berkson, E.; Porta, H. (1978). <a rel="nofollow" class="external text" href="https://doi.org/10.1307%2Fmmj%2F1029002009">"Semigroups of analytic functions and composition operators"</a>. The Michigan Mathematical Journal. 25: 101–115. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1307%2Fmmj%2F1029002009">10.1307/mmj/1029002009</a>.</cite> <cite id="CITEREFCurtrightZachos2010" class="citation journal cs1">Curtright, T. L.; Zachos, C. K. (2010). "Chaotic maps, Hamiltonian flows and holographic methods". Journal of Physics A: Mathematical and Theoretical. 43 (44): 445101. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1002.0104">1002.0104</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010JPhA...43R5101C">2010JPhA...43R5101C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1751-8113%2F43%2F44%2F445101">10.1088/1751-8113/43/44/445101</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115176169">115176169</a>.</cite>

</li>

<li id="cite_note-acz-25"><a href="#cite_ref-acz_25-0">^</a> Aczel, J. (2006), Lectures on Functional Equations and Their Applications (Dover Books on Mathematics, 2006), Ch. 6, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0486445236</bdi>.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFGill2017" class="citation web cs1"><a href="John_Gill_(climber)" title="John Gill (climber)">Gill, John</a> (January 2017). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/362010262">"A Primer on the Elementary Theory of Infinite Compositions of Complex Functions"</a>. Colorado State University.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-30" href="https://en.wikipedia.org/wiki/?title=Iterated_function&oldid=1303395161">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>