Iteration of function

Fig.1. Iterates of $T(z)=z^{2}~$: $~y\!=\!T^{n}(x)\!=\!x^{2^{n}}~$ for various $n$

Fig.2. Iterates of $T(z)=\exp(z)~$: $~y\!=\!T^{n}(x)\!=\!\exp ^{n}(x)~$ for various $n$

Fig.3. Iterates of $T(z)=z\exp(z)~$: $~y\!=\!T^{n}(x)\!=\!\mathrm {zex} ^{n}(x)~$ for various $n$

Fig.4. Iterates of $T(z)=z+\exp(z)~$: $~y\!=\!T^{n}(x)\!=\!\mathrm {tra} ^{n}(x)~$ for various $n$

The iterate of a function , or iterate (итерация), is a function, expressed as repetition of another (iterated) function; the latter may be called the iterand.

Any function by itself is considered as its first iterate.

The zeroth iterate of a function is considered to be the identity function.

The minus first iteration of a function is its inverse function

In general, iteration of a function $T$ is denoted with superscript; $T^{n}$ means the $n$th iteration of function $T$, i.e.,

(1) $~~~\displaystyle T^{n}(z)=T\!{\Big (}T\!{\big (}...T(z)...{\big )}{\Big )}$

where the function $T$ in the right hand side of equation is repeated $n$ times. (Note, however, that this abstract notation is not always used consistently for actual functions: for example, cos²θ means (cos θ)², not cos cos θ, by convention.)

This may be expressed with the equations

(2) $~~~T^{0}(z)\!=\!z~$, $~T^{n}(z)=T{\Big (}T^{n-1}(z){\Big )}~$ for $n\!>\!0$

Term iteration may have also another meaning, indicating repetition of some process (even if such a repetition does not define a new function).

Physical sense

If $T$ is transfer function of some device, say, amplifier, then, the transfer function of combination of two such devices can be expressed as $T^{2}$; and the transfer function of combination of $n$ identical devices can be written as $T^{n}$ .

Nest

In Mathematica, the special operation is incorporated, called Nest. Usually, Nest has 3 arguments. The first argument is name of iterand, which is function. The second argument is initial value of the iterations, and the last argument is number of iteration. With Nest, the iteration of function $f$ can be written as follows:

(3) $~~~f^{n}(z)=\mathrm {Nest} [f,z,n]$

In Mathematica, all arguments of all functions should be written in squared parenthesis. Up to year 2012, the Nest is implemented only for the cases, when number $n$ of iterations can be simplified to a natural number, expressed with integer constant. Attempts to call Nest to evaluate a non–integer iterate of a function are interpreted as error.

Non–integer iterate

The non–integer iterate is simple and obvious in the case when the iterand is linear function; then the iteration can be expressed through the elementary function. Let $T(z)=a+bz$. Then

(4)$~~~\displaystyle T^{n}(z)=a{\frac {1\!-\!b^{n}}{1\!-\!b}}+b^{n}z$

This expression allows the straightforward interpretation for non–integer $n$.

The case of power function is also simple. Let $T(z)=z^{a}$, where $a\!>\!1$. Then $T^{n}(z)\!=\!z^{a^{n}}$. In this expression, $n$ has no need to be integer. For $a\!=\!2$, the iterations of quadratic function are shown in Figure 1.

For more general case, the non–integer iterates of a function $T$ can be defined through its superfunction $F$ and the Abel function $G\!=\!F^{-1}$; the iterand $T$ is treated as the transfer function, and the superfunction $F$ satisfies the transfer equation

(5)$~~~F(z\!+\!1)=T\!{\Big (}F(z){\Big )}$

Then, the $n$th iteration of the transfer function can be expressed as

(6)$~~~T^{n}(z)=F{\Big (}n+G(z){\Big )}$

In this expression, parameter $n$ also has no need to be integer. Once holomorphic functions $F$ and $G$ are established, the function can be iterated even complex number of times. For a given transfer function $T$, the establishment of the superfunction $F$ and the Abel function $G$ is not trivial, but solvable problem. It had been formulated in 1950 by Helmuth Knezer for iterates of exponential, the half-iterate of the exponential appears in the title of his article. In century 21, such a function had been calculated and plotted ; the iterates of the exponential are shown in Figure 2. Then, the formalism had been extended to other functions, in particular, to the exponentials to other bases , factorial and the logistic sequence .

For the non–integer iterates, it should be assumed that $T$ is holomorphic function, and the superfunction $G$ is also expected to be holomorphic. In order to provide uniqueness of the superfunction, the certain requirements on its behavior should be assumed. Usually, $F$ approaches some fixed points of function $T$ at infinity (see the special articles Superfunction and Abel function).

Usually, the physically–meaningful transfer functions have a real fixed point and the corresponding superfunction and the Abel function can be evaluated with the regular iteration , factorial and the logistic sequence . If the fixed point are com[lex (not real), the superfunction and the Abel function can be constructed with the Cauchi iteration . With superfunction $F$ and the Abel function $G$, the non–integer (and even complex) iterate of the transfer function $T$ is straightforward; any holomorphic function can be declared as a transfer function and iterated any number of times by formula (6).

In 2011, Henryk Trappmann had suggested an example of a transfer function that has no fixed points, let it be called Trappmann function or tra; $\mathrm {tra} (z)=z\!+\!\mathrm {e} ^{z}$. Such a transfer function was supposed to be difficult to iterate. However, the superfunction (say, "SuperTrappmann") can be expressed through that for function zex, $\mathrm {zex} (z)\!=\!z\exp(z)$; then the "SuperTrappmann" can be called simply hen, and the iterates of the Trappmann function can be expressed with (6) at $F\!=\!$hen and $~G\!=\!\mathrm {ahe} =\mathrm {hen} ^{-1}$. The iterates of zex are shown in Figure 3, and the iterates of the Trappmann function tra are shown in Figure 4.

It seems, that the non-integer iterates can be defined for any holomorphic function, but the rigorous mathematical proof of such conjecture is required.

In some cases, the non–integer iterate can be defined also through the Schroeder function.

In general, the non-integer iterate is not unique, and the specific behavior of the superfunction $F$ should be assumed for the uniqueness.

Examples

This section offers a little bit more detailed description of figures, mentioned above.

Elementary functions

Let

$\displaystyle T(z)\!=\!{\frac {a^{2}+(2a\!+\!b)z}{b-z}}$

The corresponding superfunction and Abel function are

$\displaystyle F(z)={\frac {az\!+\!b}{1\!-\!z}}~$, $~\displaystyle G(z)={\frac {z\!-\!b}{a\!+\!z}}$

give the iterate

$\displaystyle T^{n}(z)\!=\!{\frac {a^{2}+{\Big (}a(1\!+\!n)+b{\Big )}z}{a+b-an-nz}}$

More "solvable" transfer functions can be constructed using Table of superfunctions. However, for some cases the solution cannot be simply expressed with elementary functions. Few of them are mentioned below.

Special functions

Figure 1 shows iterates of quadratic function, $~T(z)\!=\!z^{2}~$. In this case, the $n$th the iterate $~T^{n}(z)\!=\!z^{2^{n}}~$ can be expressed through the power function.

Figure 2 shows iterates of the exponential, $~T(z)\!=\!\exp(z)$. In this case, for non–integer $n$, the $n$th iterate $~T^{n}(z)\!=\!\mathrm {tet} {\Big (}n\!+\!\mathrm {ate} (z){\Big )}~$ is expressed through tetration tet and ArcTetration ate, and cannot be easy represented through elementary functions.

Figure 3 shows iterates of function zex, $~T(z)\!=\!\mathrm {zex} (z)\!=\!z\exp(z)~$. In this case, for non–integer $n$, the $n$th iterate

$~T^{n}(z)\!=\!\mathrm {SuZex} {\Big (}n\!+\!\mathrm {AuZez} (z){\Big )}~$

can be expressed through functions SuZex and AuZex, which are, of course, superfunction and Abel function for the transfer function zex.

Figure 4 shows iterates of the Trappmann function, $~T(z)\!=\!\mathrm {Tra} (z)\!=\!z\!+\!\exp(z)~$. In this case, for non–integer $n$, the $n$th iterate

$~T^{n}(z)\!=\!\mathrm {Tra} ^{n}(z)\!=\!\mathrm {hen} {\Big (}n\!+\!\mathrm {ahe} (z){\Big )}~$

is expressed through the Henryk function hen and the ArcHenryk function ahe , and cannot be easy represented through elementary functions. However functions hen and ahe can be expressed in terms of SuZex and AuZex mentioned above in the following way:

$\mathrm {hen} (z)=\ln {\Big (}\mathrm {SuZex} (z){\Big )}$

and

$\mathrm {ahe} (z)=\mathrm {AuZex} {\Big (}\exp(z){\Big )}$

The number of examples can be greatly extended, using the Table of superfunctions, that offers triads $T,F,G$, and general expression $T^{n}(z)=F{\Big (}n\!+\!G(z){\Big )}$. In this way, one can plot the iterates of exponential to various bases, iterates of factorial, iterates of the logistic operator, and iterates of any functions, assuming, that we can choose the appropriate superfunction and the Abel function. In principle, any holomorphic function can be iterated any number $n$ of times, even complex number of times.

This article is adopted from TORI .

Keywords

Superfunction, Abel function, Transfer equation, Abel equation