Schroeder equation

The Schroeder equation (Уравнение Шредера, Уравнение Шроедера, Уравнение Шрёдера) or Schröder equation or Schröder's equation can be written as follows:

(1) \(~ ~ ~ g\Big(T(z)\Big)= s \, g(z)\)

where function \(~T~\) is supposed to be given, while constant \(s\) and function\(~g~\) should be constructed. Usially, it is assumed that \(s\!=\!T'(0)\), and both, \(T\) and \(g\) are holomorphic at least in some vicinity of zero.

The Schroeder equation is related to the Abel equation and the transfer equation; the solutions (called Schroeder functions ) are related to the superfunctions and the Abel funcitons. These relations are especially simple for the case when function \(~T~\) has a real fixed point \(~L~\).

About the terminology
The topic about fractional iterates which seem to be the main application of the Schroeder equation had almost no advances during century 20, and there is no established terminology. The goal of this article is to compare various notations and choose the most convenient terminology for TORI.

The Schroeder equation in mathematics should not be confused with the Schroeder – Le Chatelier equation describing transitions between solid, liquid and gas , and Schroeder should not be confused with shredder ; their functions are pretty different.

Relation to the Abel equation
Consider logarithm to base \)~s~\( from both sides of equation (1), assuming that \)~s~\( and \)~g(z)~\( are not real negative number nor zero. This gives

(2) \)~ ~ ~ \log_s\Big(~ g\big( T(z)\big)~\Big) = 1 + \log_s\big(g(z)\big)~\(

Let

(3) \)~ ~ ~ G(z)=\log_s\big( g(s) \big)~\(

The substitution of (3) into (2) gives for function \)~G~\( the Abel equation

(4) \)~ ~ ~ G\big(T\big(z)\big) = 1 + G(z)~\(

The transfer from the Schroeder equation (1) to the Abel equation (4) eliminates the parameter \)~s~\(. Once the solution \)~G~\( of equation (4) is found, the solution \)~g~\( of equation (1) can be expressed through this \)~G~\(:

(5) \)~ ~ ~ g(z)=\exp_s\Big( G(z) \Big)~\(

In such a way, the logarithmic - exponential transforms (3) and (5) relate the solution \)~g~\( of the Schroeder equation (1) and the solution \)~G~\( of the Abel equation (4).

Zooming equation
While \)~g~\( is Schroeder function and solution of the Schroeder equation (1), we need also the name for inverse function, id est, \)~f=g^{-1}~\( and, the equation for this function and name for this equation. While no established system of notation can be found for these objects, let such \)~g~\( be called zoom function and such an equation be called zoom equation. The zoom equation is easy to obtain from (1).

As \)~z~\( in (1) is dummy variable, it can be replaced to any expression. Make the substitution \)~z\mapsto f(z)~\(, and apply function \)~f~\( to both sides of the resulting equation. This gives

(6) \)~ ~ ~ f\Big(~g\Big( T\big(f(z)\big)\Big)~\big) = f\Big( s\, g\big(f(z)\big) \Big)~\(

Assumption that \)~f(g(z))\!=\!z~\( and \)~g(f(z))\!=\!z~\( gives the zoom equation in the following form:

(7) \)~ ~ ~ T\big(f(z))= f(s\, z)~\(

This equation determines, how does value of function \)~f~\( transform at the zooming of the argument with factor \)~s~\(. This is reason to refer equation (7) as zooming equation. The \)~n\(th iterate of function \)~T~\( determines the the zooming of the argument with factor \)~s^n~\(:

(8) \)~ ~ ~ T^n\big(f(z))= f(s^n\, z)~\(

This gives way to express the \)~n\(th iteration of function \)T\(,

(9) \)~ ~ ~ T^n (z) = f\big(s^n\, g(z)\big)~\(

In this expression, \)~n~\( can be interpreted as any complex number. Then, for given \)~f~\(, \)~g~\( and \)~s~\(, equation (9) provides a tool to evaluate the non-integer iterates of function \)T\(. Following the work by I.N.Baker , Henryk Trappmann suggests for such an iterate name regular iteration or regular iterate , and this name will be used in TORI as one of meanings of term regular iterate or regular iteration.

Regular iterate
The zooming equation (7) and the Schroeder equation (1) should have sense for evaluation of iterates of function \)~T~\(, that have fixed point \)~L\!=\!0~\(, and is supposed to be regular in vicinity of this point. The corresponding iterate of function \)~T~\( is called regular iterate or regular iteration. While these two terms are considered as equivalent, but in future, the term regular iteration may be reserved for the tool of construction of the specific non–integer iterate, letting the term regular iterate in sense of noon to specify the result of such a construction.

Fractional iterates
The approach to the regular iterate can be based on the concept of fractional iterate.

For a given function \)~T~\(, holomorphic in vicinity of its fixed point \)~L~\(, the function \)t_r=T^r\( is considered as \)r\( fractional iterate, if \)~r\!=\!m/n~\( for some integer numbers \)~m, n~\( and

(10) \)~ ~ ~ t_c^m(z)=T^n(z)~\(

for all \)~z~\( in some vicinity of \)~L~\(. If such a function \)t_r\( is also regular in vicinity of \)~L~\(, then such a fractional iterate is called regular iterate.

Example with iterates of exponential
In particular, for base \)b<~\(HenrykEta, the tetration to base \)b\( can be considered as regular iterate of exponential at its lower fixed point \)~L_1=\mathrm{Filog}(b\!+\!\mathrm i\, o)\(, where Filog is function that determined the fixed points of logarithm,

(11) \)~ ~ ~\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1- \pi \mathrm{i}\big)}{-z} = \frac{\mathrm{WrightOmega}\!\big(\ln(z)-\pi \mathrm{i}\big)}{-z}\(

The descriptions and numerical implementations of the Tania function and WrightOmega and Filog are available.

Another fixed point \)~L_2=\mathrm{Filog}(b\!-\!\mathrm i\, o)\( indicates another regular iterate of exponential to base \)b\(, and these two regular iterates are not the same. However, in vicinity or the real axis, the regular iterates, corresponding to different fixed point, may look pretty similar. For example, for \)b=\sqrt(2)\(, half–iterates of the exponential, constructed at fixed points \)L_1$

Keywords
fractional iterate, iteration, transfer function, superfunction