Exotic iteration

Exotic iteration refers to the construction of superfunction, the Abel function and corresponding iterates of the transfer function \(T\) at its fixed point \(L\) such that \(T'(L)\!=\!1\). For this case, the Regular iteration is not possible: in the leading term of the expansion of superfunction with exponentials, the increment \(k\!=\!\ln(T'(L)\) becomes unity, giving superfunction that is constant.

However, even if the \(T'(L)\!=\!1\), construction of superfunction is possible. Few examples are considered in this article.

 This article is under construction 

Introduction
For given transfer function \(T\), superfunction is solution \(F\) of the Transfer equation

\( F(z\!+\!1)=T(F(z)) \)

The fixed point \(L\) is solution of equation \(T(L)\!=\!L\). The goal is to construct suterfunction \(F\) such that

\( \displaystyle \lim_{z\rightarrow \infty} F(z)=L\)

at least for some directions of approaching \(z\) to infinity. For the transfer function, regular at the fixed point and such that \(T'(L)\ne 1\), the superfunction can be constructed with Regular iteration, and such a superfunction exponentially approaches the fixed point \(L\).

This article deals with case when \(F'(0)=1\) and the regular iteration in its original form cannot be applied. The increment of the leading exponential would be just zero. However, in many cases the superfunction, that approaches \(L\), still can be constructed.

About the terminology
The name exotic iteration is suggested in order to distinguish the superfunction, abelfunction and the resulting non-integer iterates of the transfer function from the case of regular iteration, while the iterates are regular (id est, holomorphic) functions at least in some vicinity of the fixed point.

Names Irregular iteration or singular iteration could be also used to define the same functions. Term exotic iteration is shorter, and for this reason it is chosen for TORI. This term indicates, that exotic iteration happen not so often, as the condition \(T'(L)\!=\!1\) is only specific occasion, that may happen for some physical dependences, that can be interpreted as transfer functions. The ability to construct superfunction, abelfunction and the non-integer iterates had been declared; therefore, the special case \(T'(L)\!=\!1\) also should be considered, even if it is qualified as "exotic".

Conjugation
Consider the transfer finction \(T\) with fixed point \(L\); id est, \(T(L)\!=\!L\).

WIthout loss of generality, we may consider the case of \(L\!=\!0\).

If initially some transfer function \(t\) has fixed point \(\ell\), id est, \(t(\ell)=\ell\), then consider the conjugation below. let

\(T(z)=t(z+\ell)-\ell\)

While \(\ell\) is fixed point of \(t\), zero is fixed point of \(T\). The superfunction \(f\) for the transfer function \(t\), that approaches \(\ell\) at infinity, can be constructed as follows:

\(f(z)=F(z\!-\!\ell)+\ell\)

where \(F\) is superfunction of \(T\), constructed at it fixed point zero, that approaches zero at infinity. Through the corresponding Abel function \(G=F^{-1}\), the abelfunction \(g\!=\!f^{-1}\) of \(t\) can be exressed in the similar way

\(g(z)=G(z-\ell)+\ell\)

In such a way, it is sufficient to consider case of fixed point equal to zero, \(L\!=\!0\).

Regular transfer function
Let \(~ T(z)=z+b z^2 + c z^3+..\)

where \(b\!\ne\!0\).

As the fixed point of the transfer function is zero, the letter \(L\) used to denote it in the previous section can be used for other purposes.

Let \(~ \displaystyle P_m(L)=\sum_{n=0}^n a_{m,n}\, L^n\)

wehre \(m>0\) is positive integer, and \(a\) are constant coefficients, that may (and should) depend on the coefficients \(b\), \(c\), .. of expansion of the transfer function \(T\) at zero. For some fixed natural number \(M>0\), the asymptotic expansion of the superfunction \(f\) can be constructed as follows:

\(\displaystyle f_M(z)= \frac{A}{-z} \left( 1+ \sum_{m=1}^{M-1}  \frac {P_m(\ln(-z))} {(-z)^n} \right)\)

Then superfunction

\(\displaystyle F(z)=f_M(z)+ O\left( \frac{\ln(-z)^M}{(-z)^{M+1}} \right)\)

For simplicity, it can be assumed that \(a_{1,0}=0\). Then, the first 10 coefficients \(a\) can be calculated with the Mathematica routine below: T[z_] = z + b z^2 + c z^3; P[m_, L_] := Sum[a[m, n] L^n, {n, 0, m}] f[M_, z_] := 1/(-b z) (1 + Sum[ P[m, Log[-z]]/z^m, {m, 1, M}] ) v[0] = {a[1, 0] -> 0};

m = 1; s[m] = Simplify[ ReplaceAll[Series[f[m, -1/x + 1] - T[f[m, -1/x]], {x,0,m+2}], v[m-1]]] t[m] = Expand[Numerator[Coefficient[s[m], x^(m+2)]]]; u[1] = Extract[Solve[t[m] == 0, a[1, 1]], 1] v[m] = Join[v[m - 1], u[m]];

For[m=2, m<11, m++, s[m] = ReplaceAll[ Simplify[ ReplaceAll[Series[f[m,-1/x+1] - T[f[m,-1/x]], {x,0,m+2}], v[m-1]]], Log[x]->L]; t[m] = Numerator[Coefficient[Normal[s[m]], x^(m+2)]]; tab[m] = Table[Coefficient[t[m] L, L^n] == 0, {n, 1, m + 1}]; par[m] = Table[a[m, n], {n, 0, m}]; u[m] = Extract[Solve[tab[m], par[m]], 1]; Print["m=", m, ", ", "u[m]=", u[m]]; v[m] = Join[v[m - 1], u[m]];]

Expansion of the transfer finction
Assume, that expansion of the transfer function begins with

\( T(z)=z+p z^{r+1} ..\)

where \(p\!\ne\!0\) and \(r\!>\!0\) are constants, and .. denote terms that decay at zero faster, than the precedent term. In principle, one can consider also complex values \(p\) and even complex \(r\), but the explicit plots for the not-real function are more complicated; so, for simplicity, it is convenient to consider case of real \(p\) and \(r\).

Then, the superfunction \(F\) may have the following expansion at infinity:

\( F(z)=(-zpr)^{-1/r} + o(|z|^{-1/r})\)

Then, \(F(z\!+\!1)\) expands as follows:

\(\displaystyle F(z\!+\!1)=(-zpr)^{-1/r}(1+1/z)^-1/r +..=F(z)+ F(z) \frac{-1}{rz} +..\)

T(F(z)) expands as follows:

\(\displaystyle T(F(z)=F(z)+ p (-zpr)^{-1} +.. = F(z)+ F(z) \frac{-1}{rz} +..\)

providing asymptotic agreement with the transfer equation (1).

Similar asymptotic representation can be written for the Abel function \(G=F^{-1}\)

\(G(z)=-pr z^{-r}\)

Specification of the transfer function allows to write out more terms in the expansions of the superfunction and the Abel function.

Exponential to base exp(-1/e)
Historically, the first exotic expansion had been reported for the exponential to base \(\exp(1/\mathrm e)\) .

Let

\(\displaystyle T(z)=\exp(z/\mathrm e+1) - \mathrm e ~ \approx~ z+\frac{1}{2\mathrm e} z^2 + O(z^3)\)

In the notations above, this corresponds to \(p\!=\!\frac{1}{2\mathrm e}\) and \(r\!=\!1\). In this case, the superfunction can be expanded as follows:

\(\displaystyle F(z)=-\frac{2\mathrm e}{z}\left( 1 - \frac{\ln(\pm z)}{3z} + ..\right)\)

The leading term of this expansion corresponds to the general formula above. Sign + or - should be chosen in order to construct superfunction that decays or grows along the real axis, correspondently; however, in the last case, the only in certain directions in the complex plane the superfunction approaches zero at infinity, and the positive direction of the real axis does not belong to this range.

Zex
Let \(T(z)=z \exp(z)=\mathrm{zex}(z)~\). This transfer function is considered in articles zex, SuZex and AuZex. Expansion of function zex is obvious:

\( \mathrm {zex}(z) = z+ z^2 + ..\)

In this case, \(p\!=\!1\), \(r\!=\!1\). The asymptotic expansion of superfunction for zex can be written as follows:

\(\displaystyle F(z)=\frac{-1}{z} \left(1+\frac{1}{2} \frac{\ln(\pm z)}{z} +O\left(\frac{\ln(|z|)}{z}\right)^2 \right)\)

Again, the choice of the sign + or - depends on the behaviour of superfunction: should it decay to zero at \(-\infty\) and to grow along the real axis, or should it approach zero at \(+\infty\).

sin
For \(T\!=\!\sin\), the parameters have following values:

\(~r\!=\!2~\), \(~p\!=\! -1/6~\)

These values correspond to expansion

\( \sin(z)=z-z^3/6+O(z^5) \)

This case is considered in article Super sin .

The expansion of superfunction of sin has the following form:

\( ~ ~ ~ \displaystyle \mathrm{SuSin}(z)= \sqrt{\frac{3}{z}} \left(1 - \frac{3}{10} \frac{\ln(z)}{z}+ O\left(\frac{1}{z}\right) \right) \)

The leading term \(\sqrt{3/z}\) agrees with the more general formula.

Function, similar to sin
Only the leading term of the asymptotic expansion of superfunction is determined by parameters \(p\) and \(r\) of the expansion of the transfer function. For the transfer function

\(\displaystyle T(z)=z-\frac{1}{6} z^3\),

the expansion of the superfunction \(F\), similar to SuSin, can be written as follows:

\( ~ ~ ~ \displaystyle F(z)= \sqrt{\frac{3}{z}} \left(1 - \frac{3}{8} \frac{\ln(z)}{z}+ O\left(\frac{1}{z}\right) \right) \) More terms fd this expansion can be calculated with the pbcious modification of the Mathematica code, suggested in the article SuSin for the superfunction of sin.

Conclusion
The strong conjectures about superfunctions, namely, Conjecture on superfunctions, had beed formulated in 2013–2014. In order to support them, the special case of non–regular iteration above needed to be considered.

In principle, the superfunction for the "exotic" transfed functions can be constructed and evaluated, using the only leading term of the asymptotic expansion and the transfer equation. However, for the efficient numerical implementation, several terms of the asymptotic expansion should be calculated. For a specific transfer function, Mathematica can calculate of order of 10 coefficient of the expansion automatically. Examples of the code to do this are suggested in articles SuZex and SuSin.

Keywords
Iteration, Superfunction, Abel function, Transfer function,