Nemtsov function and its iterates

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Nemtsov function and its iterates is article about the Nemtsov function, adaptation from version, prepared for publication, with goal to check the cross-references and to catch misprints, if any.

Author: Dmitrii Kouznetsov.

Abstract: The Nemtsov function appears as polynomial \(~\mathrm{Nem}_q(z)=z+z^3+qz^4 \,\); \(~\) \(q\) is parameter. The Superfunction, Abelfunction and iterates \(\mathrm{Nem}_q^n\) for complex \(n\) are constructed.

Keywords: Nemtsov function, Superfunction, Abelfunction, Iterate

Introduction

Overview

In 1950, the interest to the non-integer iterates had been boiled-up with iterates of exponential and, in particular, iterate half of the exponential [1], id est, function \(\varphi\) such that \(\varphi(\varphi(z))\!=\!\exp(z)\). The problem of iteration [2][3][4][5] of holomorphic function had been formulated, although until year 2009, no efficient algorithm for computation of non-integer iterates (except few special functions) had been suggested. Then, such algorithms had been reported; and not only for the exponent to various bases [6][7][8][9][10][11][12][13][14], but also for other holomorphic functions: for factorial by [15], for the ``logistic operator \(z \!\mapsto\! s\,z\,(1\!-\!z)\) by [16] for the Trappmann function \(z \!\mapsto\! z\!+\!\exp(z)\) and function \(z \!\mapsto\! z \exp(z)\) by [17] and for sin by [18].

For various examples of a holomorphic functions \(T\), called transfer function, the iterates can be expressed (and evaluated) through the superfunction \(F\), which is solution of the transfer equation

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

and the corresponding abelfunction \(G=F^{-1}\):

\(\!\!(2)~~ T^n=F\big(n\!+\!G(z)\big) \)

Here, the superscript after the name of function indicates the number of its iterate; this notation had been suggested in 1993 by W.Bergweiler [4]. In these notations,

\(\!\!(3)~~T^0(z)=z\)
\(\!\!(4)~~T^1(z)=T(z)\)
\(\!\!(5)~~T^2(z)=T(T(z))\)
\(\!\!(6)~~T^3(z)=T(T(T(z)))\)

and so on; in particular, \(\sin^2(x)\) denotes \(\sin(\sin(x))\), but neither \(\sin(x)^2\) nor \(\sin(x^2)\). In the representation \rf{Tn}, the number \(n\) of iterate has no need to be integer. It can be a fractal and even a complex number.

The abelfunction \(G=F^{-1}\) satisfies the Abel equation

\(\!\!(7)~~ G(T(z))=G(z)+1\)

In addition to equations (1) and/or (7), some supplementary requirements on the asymptotic behaviour of \(F\) and/or \(G\) are applied in order to provide the uniqueness [11][12][19].

Alternatively, the explicit way of the computation may be postulated in the definition of the superfunction \(F\) and/or the Abel function \(G\); then, this way of computation determines the asymptotic behaviour.

The success of construction of superfunctions for various transfer functions provoke the attempts to construct a holomorphic transfer function \(T\) such that its iterates cannot be determined in natural intuitive way, nor the efficient algorithm for the superfunction and abelfuctions can be constructed. Consideration of the Nemtsov function describes the failure of such an attempts; the superfunciton and the abelfunction for the Nemtsov function can be defined, calculated and supplied with the efficient numerical implementation for the evaluation.

Previously published methods: Regular iteration

For the case of a real-holomorphic transfer function \(T\) with real fixed point \(L\), (id est, \(T(L)\!=\!L\)), such that \(T'(L)>0\), the most important construction is the Regular iteration.

Keeping in mind the reading by the colleagues, who did not read the previous articles (an in order not to force the reader to dig the previous publications), the regular iteration is shortly repeated here.

Search for the asymptotic solution of the transfer equation (1) in the following form:

\( \displaystyle F(z) = L+\sum_{n=1}^{N} a_n \varepsilon^n + o(\varepsilon^N)\)

where \(N\) is natural number, \(\varepsilon=\exp(kz)\) for some constant \(~k~\) and \(~a~\) are constant coefficients.

The substitution into the transfer equation and the asymptotic analysis at \(\varepsilon \ll 1\) gives

\(k=\log(K)=\log(T′(L))\)

and set of equations for coefficients \(a\). It is convenient to set \(a_1=1\) Then, other coefficients appear as solutions of equations

\(a_2 K - a_2 K^2 + {T\,''(L)}/{2} =0 \)

\(a_3 K - a_3 K^2 + 2 a_2 {T\,''(L)}/{2}+ {T\,'''(L)}/{6} = 0\)

and so on.

The solution is singular at \(K=\log(T′(L)=0\).

Namely this case is of interest in this article. The Nemtsov function is example of such function:

\(\mathrm{Nem}_q(z)=z+z^3+qz^4\)

\(L=0\),

\(\mathrm{Nem}_q'(L)=1\)

Exotic iteration: \(T'(L)=1\), \(T''(L) \ne 0\)

At the regular iteration of a real–holomorphi function \(T\) the expansion of the superfunction \(F\) begins with constant (fixed point \(L\)) and the exponential term.

I the case \(T'(L)=0\) the expansion of super function begins with with constant (fixed point \(L\)) and power function, and also includes the logarithmic term. This can be written as follows:

\(T(L)=L\)

\(\displaystyle F[z] \sim L + \frac{a}{z}+ \sum_{m=1}^M \frac{P_m(\ln(\pm z))}{z^{m+1}}+O\left(\frac{\ln(\pm z)^{M+1}}{z^{M+1}}\right)\)

for any positive integer \(M\). Here, \(P_m\) is some polynomial of \(m\)-th order. Coefficients of this polynomials can be calculated, substituting the asymptotic representation above into the transfer equation. Sign + or - should be chosen, dependently, should the iterate \(T^n(z)\) be holomorphic at \(z>L\) to at \(z<L\). In general cases, both seem to be impossible; at the fixed point \(L\), the non–integer iterate have the branch point; In order to get iterates \(T^n(z)\), growing at \(z>L\), we should choose sign -. Then, truncation of the asymptotic expansion at some \(M\) gives the approximation, valid for large negative values of the real part of the argument. From these values, the superfunction can be evaluated with any required precision. Then, the solution can be extender to the whole complex plane, iterating the transfer equation. In such a way, the asymptotic solution determines (and gives way for the efficient evaluation) of the superfunction.

For case \(T=\exp^2(-1)\), id est \(T(z)=\exp(z/e)\), the superfunctions, abelfunctions and iterate are described in 2011 [9].

The leasing coefficient \(a\) in the expansion above is expressed as follows:

\(a=-2/T''(L)\)

In such a way, for this method, condition \(T''(L) \ne 0\) is essential.

Exotic iteration: \(T'(L)=1\), \(T''(L) = 0\), \(T'''(L) \ne 0\)

The restriction on the second derivative at the fixed point indicates the candidate for the transited function, that is supposed to be difficult to iterate. The simples case is \(T'(L)=1\), \(T''(L) = 0\), \(T'''(L) \ne 0\).

Actually, this case is not very exotic; this takes place, for example, for \(T=\sin\), considered in 2014; the superfunction SuSin and abelfunction AuSin are constructed [18]:

\(T(L)\!=\!L\!=\!0\), \(T'(L)\!=\!1\), \(T''(L)\! =\! 0\), \(T'''(L) \!=\!-1/6\)

The consideration of sin, and the straightforward generalisation for other functions \(T\) with non–zero \(T'''(L)\) [18] explicitly uses the symmetry \(T(-z)=-T(z)\).

The "minimal" distortion, of the symmetry seemed to give an example to break the pretentious statement
for any real-holomorphic transfer function, the superfunction, abelfunction and non–integer iterates can be constructed!

The Nemtsov function seemed to be the simplest example of function of such a kind. For this reason, the Nemsov function

\(\mathrm{Nem}_q(z)=z+z^3+q z^4\)

is especially interesting.

This gives sway to iterate also some other polynomials, that are expressed through the Nemtsov function with linear conjugation

\(T(z)= U(\mathrm{Nem}_q(V(z)))\)

while \(V=U^{-1}\); the simple conjugation appear with

\(U(z)=\mu+\nu z\)

\(V(z)=(z-\mu)/\nu\)

For this reason, the coefficients at the linear and cubic term in the Nemtsov function are chosen to be unity. As for the 4th order coefficient, it is not so easy to adjust with the linear conjugation; so, is is kept as a free parameter. In particular, it helps to reproduce results for the symmetric function (that had been implemented previously) and serves as a test of the algorithm of calculation of the Nemtsov function and its iterates.

Inverse function

Inversion of the Nemtsov function is not so easy, as it seems to be. At given \(x\), The 4th order equation

\(y+y^3+q y^4=x\)

has 4 solutions. In order to avoid unwanted cut lines in the maps of the Abelfunction and iterates of the Nemtsov function, the branches for the inverse function should be chosen. In particular, it is convenient, that the cut lines are straight line. Any of solutions, provided with code

Solve[y + y^3 + q y^4 == x, y]

has curvilinear cut lines. For this reasons, the 3 inverse functions are constricted:

ArcNem\(_q\)

ArkNem\(_q\)

ArqNem\(_q\)

They coincide in vicinity of the positive part of the real axis, but have different cuts in the complex plane.

References

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