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Questions tagged [fractional-iteration]

The study of fractional self-iterations of a map. A basic example is the analysis of functional square roots of a map $g$, i.e. solutions $f$ to the functional equation $f\circ f=g$ are functional square roots and solutions to $f^n=g$ are functional nnth roots.

The continuous version of fractional iteration concerns maps which have flows. This case is also known as continuous iteration. A classic example is the problem of extending tetration to the real and complex numbers.

wikipedia article about fractional iterations

wikipedia article about functional square root

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half iterate of $x^2+c$

I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial. $$h(h(x))=x^2$$ $$h(x)=x^{\sqrt{2}}$$ The question is, for $c>0,$ and $x>1$, when is the half iterate of $x^2+c$ smaller than…
Sheldon L
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Find $f(f(\cdots f(x)))=p(x)$

$\newcommand{\nest}{\operatorname{nest}}$Let's define a function $\nest(f, x, k)$, which takes a function $f$, an input $x$, and a non-negative integer $k$, and calls $f$ on $x$ repeatedly ($k$ times). For example, $$ \nest(f, x, 0) = x\\ \nest(f,…
Antonio Perez
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Fourier series of iterated sin / Diagonalization of an infinite matrix of Bessel functions

We define the iterated sine function as : $$ \sin^n(x) = \sin(\sin(.... \sin(x)))\:\:n\:\text{times.} $$ We know the "Frequency Modulation" formula based on Bessel functions :$$ \sin( p\, \sin(x) ) = \sum_{k=1, \,odd}^{\infty} 2 \, J_k(p) \, \sin( k…
al4085
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Find the curve of the iterated symmedian points

The symmedian point of the triangle with vertices $A=(-1,0),B=(1,0),C=(x,y)$ is $$F(x,y)=\left(\frac{4 x}{x^2+y^2+3},\frac{2 y}{x^2+y^2+3}\right)$$ By this question $F$ maps $\mathbb{R}^2$ onto an ellipse with foci $A,B$ and eccentricity…
hbghlyj
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A method to iterate the exponential function a non-integer number of times?

Notation We employ the following notation: $$ a_1 = e^{x} $$ $$ a_2 = e^{e^{x}} $$ $$ a_3 = e^{e^{e^{x}}} $$ $$ a_n = e^{\vdots^{e^{x}}}$$ We also use define $c$ by: $$ e^c =c$$ Motivation Let us try to define $a_{1.5}$ Possible method? Now,…
More Anonymous
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Fourier transform and fourth root

Given a well-behaved convex function $f(t):\mathbb{R}\to \mathbb{R}$, its Fourier transform (FT) $\hat{f}(\omega)=\mathcal{F}[f(t)](\omega)$ is positive (and decreasing) proof here. It follows that the fourth square root of the FT…
Chaos
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If $q(x)=x^2+1$, does $q^{\circ 1/2}$ exist?

I've been doing a lot of research about functional half-iteration, and I posed the following question to myself: Consider the function $q:\mathbb R\mapsto\mathbb R$ defined as $$q(x)=x^2+1$$ Does $q^{\circ 1/2}$ exist? Does a continuous…
Franklin Pezzuti Dyer
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Iterated exponential function

Is it possible to infinitely differentiably extend the function defined as $f(x+1,a)=e^{f(x,a)}$, $f(0,a)=a$ to non-integers? What I’m trying to do is derive a sort of «half logarithm», a function that if applied twice gives the natural logarithm.
Fernando Martínez
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Does this nested log function exist?

I was wondering if the following function exists: lets say you have $$f^1(x) = \ln(x)$$ $$f^2(x) = \ln(\ln(x))$$ $$f^3(x) = \ln(\ln(\ln(x)))$$ $$f^4(x) = \ln(\ln(\ln(\ln(x))))$$ and so forth is there a way to generalize $$f^n(x)$$ when $n$ can be a…
Sasha1296
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Functional Square Root of Digit Sum?

Define $\text{sdig}(n)$ to be the sum of the decimal digits of $n$, where $n$ is a positive integer. My question is as follows: Does there exist a function $h:\mathbb Z^+\mapsto\mathbb Z^+$ such that $$(h\circ h)(n)=\text{sdig}(n)$$ for all…
Franklin Pezzuti Dyer
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Construct a function $f$ that $f(f(z))=-z(1-z)$ without using any information linked to $f(f(2))=2$

Consider the analytic function $g(z)=-z(1-z)$ which is a generator of Logistic Sequence with multiplier $-1$, having 2 global fixed point, $g(0)=0$ and $g(2)=2$. From classical dynamics, whenever a multiplier, or the derivative $\lambda=g'(L)$ at…
Leo Warvin Peng
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Is there an method similar to FFT for Fractionally Iterated Fourier Transforms?

FFT is one of the 20th Century's greatest inventions, running as $O(n \log(n))$ rather than as $O(n^2)$ as a simple implementation of a discrete Fourier transform would. But what about half-order Fourier transforms, or arbitrary real order iterated…
DarenW
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Uniquely extended fractional iterations of $\exp$

Let us define the following basic conditions for an iterated exponential function: $$\exp^1(x)=e^x\tag{$\forall x$}$$ $$\exp^{a+b}(x)=\exp^a(\exp^b(x))\tag{$\forall a,b,x$}$$ I then pondered what sort of additional conditions could be applied. Using…
Simply Beautiful Art
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A conjectured identity for the fractional iterates of $\sqrt{x}+2$

Deeply impressed by the Fractional iterates of $x²-2$, a problem by Ramanujan post one month ago, I decided to investigate a little by myself on the topic and discovered that using Carleman matrices would be of great help for that purpose. After…
Thomas Baruchel
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Tetration Power Series

While reading through the Citizendium article on tetration, iterated exponentation, I came across a power series approximate of tetration. The article said that it got the series coefficients from cauchy’s integral formula for the nth derivative of…
Anthony Corsi
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