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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 $f \circ f = g$. The continuous version of fractional iteration concerns maps which have flows. This case is also known as ...

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2k views

$f\circ f=g$ revisited

This may be related to solving $f(f(x))=g(x)$. Let $C(\mathbb{R})$ be the linear space of all continuous functions from reals to reals, and let $\mathcal{S}$ $:=$ { $g\in C(\mathbb{R})$ $;$ $\exists$ $...
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305 views

elementary Abel function of a polynomial

Is there an elementary real function $F$ such that $F(1+F^{-1}(x))$ is a polynomial of degree at least 2 without real fixpoints.
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0answers
169 views

Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying. Carleman linearization is a technique used to embed a finite ...
2
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0answers
116 views

Fractional iteration of a variant of the $\sin()$ function - how to fractionally iterate $ f(x)=\sum_{k=1}^\infty (-1)^k a_{2k}x^{2k}$?

I was reconsidering the fractional iteration of the sine-function and remembering that the power series for the fractional iterates have convergence radius zero I looked at the variant of the sine ...
1
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0answers
121 views

Fractional exponentiation with different bases

The standard analytic tetration base $b>e^{1/e}$, $F_b$ is the unique analytic function defined everywhere on $ℂ$ except on the ray from -2 to -∞ such that $F_b(0)=1$, $F_b(\overline z) = \overline{...
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159 views

Analytic Tetration and Natural Rates of Growth

What evidence do we have that the standard analytic tetration $F$ has a natural asymptotic rate of growth even at noninteger arguments? Intuitively, $y=x$ has a natural growth rate, while $y=x+\sin(x)...
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205 views

Generating a series representation for the inverse of the operator $f(f)$

I am considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...