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 $f \circ f = g$. 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.
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$f\circ f=g$ 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
$\mathbb{R}$ to $\mathbb{R}$, and let $\mathcal{S}:=\{g\in C(\mathbb{R}) ; \...
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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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Understanding a more intricate Schwarz reflection principle--A question about Tetration
everyone. This is going to be a long question as it requires a good amount of back story in theory. This question is mostly along the lines: "I think this should happen, and I think my proof is ...
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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
...
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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 ...
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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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Fractional power of the operator $\mathcal{L}_t[t f(t)](x)$ and equivalence of divergent integrals
I wonder whether an expression for fractional power of operator $\mathcal{L}_t[t f(t)](x)$ that involves Laplace transform can be derived?
I am asking this because this operator preserves the area ...
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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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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 ...
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Does the functional square root of the cosine admit a vector-based interpretation?
In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$
The functional square root of the cosine has at ...
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