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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 $f \circ f = g$. 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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What are the iterates of $x \mapsto 1 - \sqrt{1-x^2}$?

Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be ...
Gro-Tsen's user avatar
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4 votes
2 answers
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Borel summation and the Abel function of $e^z-1$

This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...
Richard Diagram's user avatar
4 votes
2 answers
326 views

Is there half an iteration of the QR algorithm?

Every real square matrix $M$ has a QR decomposition $M = QR$ where $Q^{-1}=Q^T$ and $R$ is an upper triangular matrix with non-negative reals on the diagonal. Call the function $f(QR)=RQ$ the Francis ...
wlad's user avatar
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4 votes
1 answer
239 views

Finding closed forms/related constants to a limit involving tetration

I was working on finding a series expression for a function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(x^y) = f(x)^{f(y)}$ along the way for construction of such a function I came across a ...
Sidharth Ghoshal's user avatar
2 votes
1 answer
371 views

Fractional Laplacian on closed manifolds

Naturally given any $s\in (0,1)$, the fractional Laplacian, $(-\Delta_g)^s u$ on a closed Riemannian manifold can be defined via spectral decomposition of $-\Delta_g$. There is another formulation of ...
Ali's user avatar
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2 votes
0 answers
288 views

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 ...
Richard Diagram's user avatar
0 votes
1 answer
359 views

On the relevance of the property $\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ a+b}(z)$ for the *fractional* iteration ("tetration")

In the concept of fractional iteration of the exponential function ("tetration") the property of $$\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ b}(\exp^{\circ a}(z))=\exp^{\circ a+b}(z) \...
Gottfried Helms's user avatar
1 vote
0 answers
65 views

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 ...
Anixx's user avatar
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1 vote
2 answers
368 views

Finding the “root” of a monotone function (in the sense of composition)

Let $f:[0,\infty)\rightarrow [0,\infty) $ be a smooth and monotone function s.t $f(0)=0$. Let $N\in\mathbb{N}$. Can we find a function $g: [0,\infty) \rightarrow [0,\infty) $ s.t $g\circ\cdots\circ g$ ...
JustSomeGuy's user avatar
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0 answers
190 views

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 ...
Max Muller's user avatar
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3 votes
2 answers
494 views

if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?

This question is related to solving $f(f(x))=g(x)$. Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
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1 vote
1 answer
353 views

Moving between Abel's and Schroeder's Functional Equations [closed]

I see a reoccurring tendency in attempts to extend tetration where the author moves easily between Abel's and Schroeder's Functional Equations. The premise with them being topologically conjugate is ...
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3 votes
0 answers
462 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 ...
Hyeonseo Yang's user avatar
1 vote
0 answers
193 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{...
Dmytro Taranovsky's user avatar
3 votes
0 answers
290 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)...
Dmytro Taranovsky's user avatar
2 votes
0 answers
342 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 ...
Gottfried Helms's user avatar
3 votes
1 answer
240 views

Reconstructing analytic tetration with a complex height from a thinner set of points

This is a follow-up to my previous question An explicit series representation for the analytic tetration with complex height. Recall the definition $(11)$ from there: $$t(z) = \sum_{n=0}^\infty \sum_{...
Vladimir Reshetnikov's user avatar
29 votes
3 answers
3k views

An explicit series representation for the analytic tetration with complex height

Tetration is the next hyperoperation after more familiar addition, multiplication and exponentiation. It can be seen as a repeated exponentiation, similar to how exponentiation can be seen as a ...
Vladimir Reshetnikov's user avatar
1 vote
3 answers
924 views

Existence of a square root of a functional equation

We can define the iterates $f^{n+1}=f\circ f^n$ for a given smooth map $f:X\to X$, where $X$ could be a finite interval, the real line $\mathbb{R}$, or the circle $S^1$, or any general smooth manifold....
Pengfei's user avatar
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0 votes
1 answer
333 views

Given $g(h(z))$ is convergent, what can be said about the convergence of $g(z)$ and $h(z)$? [closed]

Consider the iterated function $f^t(z)=f(f(f(...f(z))...))$ where $t \in \mathbb Z$ and $f(z)$ is convergent. Then the iterates of $f(z)$ such as $f^2(z), f^3(z), f^4(z)$ are convergent. Now let $r \...
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1 vote
0 answers
367 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 ...
Sidharth Ghoshal's user avatar
3 votes
4 answers
527 views

Convergence of expansion for fractional iteration

I was reading about tetration here. The site mentions that the convergence of the expansion for fractional iteration is unproven. However, I was interested in reading more literature about convergence ...
Zachary W. Robertson's user avatar
5 votes
1 answer
428 views

Smoothness in Ecalle's method for fractional iterates

Some four years ago I answered my own question on fractional iteration, concluding that there is a half iterate of sine, that is $f(f(x)) = \sin x,$ which is real analytic for $0 < x < \pi$ but ...
Will Jagy's user avatar
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2 votes
0 answers
365 views

How to solve $f(f(x))=x^2+x$ [duplicate]

Now I just have the equation $f(f(x))=x^2+x$. How can I find $f(x)$? I have already tried many times, but I cannot solve it by any way I know. Is a solution possible?
Meng Du ghost's user avatar
4 votes
1 answer
201 views

Homeomorphisms that admit a decomposition

Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$. If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ \text{...
Lucy's user avatar
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5 votes
1 answer
659 views

solution of functional equation $f^{\circ k}(x) = x$

The equation $f^{\circ k}(x) = \mathrm{Id}$ for $x\in E$ is called the Babbage equation and the general solution is given in the following way [M. Kuczma, Functional equations in a single variable]: ...
Lucy's user avatar
  • 183
2 votes
3 answers
527 views

How this expression leads to the given sequence

Here given is a sequence from OEIS. The sequence is triangle of coefficients from fractional iteration of $e^x - 1$. Few terms are: 1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, ...
Nadia's user avatar
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29 votes
3 answers
2k views

Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$, ...
Alexandre Eremenko's user avatar
5 votes
2 answers
881 views

Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point

Lets say I have $f(z)=z^2+c$, with $c=0.35676274578 + 0.32858194507i$. Then $f(z)$ has a fixed point $\kappa_0=0.15450849719 + 0.47552825815i$, which is rationally indifferent with a period $m=5$. ...
Sheldon's user avatar
  • 159
26 votes
5 answers
3k views

Do complex iterates of functions have any meaning?

Using a method explained in this answer to How to solve $f(f(x)) = \cos(x)$?, it is possible to calculate not only integer and real iterates of functions but also complex ones, for example, the $i$-th ...
Anixx's user avatar
  • 9,737
13 votes
6 answers
4k views

Finding f such that f(f(x))=g(x) given g

Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...
David Corwin's user avatar
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48 votes
6 answers
6k views

Is there an "elegant" non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway... Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "...
1 vote
1 answer
613 views

fractional iteration of $xe^x$ has zero convergence radius?

The equation $f(f(x))=xe^x=x+x^2+\frac{x^3}{2}+\frac{x^4}{6}+\dots$ has a unique formal powerseries solution. Is its convergence radius 0 as was shown by Baker for the equation $f(f(x))=e^x-1$? Or ...
bo198214's user avatar
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54 votes
8 answers
9k views

Does the formal power series solution to $f(f(x))= \sin( x) $ converge?

I have spent some time using gp-pari. There is, of course, a formal power series solution to $ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
Will Jagy's user avatar
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45 votes
2 answers
8k views

"Closed-form" functions with half-exponential growth

Let's call a function f:N→N half-exponential if there exist constants 1<c<d such that for all sufficiently large n, cn < f(f(n)) < dn. Then my question is this: can we prove that no ...
Scott Aaronson's user avatar
2 votes
0 answers
315 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.
bo198214's user avatar
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7 votes
4 answers
1k views

The non-convergence of f(f(x))=exp(x)-1 and labeled rooted trees

This question is closely related to MO f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential. Consider $e^{e^x-1}$, this is the generating function of the Bell ...
user avatar
6 votes
2 answers
908 views

Do maps have flows?

In A New Kind of Science: Open Problems and Projects(pg. 36). How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? The ...
user avatar
14 votes
6 answers
3k views

What's a natural candidate for an analytic function that interpolates the tower function?

I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
John Jiang's user avatar
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32 votes
0 answers
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 $\mathbb{R}$ to $\mathbb{R}$, and let $\mathcal{S}:=\{g\in C(\mathbb{R}) ; \...
Ady's user avatar
  • 4,050
107 votes
9 answers
36k views

solving $f(f(x))=g(x)$

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
Kevin Buzzard's user avatar
122 votes
12 answers
28k views

How to solve $f(f(x)) = \cos(x)$?

I found the following equation on some web page I cannot remember, and found it interesting: $$f(f(x))=\cos(x)$$ Out of curiosity I tried to solve it, but realized that I do not have a clue how to ...
user4503's user avatar
  • 1,561
52 votes
11 answers
24k views

Does the exponential function have a (compositional) square root?

(asked by Nathaniel Hellerstein on the Q&A board at JMM) Is there a "half-exponential" function $h(x)$ such that $h(h(x))=e^x$? Is it unique? Is it analytic? Related question: Is there an ...
2010 Joint Meetings's user avatar
74 votes
15 answers
18k views

$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential

The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
Gil Kalai's user avatar
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