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.
44
questions
11
votes
2
answers
828
views
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 ...
4
votes
2
answers
387
views
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,...
4
votes
2
answers
310
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 ...
4
votes
1
answer
231
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 ...
2
votes
1
answer
329
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 ...
2
votes
0
answers
256
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 ...
0
votes
1
answer
351
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 $$\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ b}(\exp^{\circ a}(z))=\exp^{\circ a+b}(z) \...
1
vote
0
answers
63
views
Fractional power of the operator $\mathcal{L}_t[t f(t)](x)$ and equivalence of divergent integrals
I wonder whether an expression for fractional power of operator $\mathcal{L}_t[t f(t)](x)$ that involves Laplace transform can be derived?
I am asking this because this operator preserves the area ...
1
vote
2
answers
355
views
Finding the “root” of a monotone function (in the sense of composition)
Let $f:[0,\infty)\rightarrow [0,\infty) $ be a smooth and monotone function s.t $f(0)=0$. Let $N\in\mathbb{N}$. Can we find a function $g: [0,\infty) \rightarrow [0,\infty) $ s.t $g\circ\cdots\circ g$ ...
0
votes
0
answers
188
views
Does the functional square root of the cosine admit a vector-based interpretation?
In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$
The functional square root of the cosine has at ...
3
votes
2
answers
491
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 $...
1
vote
1
answer
346
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 ...
2
votes
0
answers
431
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
...
1
vote
0
answers
190
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{...
3
votes
0
answers
285
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)...
2
votes
0
answers
328
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 ...
3
votes
1
answer
237
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_{...
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 ...
1
vote
3
answers
880
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....
0
votes
1
answer
330
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 \...
1
vote
0
answers
362
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 ...
3
votes
4
answers
509
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 ...
5
votes
1
answer
421
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 ...
2
votes
0
answers
356
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?
4
votes
1
answer
200
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{...
5
votes
1
answer
655
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]:
...
2
votes
3
answers
521
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, ...
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$,
...
5
votes
2
answers
871
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$. ...
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 ...
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 ...
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
610
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 ...
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 ...
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 ...
2
votes
0
answers
314
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.
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 ...
6
votes
2
answers
905
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 ...
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 ...
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}) ; \...
106
votes
9
answers
35k
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 ...
121
votes
12
answers
27k
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 ...
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 ...
74
votes
15
answers
17k
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://...