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

I found the following interesting equation on some web page I cannot remember:
$f(f(x))=\cos(x)$
Out of curiosity I tried to solve it, but realized that I do not have a clue how to approach such an ...

**87**

votes

**9**answers

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### 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 ...

**64**

votes

**15**answers

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### 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://scottaaronson....

**44**

votes

**8**answers

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### 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 ...

**43**

votes

**2**answers

5k 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 ...

**41**

votes

**5**answers

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### 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 "...

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**10**answers

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### 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 ...

**28**

votes

**3**answers

1k 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 $m,n$,
...

**27**

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
reals to reals, and let $\mathcal{S}$ $:=$ { $g\in C(\mathbb{R})$ $;$ $\exists$ $...

**19**

votes

**3**answers

1k 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 ...

**15**

votes

**3**answers

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### Do complex iterates of functions have any meaning?

Using a method explained in this answer it is possible to calculate not only integer and real iterates of functions but also complex ones, for example, the $i$-th iterate, where $i=\sqrt{-1}$. Here ...

**14**

votes

**6**answers

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### 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 ...

**11**

votes

**5**answers

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### 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 ...

**7**

votes

**4**answers

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### 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

818 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 ...

**5**

votes

**1**answer

275 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 ...

**5**

votes

**2**answers

632 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 $m=5$. ...

**4**

votes

**3**answers

367 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 ...

**4**

votes

**1**answer

182 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{...

**4**

votes

**1**answer

542 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]:
...

**3**

votes

**1**answer

132 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_{...

**3**

votes

**0**answers

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.

**2**

votes

**3**answers

349 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....

**2**

votes

**1**answer

574 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 ...

**2**

votes

**0**answers

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**

votes

**0**answers

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 ...

**2**

votes

**0**answers

241 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?

**1**

vote

**2**answers

431 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, ...

**1**

vote

**0**answers

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{...

**1**

vote

**0**answers

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)...

**1**

vote

**0**answers

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 ...

**0**

votes

**1**answer

294 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 \...