**3**

votes

**3**answers

223 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

171 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 \ ...

**4**

votes

**1**answer

378 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]:
...

**1**

vote

**2**answers

386 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, ...

**5**

votes

**1**answer

464 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 $m=5$. ...

**15**

votes

**3**answers

1k views

### 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 ...

**3**

votes

**3**answers

1k 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 ...

**2**

votes

**1**answer

482 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 ...

**28**

votes

**2**answers

3k 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 ...

**3**

votes

**0**answers

293 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.

**8**

votes

**4**answers

980 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 ...

**5**

votes

**1**answer

516 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 ...

**67**

votes

**9**answers

19k 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 ...

**16**

votes

**8**answers

8k views

### Does the exponential function have a 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))=ex? Is it unique? Is it analytic?
Related question: Is there an invertible ...

**46**

votes

**16**answers

10k 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 ...