# Tagged Questions

**2**

votes

**1**answer

79 views

### Constructing an interval exchange given a prescribed trajectory

Given a prescribed trajectory, is it possible to construct an interval exchange having this trajectory?
For example, given a 3-letter word (like aaabbbccabcaaa ), is it possible to construct a 3- ...

**0**

votes

**0**answers

99 views

### excplicit formula of iterates of an interval exchange

Let $f$ be an interval exchange transformation of $[0,1]$. Is there an explicit formula giving $f^k(0)$ in function of $k$?
If not, are there particular cases where this formula is simple? (except ...

**21**

votes

**3**answers

4k views

### Status of the 196 conjecture?

A palindrome is a number which remains the same when reversing it, for instance 34143. Now pick an arbitrary number, say 26: then 26+62=88 is a palindrome. If the number was 57, then 57+75=132 is not ...

**2**

votes

**1**answer

291 views

### Bohr sets, Coin-flip sets and Roth's theorem

I have been learning about Roth's theorem, trying to understand how Fourier series and dynamical systems (or even graph theory and binary sequences)are involved in counting arithmetic sequences in ...

**5**

votes

**1**answer

211 views

### constant averages along orbits

What should one say to describe the situation in which a function $T$ from some set $X$ to itself, and a function $f$ from $X$ to some characteristic-zero field $K$, have the property that the average ...

**13**

votes

**5**answers

1k views

### subwords of the fibonacci word

The Fibonacci word is the limit of the sequence of words starting with "0" and satisfying rules $0 \to 01, 1 \to 0$. It's equivalent to have initial conditions $S_0 = 0, S_1 = 01$ and then recursion ...

**3**

votes

**0**answers

191 views

### How many set partitions on a big cube’s boundary arise from cubomino decompositions of the solid cube?

Introduction. This is a counting question about configurations that can appear on the outside of assembled Soma cube-like puzzles. More specifically, it’s about the ways in which the pieces of an ...

**6**

votes

**2**answers

723 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 ...

**26**

votes

**3**answers

2k views

### Is there a reset sequence?

There is a question someone (I'm hazy as to who) told me years ago. I found it fascinating for a time, but then I forgot about it, and I'm out of touch with any subsequent developments. Can anyone ...

**16**

votes

**3**answers

658 views

### Counting certain arrangements of n triangles. Does the count grow superexponentially?

Overview
For integers n ≥ 1, let T(n) = {0,1,...,n}n and B(n)= {0,1}n. Note that |T(n)|=(n+1)n and |B(n)| = 2n.
A certain set S(n) ⊂ T(n), defined below, contains B(n). The question is about ...

**10**

votes

**3**answers

842 views

### Periodic orbits and polynomials

There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
fact 1 Consider the "tent map" ...

**10**

votes

**4**answers

1k views

### reversible Turing machines

Hello,
Let T be a Turing machine such that
1) it operates on the alphabet {0,1},
2) its set of states is A
3) the language it accepts is $L$ .
Does there exists a Turing machine S which also ...

**9**

votes

**4**answers

1k views

### Uniqueness in Composition of Polynomials

The following situation came up in my research:
Suppose two functions $f$ and $g$ map $[0,\infty)$ to (a subset of) itself. The function $f$ is linear and $g$ is quadratic, but $g$ is one-to-one on ...

**12**

votes

**2**answers

620 views

### Is there an analogue of the Lefschetz fixed point theorem for discrete dynamical systems?

Background/Motivation
Let $(X, f)$ be a discrete dynamical system. For now, $X$ is just a set and $f$ is just a function $f : X \to X$. Suppose that $f^n$ has a finite number of fixed points for ...