Diameter of the range of composition of random maps on the circle

Question

My questions are related to the paper https://hal.science/hal-03933493v1 (accepted with corrections in Ergodic Theory and Dynamical Systems). I fix an irrational number $\theta \in [0,1[$. I define the maps $f_0$ and $f_1$ from $[0,1[$ to $[0,1[$ by $$f_0(y) := y-\theta 1_{[\theta,1[}(y).$$ $$f_1(y) := y+(1-\theta)1_{[0,\theta[}(y).$$ The union of their graphs is also the union of the graph of the identity map and the graph of the rotation $y \mapsto (y-\theta) \mod 1$.

If $(i_1,\ldots,i_n) \in \{0,1\}^n$, the map $f_{i_n} \circ \cdots \circ f_{i_1}$ is a piecewise translation. There are at most $n+1$ pieces, corresponding to the $n+1$ intervals defined by the points $x_k = (k\theta) - \lfloor k\theta \rfloor$ for $0 \le k \le n$, and the point $1$.

Now, take a sequence $(I_n)_{n \ge 1}$ of i.i.d. uniform random variables with values in $\{0,1\}$, and set $F_n = f_{I_n} \circ \cdots \circ f_{I_1}$. I have proved that :

• given two points $y$ and $y'$, almost surely, the Cesàro means of the distance between the $F_n(y)$ and $F_n(y')$ goes to $0$. The distance itself cannot go to $0$ almost surely (excepted if it is $0$ eventually) since the orbits must separate for a while before getting closer for a long time.
• the Lebesgue measure of the range $F_n([0,1[)$ goes to zero as $n$ goes to infinity. Informally, it is because for every $y \in [0,1[$, almost surely, the images of $y$ and $T_\theta(y)$ by $F_n$ coincide for all $n$ large enough.

I conjecture that the diameter (for the distance on the circle) of the range $F_n([0,1[)$ goes to zero in $L^1$ as $n$ goes to infinity.

Equivalently, we could couple from the past, i.e. take an i.i.d. uniform sequence $(I_n)_{n \ge 1}$ and look at $P_n = f_{I_0} \circ \cdots \circ f_{I_{-n+1}}([0,1[)$. This gives a non-decreasing sequence of sets, and I expect that diameter goes to $0$ almost surely.

At present, I have no idea for proving this. I am interested by suggestions. And I have a simpler question of programming. If one make simulations to see the long-time behavior of the diameter of $F_n([0,1[)$, what is a good strategy to compute the diameter for the distance in the circle $\mathbb{R}/\mathbb{Z}$? Of course, determining the diameter for the usual distance in $\mathbb{R}$ is easy: it is $\sup F_n([0,1[) - \inf F_n([0,1[)$.

Answer 1

One way of computing the diameter of a (finite) explicit union $S$ of intervals in the circle $\mathbb R / \mathbb Z$ is to look at the complement $S^c$, itself a finite union of intervals of the circle: when $S$ is small, the length of the largest connected component of $S^c$ is exactly equal to $1-\mathrm{diam}(S)$, and this maximal length is be easy to compute given the decomposition of $S$ into intervals.

Of course this probably gives no insight into your mathematical question.