Notation: Here $S^1$ denotes the circle, which we view as $[0, 1]$ with its endpoints identified under the Lebesgue measure. We equip it with its natural length metric $d(x, y) = \text{min}(|x - y|, |y-x|).$

Let $\{\epsilon_i\}_{i \geq 1}$ be iid uniformly distributed random variables on $[0, 1]$, and define the random maps $T_i: \Omega \times S^1 \to S^1$ by

$$T_i (x) := 2 \epsilon_ix \text{ mod } 1.$$

For $x \in S^1$, we write for short $D_n (x) := T_n \dots T_1 (x).$

Question: Is it true that for all $x, y \in S^1$, we have

$$d(D_n (x), D_n (y)) \to 0$$

almost surely?