Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric.

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

$$T_i (e^{i \theta}) := e^{2 \epsilon_i \theta}.$$

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?

Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric.

Let $\{\epsilon_n\}_{n \geq 1}$ be iid uniformly distributed random variables on $[0, e]$ (with underlying probability space $\Omega$), and define the random maps $T_n: \Omega \times S^1 \to S^1$ by

$$T_n (e^{i \theta}) := e^{i \epsilon_n \theta}.$$

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?

Notation: Here $S^1$ denotes the circle, which we view as $[0, 1]$ with its endpoints identified under the Lebesgue measure. We equip the circle with its natural length metric.

Let $\{\epsilon_i\}_{i \geq 1}$ be iid uniformly distributed random variables on $[0, 1]$ (with underlying probability space $\Omega$), 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?

Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric.

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

$$T_i (e^{i \theta}) := e^{2 \epsilon_i \theta}.$$

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?

Notation: Here $S^1$ denotes the circle, which we view as $[0, 1]$ with its endpoints identified under the Lebesgue measure. We equip the circle with its natural length metric.

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?

Notation: Here $S^1$ denotes the circle, which we view as $[0, 1]$ with its endpoints identified under the Lebesgue measure. We equip the circle with its natural length metric.

Let $\{\epsilon_i\}_{i \geq 1}$ be iid uniformly distributed random variables on $[0, 1]$ (with underlying probability space $\Omega$), 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?