Let be $(\Omega, \mathcal{F}, \mathbb{P})=([0,1), \mathcal{B}([0,1)), \lambda)$. We define $Y_n : \Omega \rightarrow \Omega$ by $Y_n := 2Y_{n-1} \mod 1$. Many sources claim that this is a stationary process with help of the following theorem:
Let $(X_n)_n$ be stationary with values in $\mathcal{S^{\mathbb{N}}}$, then $(Y_k)_k$ with $Y_k=g(X_k, X_{k+1},..)$ for measurable $g : \mathcal{S^{\mathbb{N}}} \rightarrow T$ is stationary too.
I understand some ideas of the proof. To be precise, we start with a construction of iid sequence of bernoulli random variables $X_i$ with values in $\mathcal{S}=\{0,1\}$, $\mathbb{P}^*(X_i=0)=\mathbb{P}^*(X_i=1)=1/2$. Lets denote with $(\Omega^*,\mathcal{P}^* ,\mathbb{P}^*)$ the product space. Further we define $g$ as follows
$g: \Omega^* \rightarrow \Omega \\ g((w_n)_n ) = \sum\limits_{n=0}^\infty w_n 2^{-(n+1)}$
I managed to show that $(\mathbb{P}^*)^g=\lambda$. I can use the theorem and show that $Z_k=g(X_k, X_{k+1},..)$ are stationary. I was able even to show that $2Z_{k-1} = Z_k$, but $Z_k$ are not $Y_k$ they live on different space. No idea how to conclude that $Y_k$ are stationary too.