Showing the connection between the doubling map and Bernoulli shift map.

Question

Here is a remaining part of the question I asked here Showing that a set has measure zero? :

Let $S : [0,1) \to [0,1) $ be the doubling map, $S(x) = 2x \pmod{1}.$ I want to show that $\phi(T(\textbf{x})) = S(\phi(\textbf{x})).$

I am just wondering how can I do this? What is $S(\phi(\textbf{x}))$?

Note that the map $\phi$ is defined as: Consider the map $\phi: X\setminus X_{\infty} \to [0,1)$ defined by $$\phi(a_1, a_2, \dots , a_n) = \sum_{n=1}^{\infty}a_n2^{-n}$$ so $\phi$ sends each sequence $(a_1, a_2,\dots)$ to the point $x$ in $[0,1)$ with that binary decimal expansion.

I also, want to know how can I use the fact that $S$ is ergodic to prove that $T$ is ergodic?

Any hints for the above two questions will be greatly appreciated.

Here is the general setting of the problem that is given in the link mentioned above:

Consider the probability space $X = \{0,1\}^{\mathbb N}$ of sequences $x = (a_1, a_2, a_3, \dots)$ where each $a_i$ is either $0$ or $1.$ Take $\sigma$ to be the smallest $\sigma$-algebra that contains all cylinder sets $$C(n,A) = \{\textbf{x}| x_n \in A, A \subset \{0,1\}\}$$ that is, those sequences whose $n^{th}$ term belongs to some subset $A$ of $\{0,1\}.$ Let $\mu$ be the measure so that, given a finite collection $A_1, \dots , A_n \subset \{0,1\},$ we have $$\mu (\{\textbf{x} \,|\, x_i \in A_i, i = 1, \dots , n \}) = \Pi_{i=1}^{n}\frac{|A_i|}{2}.$$ Consider the map $\phi: X\setminus X_{\infty} \to [0,1)$ defined by $$\phi(a_1, a_2, \dots , a_n) = \sum_{n=1}^{\infty}a_n2^{-n}$$ so $\phi$ sends each sequence $(a_1, a_2,\dots)$ to the point $x$ in $[0,1)$ with that binary decimal expansion.We know that $\phi$ is a bijection. We may assume that $\phi$ is measurable with measurable inverse. Also, we know that if $V$ is a measurable subset of $[0,1)$ and if $\lambda$ is Lebesgue measure on $[0,1).$Then $\mu(\phi^{-1}(V)) = \lambda (V).$

Answer 1

I am just wondering how can I do this? What is $S(\phi(\textbf{x}))$?

For $x$ a sequence, $\phi(x)$ is the real number (one of) whose binary expansion(s) is given by the entries of $x$. $S$ is the doubling map; so

\begin{align*} S(\phi(x))=S(x_1/2+x_2/4+x_3/8+\cdots) &= x_1+x_2/2+x_3/4+\cdots\mod 1 \\ &= x_2/2+x_3/4+\cdots. \end{align*}

I also, want to know how can I use the fact that $S$ is ergodic to prove that $T$ is ergodic?

Hint: Ergodicity is invariant under measure-preserving conjugacies.