How can I find a semiconjugacy? 
Let me consider the maps $$T:\Bbb{R}/\Bbb{Z}\rightarrow \Bbb{R}/\Bbb{Z}; (x+\sqrt{2}) \operatorname{mod} 1$$ and $$S:\prod_{i\geq 1} \{0,1,2,3\}\rightarrow \prod_{i\geq 1} \{0,1,2,3\}: (x_i)_{i\geq 1}\mapsto (x_{i+1})_{i\geq 1}$$
I want to check if there is a semiconjugacy $s$ from $T$ to $S$.

I know that by definition this is a map $s:\Bbb{R}/\Bbb{Z}\rightarrow \prod_{i\geq 1} \{0,1,2,3\}$ which is continuous  and surjective s.t. for all $x\in \Bbb{R}/\Bbb{Z}$ $$s(T(x))=S(s(x))$$
My problem is that I don't know where to start. I have no intuition if there should be such a semiconjugacy or not. Therefore I can't really focuse on finding such an $s$ or finding a counterexample. What I understand is that $T$ takes some real number and eliminates it's "integer" part, so that $T(x)\in [0,1]$.
If there were such an $s$ then I should have $s((x+\sqrt{2}) \operatorname{mod} 1)=S(s(x))$ so if I apply $s$ again to $(x+\sqrt{2}) \operatorname{mod} 1$ then It should be the same as shifting $s(x)$.
Can someone help me further?
 A: Let all real numbers be implicitly considered modulo $1$, to save on writing $\bmod1$ all the time.
Maps $s:\Bbb R/\Bbb Z\to\prod_{i\ge1}[3]$ are in direct correspondence with continuous maps $s_i:\Bbb R/\Bbb Z\to[3]$, corresponding to the components. To be continuous, it is the same to say that $s_i^{-1}(j)$ is open for every $j\in[3]$.
The semiconjugacy condition (which my text called conjugate, iirc - what's the "semi" about?) is saying that $s_i(x+\sqrt{2})=s_{i+1}(x)$ for all $x$ and all $i$. So, knowing the map $s_i$ uniquely determines the map $s_{i+1}$. That is, given the map $s_1$, all the other maps $s_i$ for $i>1$ are determined. It's also clear that if $s_i$ is continuous then $s_{i+1}$ must be too.
It then remains to find a continuous map $s_1:\Bbb R/\Bbb Z\to[3]$. Are there any conditions on this map other than continuity? No. But continuity is very restrictive: do notice that $s_1^{-1}(j)$ must be open and disjoint from $s_1^{-1}(k)$ if $j\neq k$. For connectivity reasons, we actually see that $s_1$ has to be constant.
There should then be exactly four semiconjugacies $(\Bbb R/\Bbb Z;T)\to(\prod_{i\ge1}[3];S)$, corresponding to the constant maps to $0,1,2,3$, since the constancy of $s_1$ implies the constancy of $s_2,s_3$, etc.
The shift by $\sqrt{2}$ is not relevant either. In fact, the same argument shows that for any connected topological dynamical system $(K;\varphi)$ and any discrete space $F$, there are exactly $|F|$ semiconjugacies $(K;\varphi)\to(\prod_{i\ge1}F;S)$ corresponding to the constant maps to a particular $f\in F$.
