Topologic Entropy of the the free-square flow S someone knows how to prove that $\eta(n)= \mu^2(n)$ and a-fortiori μ(n), is not deterministic? I'm prove that the system associated to the flow $(X_S,T_S)$ is ergodic, where $X_S$ ís a closure the $T_S$-orbit of $\eta$ in $\\{0,1\\}^{N}$, but still I did not get to prove that a topologic entropy is $\frac{6}{\pi^2} \log 2$ like ensures Sarnak on his lecture about the ramdomness and Dynamic of the Möbius function. Any suggestion for to calculate this entropy?

I assume that the notation you are using comes from Sarnak's lecture on Möbius randomness and dynamics.

In that case, the result you are looking for appears as Theorem 10 on page 9 of this paper of Sarnak: Three Lectures on the Mobius Function Randomness and Dynamics.