We consider (topological) dynamical systems $(\Omega, S)$, where $S$ is the shift $(Sx)_n=x_{n+1}$, and $\Omega\subset[0,1]^{\mathbb Z}$ is a compact, shift invariant subspace. I call such a system $(\Omega, S)$ deterministic if $x_n=y_n$ for all $n<0$ implies that $x=y$ for any two points $x,y\in\Omega$. (Of course, this is a non-standard use of the term.)
We now consider conjugacies between shifts, that is, homeomorphisms $\varphi: \Omega_1\to\Omega_2$ such that $\varphi S= S\varphi$. Determinism in the above sense is a property of the space $\Omega$, not of the dynamical system, and it is not invariant under conjugacy: for an easy example, take the full shift on two symbols $\Omega=\{0,1\}^{\mathbb Z}$. This is clearly non-deterministic, but we can map $f:\Omega\to C\subset[0,1]$ homeomorphically and then define $\varphi(x)$ as the sequence whose $n$th entry is $f(S^nx)$. Then $\varphi$ is an isomorphism (of dynamical systems) onto a deterministic system; in fact, in this new realization, $x$ is already determined by $x_0$.
Two questions: (1) Are there systems of this type that are not isomorphic to a deterministic shift? Put differently, is the full shift on $[0,1]^{\mathbb Z}$ isomorphic to a deterministic shift?
(2) Can one say non-trivial things about systems that have only deterministic realizations in this sense? (For example, distal systems have this property, but that's quite obvious.)
