Let $X = \{0,1\}^{\mathbb{N}}$ with the product topology. Given a Turing machine $M$ and $x \in X$, define $M(x) \in \{0,1\}^* \cup X$ as the sequence of bits output by $M$ when given an oracle for $x$.
Given two Turing machines $M_1, M_2$, how difficult is it to determine whether they define homeomorphisms $f_1, f_2 : X \to X$ such that the topological dynamical systems $(f_1, X)$ and $(f_2, X)$ are isomorphic?
By $M_i$ defines a homeomorphism I mean $M_i(x) \in X$ for all $x \in X$ and that this map is a homeomorphism on $X$ (equivalently bijective). By isomorphism I mean a homeomorphism $\phi : X \to X$ that intertwines the actions. By how difficult I mean the smallest level of the arithmetical/hyperarithmetical/analytical hierarchy (of sets of natural numbers) that contains the set of Gödel numbers of such pairs of machines $(M_1, M_2)$.
The set of pairs such that both $M_i$ are homeomorphisms is low in the arithmetical hierarchy: having image in $X$, being injective and being surjective are all $\Pi^0_2$ properties, so defining a homeomorphism is too. I am mainly interested in how difficult isomorphism is. Clearly it is $\Sigma^1_1$, is it $\Sigma^1_1$-complete?
