It is well-known that with respect to Borel reducibility the class of Borel equivalence relations on a standard Borel space does not admit a maximal element. We can use the well-known Friedman-Staley jump (as defined here) to show this.
For any Borel equivalence relation $E$ on a standard Borel space $X$, let $E^{\mathcal{N}}$ be the Borel equivalence relation on $X^{\omega}$ defined by $(x_n) E^{\mathcal{N}} (y_n) \leftrightarrow \{n \in \omega: x_n E\ y_n\} \in \mathcal{N}$ where $\mathcal{N}$ denotes the cofinite filter on $\omega$. Two sequences are $E^{\mathcal{N}}$-equivalent if and only if their entries are eventually $E$-equivalent.
In his paper "On the reducibility order between Borel equivalence relations", Alain Louveau proves that $E \mapsto E^{\mathcal{N}}$ defines a jump operator, that is $E <_B E^{\mathcal{N}}$, for every Borel $E$ with at least two equivalence classes. I will refer to this jump operator as the Louveau jump.
If we let $E$ be the equality relation $\Delta_{\omega}$ on $\omega$, then the Louveau jump would be Borel bireducible with $E_0$. If we let $E$ be the equality relation $\Delta_{2^{\omega}}$ on $2^{\omega}$, then the Louveau jump would be Borel bireducible with $E_1$.
Recalling the dichotomy theorems due to Silver, Harrington-Kechris-Louveau and Kechris-Louveau, we see that $\Delta_{2^{\omega}}$ is the immediate $\leq_B$-successor to $\Delta_{\omega}$, $E_0$ is the immediate $\leq_B$-successor to $\Delta_{2^{\omega}}$ and $E_1$ is an immediate $\leq_B$-successor to $E_0$. (Here, "immediate successor" just means that there is no Borel equivalence relation in between with respect to $\leq_B$.)
Are these examples just coincidences or does this pattern persist? Is there a Borel equivalence relation strictly between $E_1$ and $E_0^{\mathcal{N}}$ with respect to $\leq_B$? More generally, if we keep taking Louveau jumps along these relations, do we get a sequence of Borel equivalence relations that are immediate successors of each other?
Unfortunately, I was not able to find much on this jump operator except the paper I linked above and would appreciate being directed to a reference if this has been studied.
