# On typicality of translation flows which are disjoint with their inverse

## TL;DR

This paper demonstrates that in most non-hyperelliptic translation structures, the vertical flows are weakly mixing and disjoint from their inverses, contrasting with hyperelliptic cases where flows are isomorphic to their inverses.

## Contribution

It establishes that such disjoint and weakly mixing translation flows form a dense set in moduli space, using limits of off-diagonal joinings and a novel embedding approach.

## Key findings

- Disjointness and weak mixing are generic in non-hyperelliptic components.
- Hyperelliptic translation structures have flows isomorphic to their inverses.
- A new embedding of moduli space into measure-preserving flows was constructed.

## Abstract

In this paper we prove that translation structures for which the corresponding vertical translation flows is weakly mixing and disjoint with its inverse, form a $G_\delta$-dense set in every non-hyperelliptic connected component of the moduli space $\mathcal M$. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is isomorphic to its inverse. To prove the main result, we study limits of the off-diagonal 3-joinings of special representations of vertical translation flows. Moreover, we construct a locally defined continuous embedding of the moduli space into the space of measure-preserving flows to obtain the $G_\delta$-condition.