Existence of arbitrary large numbers of non-$\mathbb R$-covered Anosov flows on hyperbolic $3$-manifolds

The purpose of this paper is to prove that, for every $n\in \mathbb N$, there exists a closed hyperbolic $3$-manifold $M$ which carries at least $n$ non-$\mathbb R$-covered Anosov flows, that are pairwise orbitally inequivalent. Due to a recent result by Fenley, such Anosov flows are quasi-geodesic. Hence, we get the existence of hyperbolic $3$-manifolds carrying many pairwise orbitally inequivalent quasi-geodesic Anosov flows. One of the main ingredients of our proof is a description of the clusters of lozenges that appear in the orbit spaces of the Anosov flows that we construct. The number of lozenges involved in each cluster, as well as the orientation-type of this cluster, provide powerful dynamical invariants allowing to prove that the flows are orbitally inequivalent. We believe that these dynamical invariants could be helpful in a much wider context.