Infinitely many closed paths in the graph of Anosov flows

Given an Anosov flow on a closed 3-manifold, we are interested in the problem of whether or not making non-trivial Fried surgeries along a finite set of periodic orbits can produce a flow equivalent to itself. We show that for some suspension Anosov flows, there exist infinitely many pairs of periodic orbits satisfying this property.

💡 Research Summary

The paper investigates the effect of Fried surgery on Anosov flows on closed three‑dimensional manifolds, focusing on whether a finite collection of non‑trivial surgeries can return a flow to a flow equivalent to the original. By encoding Anosov flows as vertices and Fried surgeries as directed edges, the authors obtain a graph G whose topology reflects the combinatorial complexity of these operations.

A Fried surgery is performed by first blowing up a periodic orbit γ of an Anosov flow (φ,M) to obtain a manifold M* with torus boundary, then collapsing a foliation C on the boundary whose homology class is a+ m b (a meridian, b a longitude). The resulting manifold N carries a new flow ψ that coincides with φ away from the surgery region. Although ψ is only a “topological Anosov flow” in general, the Bonatti‑Wilkinson conjecture (proved for transitive flows) allows the authors to treat ψ as a genuine smooth Anosov flow.

The graph G is defined as follows: vertices are orbital‑equivalence classes