Almost equivalence of suspension Anosov flows

TL;DR

This paper provides a detailed proof that all suspension Anosov flows generated by hyperbolic matrices with positive trace are almost equivalent, using explicit calculations of first return maps and bounds on flow distances.

Contribution

It offers a rigorous written proof of Minakawa's almost equivalence result and explicitly computes first return maps to improve understanding of flow similarities.

Findings

All suspension Anosov flows with positive trace are pairwise almost equivalent.

Explicit bounds on distances between suspension flows are established.

Constructed genus-one Birkhoff sections with fewer fixed points than original maps.

Abstract

We provide a written proof of a result due to H. Minakawa, which states that all suspension Anosov flows generated by hyperbolic matrices with positive trace are pairwise almost equivalent. The proof relies on constructing, for any given suspension flow, a genus-one Birkhoff section whose first-return map has fewer fixed points than the original map. We improve Minakawa's result by explicitly calculating the first return map onto this section, which leads to explicit bounds on the distances between suspension Anosov flows within the graph of Anosov flows.