Diversified homotopic behavior of closed orbits of some \(\mathbb{R}\)-covered Anosov flows (Q2805057)

The author considers Anosov flows \(\varphi_{t}\) on 3-manifolds, focusing on the number of closed orbits in a free homotopy class of closed curves. If \(c(t)\) is a closed orbit of period \(T > 0\), then the reverse orbit \(d(t) = c(T-t)\) is also a closed orbit of period \(T\). By convention the closed orbits \(c\) and \(d\) are regarded as freely homotopic, and as a consequence, every closed orbit is freely homotopic to at least two closed orbits. If \(\varphi_{t}\) is an Anosov geodesic flow, then every closed orbit is freely homotopic to exactly two closed orbits.NEWLINENEWLINE\noindent A flow \(\varphi_{t}\) on a compact 3-manifold \(M\) is Anosov if \(M\) admits two 1-dimensional foliations \(E^{s}\) and \(E^{u}\) transverse to the flow such that \((\varphi_{t})_{*}\) exponentially contracts vectors tangent to \(E^{s}\) and exponentially expands vectors tangent to \(E^{u}\). A classical example is the geodesic flow on the unit tangent bundle \(T^{1}M\) of a compact surface \(M\) with negative Gauss curvature and without boundary. NEWLINENEWLINENEWLINEThe main result of the paper is as follows : There are infinitely many examples of Anosov flows on compact 3-manifolds so that the set of closed orbits is partitioned into two infinite subsets \(A\) and \(B\) with the following properties. Every closed orbit in \(A\) has infinitely many closed orbits in its free homotopy class. Every closed orbit in \(B\) has exactly two orbits in its free homotopy class. NEWLINENEWLINENEWLINESuch examples are obtained by Dehn surgery on closed orbits of geodesic flows. Moreover, there are both Seifert and atoroidal pieces in the torus decomposition.