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We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations
are transverse to a common one-dimensional foliation
whose monodromy on the non-compact leaves of
exhibits the nice dynamics of Handel–Miller theory, then
also induces monodromy on the non-compact leaves of
exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends.
We produce infinitely many examples of Anosov flows in closed
-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one other closed orbit. The examples are obtained by Dehn surgery on geodesic flows. The manifolds are toroidal and have Seifert pieces and atoroidal pieces in their torus decompositions.
In this article we analyze totally periodic pseudo-Anosov flows in graph 3-manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we previously constructed in Barbot and Fenley (Pseudo-Anosov flows in toroidal manifolds. Geom. Topol.17 (2013), 1877–1954). A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (i.e. there is a isotopy of
mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.
We consider Anosov flows in 3-manifolds.
Suppose that there is a rank-two free abelian subgroup
of the fundamental group of the manifold, so that none of its
elements can be represented by a closed orbit of the flow.
We then show that the flow is topologically conjugate to a suspension
of an Anosov diffeomorphism. As a consequence we prove that if $T$
is an incompressible torus so that no loop in
$T$ is freely homotopic to a closed orbit of the flow,
then $T$ is isotopic to a transverse torus.
Finally, we show that if $T$ is an incompressible torus
transverse to the stable foliation, then either
there is a closed leaf in the induced foliation
in $T$, or the flow is topologically conjugate to
a suspension Anosov flow.
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