UC Berkeley

We prove a correspondence theorem between veering triangulations and pseudo-Anosov flows that takes a more dynamical approach compared to Schleimer and Segerman's work. We introduce the notion of a veering branched surface, which provides an equivalent, yet more natural way of studying veering triangulations with pseudo-Anosov flows in mind. This sets up a theory that allows one to study questions regarding the dynamics of pseudo-Anosov flows using the combinatorial tool of veering triangulations/branched surfaces.

We demonstrate some applications of this theory.\begin{enumerate}[label=(\roman*)] \item We compute explicit Markov partitions for geodesic flows on the unit tangent bundle of negatively curved surfaces. \item We prove that any transitive pseudo-Anosov flow admits a Birkhoff section with at most two boundary components. \item We show that if $f$ is a fully-punctured pseudo-Anosov map with normalized dilatation $\lambda^{-\chi}$, then the mapping torus of $f$ admits a veering triangulation with at most $\frac{\lambda^{-2\chi}}{2}$ tetrahedra. \end{enumerate}

In joint work with Eriko Hironaka, we show that if $f$ is a fully-punctured pseudo-Anosov map with at least two puncture orbits, then the normalized dilatation of $f$ is greater or equal to $\mu^4 \approx 6.854$, where $\mu = \frac{1+\sqrt{5}}{2}$ is the golden ratio. Combining this result with a refinement of the argument in (iii), as well as an exhaustive computation, we show that the minimum element of the set of normalized dilatations of fully-punctured pseudo-Anosov maps is $\mu^2$, and that the minimum accumulation point of the set is $\mu^4$.

In joint work with Michael Landry, we generalize the theory of veering branched surfaces to the setting of sutured manifolds. In particular, we show that for every endperiodic map, the unstable Handel-Miller lamination on its compactified mapping torus is carried by a veering branched surface which is unique up to an explicit set of moves. Furthermore, from this veering branched surface, one can compute the foliation cone in which the associated depth one foliation belongs to.