arXiv subjects
Publications and source records attributed to Neige Paulet.
In this article, we give a characterization of when two pseudo-Anosov flows obtained via gluings of pieces of pseudo-Anosov flows are orbit equivalent. As an application of this work, and the description of pseudo-Anosov flows in Seifert pieces due to Barbot and Fenley, we prove a ``virtual'' version of the Finiteness Conjecture for transitive pseudo-Anosov flows on graph-manifolds.
We show that the action on its orbit space induced by a pseudo-Anosov flow on a closed $3$-manifold (and more general Anosov-like actions) can be seen as an isometric action on a Gromov-hyperbolic space. When the flow is not $\R$-covered, we show that this action admits elements that are weakly properly discontinuous and deduce that elements of $\pi_1(M)$ that do \emph{not} represent a periodic orbit of the flow are generic for any word metric coming from a finite generating set. We also give a number of other geometric group-theoretic results for Anosov-like group actions on bifoliated planes.
We study obstructions preventing a three-dimensional Anosov flow from serving as the base of a fiberwise Anosov flow. We prove a non-existence result if the base flow admits infinitely many periodic orbits in the same free homotopy class. We get as a corollary that any R-covered Anosov flow serving as the base of a fiberwise Anosov flow is orbit equivalent to a suspension or a geodesic flow.
We prove a new result allowing to construct Anosov flows in dimension 3 by gluing building blocks. By a building block, we mean a compact 3-manifold with boundary $P$, equipped with a $C^1$ vector field $X$, such that the maximal invariant set $\cap_{t \in \mathbb{R}} X^t (P)$ is a saddle hyperbolic set, and the boundary $\partial P$ is quasi-transverse to $X$, i.e. transverse except for a finite number of periodic orbits contained in $\partial P$. Our gluing theorem is a generalization of a recent result of F. B\'eguin, C. Bonatti, and B. Yu who only considered the case where the block does not contain attractors nor repellers, and the boundary $\partial P$ is transverse to $X$. The quasi-transverse setting is much more natural. Indeed, our result can be seen as a counterpart of a theorem by Barbot and Fenley which roughly states that every 3-dimensional Anosov flow admits a canonical decomposition into building blocks (with quasi-transverse boundary). We will also show a number of applications of our theorem.