arXiv research

Constructs Anosov flows in hyperbolic 3-manifolds, disproving a conjecture.

Deforms quasigeodesic flows to pseudo-Anosov ones.

New invariants explain topological properties of pseudo-Anosov maps.

Shows Anosov flows with genus one sections, supporting a conjecture.

New contact structures detected by contact homology.

We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with…

Constructs infinite families of hyperbolic knots satisfying a volume conjecture.

Proves a conjecture about matrix orders for pseudo-Anosov maps.

Proves stability of geodesic flows on closed surfaces.

Study geometric manifolds in arbitrary dimensions, focusing on maps and diffeomorphisms.

Two Anosov metrics with same boundary distance are isometric.

In this paper, we prove that for every bumpy Finsler $n$-sphere $(S^n,\,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $(\fracλ{λ+1})^2<K\le 1$, there exist $2[\frac{n+1}{2}]$ prime closed geodesics. This gives a confirmed answer to a conjecture of D. V. Anosov \cite{Ano} in 1974 for a generic case.

New 3D shapes found without certain flows.

The paper proves a conjecture about the minimum number of closed geodesics on a Finsler 3-sphere.

This note gives a brief survey of the minimum dilatation problem for pseudo-Anosov mapping classes, and the first explicit train track description of an infinite family of pseudo-Anosov mapping classes with orientable stable foliations and the conjectural minimum dilatation for closed surfaces of even genus $g \ge 2$.

Geodesic flows on specific manifolds are structurally stable.

Generalizes Klein-Maskit theorem to free products of Anosov subgroups.

A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely cov…

We show that Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner's construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner's construc…

Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.

The paper constructs Anosov representations for specific types of groups.

Anosov groups' measures on limit sets are uniquely determined by their dimension.

The paper calculates intertwiners for a torus and proves a conjecture about their limits.

We find the minimum dilatation of pseudo-Anosov braids with many strands.

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliation…

We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross section and is therefore topologically conjugate to a suspension of a linear toral automorphism. This proves a conjecture of Verjovsky from the 1970's in the volume pres…

Invites study of contact structures and Reeb flows dynamics.

The paper proves a minimum number of closed geodesics on positively curved Finsler spheres.

The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank~1. This conjecture has been proved by Z. Szabó \cite{Sz} for harmonic manifolds with compact universal cover. E. Damek and F. Ricci \cite{DR} provided examples showing that in the noncompact case the conj…

In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the minimum dilatations for orientable mapping classes for genus g=2,3,4,5,8 as well as …

Reconstruct flows and manifolds from their boundary actions on circles.

We study the magic manifold $N$ which is a hyperbolic and fibered $3$-manifold. We give an explicit construction of a fiber $F_a$ and its monodromy $:F_a \rightarrow F_a$ of the fibration associated to each fibered class $a$ of $N$. Let $δ_g$ (resp. $δ_g^+$) be the minimal dilatation of pseudo-Anosovs (resp. pseudo-Ano…

Uniformizes Hodge structures, proving Lyapunov exponents and log-Anosov monodromy.

For every irreducible automorphism $φ\in\text{SL}_3({\mathbb Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $φ$, whose stretch factor is the product of the expanding eigenva…

Let $Σ$ be a compact orientable surface of finite type with at least one boundary component. Let $f \in \textup{Mod}(Σ)$ be a pseudo Anosov mapping class. We prove a conjecture of McMullen by showing that there exists a finite cover $\widetildeΣ \to Σ$ and a lift $\widetilde{f}$ of $f$ such that $\wt{f}_*: H_1(\wtΣ; \m…

We introduce a new object, the dynamical torsion, which extends the potentially ill-defined value at $0$ of the Ruelle zeta function of a contact Anosov flow twisted by an acyclic representation of the fundamental group. We show important properties of the dynamical torsion: it is invariant under deformations among con…

Study shows non-wandering, partially hyperbolic systems are ergodic.

The paper proves a relation between four types of invariants.

We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.

Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level $r$). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants $TV_r…

Geometric methods show cosets of mapping class groups for 3-manifolds.

We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the …

In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should…

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold $(M,g)$ with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a completely new stabilit…

Study partially hyperbolic dynamics on 3-manifolds with quasi-isometric center.

Classifies 3D partially hyperbolic systems, proving ergodicity.

The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.

The study computes trace fields and minimal polynomials for specific knots and links.