Pierre Dehornoy

Geodesic flow, left-handedness and templates

We establish that for every hyperbolic orbifold of type .2;q;1/ and for every orbifold of type .2;3;4gC2/, the geodesic flow on the unit tangent bundle is left handed. This implies that the link formed by every collection of periodic orbits .i/ bounds a Birkhoff section for the geodesic flow, and .ii/ is a fibered link. We also prove similar results for the torus with any flat metric. We also observe that the natural extension of the conjecture to arbitrary hyperbolic surfaces (with non-trivial homology) is false. 37D40, 57M20; 37D45, 37B50

Signature and concordance of positive knots: SIGNATURE AND CONCORDANCE OF POSITIVE KNOTS

We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.

Genus-one Birkhoff sections for geodesic flows

We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with 3 or 4 singular points admits explicit genus one Birkhoff sections, and we determine the associated first return maps.

Coding of geodesics and Lorenz-like templates for some geodesic flows

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

Almost commensurability of 3-dimensional Anosov flows

Abstract Two flows are almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwise almost commensurable.

On the 3-distortion of a path

We prove that, for embeddings of a path of length n in R^2, the 3-distortion is @W(n^1^/^2), and that, when embedded in R^d, the 3-distortion is O(n^1^/^(^d^-^1^)).