On connections, geodesics and sprays in synthetic differential geometry
On the creation of conjugate points.
On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flows
On the Entropy of the Geodesic Flow in Manifolds Without Conjugate Points.
On the Ergodicity of Frame Flows.
On the existence of escaping geodesics.
On the finite blocking property
A planar polygonal billiard is said to have the finite blocking property if for every pair of points in there exists a finite number of “blocking” points such that every billiard trajectory from to meets one of the ’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces....
On the geodesics flow of tori without conjugate points.
On the Lengths of Closed Geodesics on Almost Round Spheres.
On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus
We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham’s proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surface. For genus to , the minimum dilatation is the smallest Salem number for polynomials of degree .
On the spectral theory and dynamics of asymptotically hyperbolic manifolds
We present a brief survey of the spectral theory and dynamics of infinite volume asymptotically hyperbolic manifolds. Beginning with their geometry and examples, we proceed to their spectral and scattering theories, dynamics, and the physical description of their quantum and classical mechanics. We conclude with a discussion of recent results, ideas, and conjectures.
Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature.