Hall rays for Lagrange spectra at cusps of Riemann surfaces
Aix-Marseille Université - Site St Charles
3, place Victor Hugo - case 39
13331 MARSEILLE Cedex 03
The Lagrange spectrum is a classical object in Diophantine approximation on the real line. It can be also seen as the spectrum of asymptotic penetration of hyperbolic geodesics into the cusp of the modular surface. This interpretation yielded many generalizations of the Spectrum to non-compact, finite volume, negatively curved surfaces and higher dimensional manifolds. A remarkable property of the classical Spectrum is that it contains an infinite interval, called Hall ray. The presence of the Hall ray is a common feature of the generalizations of the Lagrange spectrum to higher dimensions. We show that the Lagrange spectrum of hyperbolic surfaces contains a Hall ray. Moreover, we show that the same result holds if we measure the excursion into the cusps with a proper function that is close in the Lipschitz norm to the hyperbolic height.
This is a joint work with L. Marchese and C. Ulcigrai.
- Mauro ARTIGIANI