Abstract

Motivated by the famous question popularized by M. Kac: "Can you hear the shape of a drum?" one can define various spectra associated with a geodesic flow on a closed surface. Is it possible to recover such a billiard or such a geodesic flow up to isometry? In the case of a geodesic flow of a negatively curved surface the answer is yes if we know a marked length spectrum. It is a classical result by Croke and by Otal. What happens when we remove markings? For a smooth expanding circle map f, the (unmarked) Lyapunov spectrum of f of order n ≥ 1 is defined as the set of multipliers along periodic orbits of period n.  We show that for a smooth expanding circle map f of degree d≥ 2, under certain assumption of sparsity of its Lyapunov spectrum, the answer is also yes. The proof uses the Whitney extension theorem, a quantitative Lifshits theorem and a novel iterative scheme. This is a joint work with K. Drach.