We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated by Dyatlov-Zahl and proved for porous sets in Bourgain-Dyatlov.

In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.

We study Schrödinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of non-uniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a “stratified sense” which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to establish that the “critical set” for the transition lies within countably many codimension one subvarieties of the (infinite-dimensional) parameter space. A more refined renormalization-based analysis shows that the critical set is rather thin within those subvarieties, and allows us to conclude that a typical potential has no critical energies. Such acritical potentials also form an open set and have several interesting properties: only finitely many “phase transitions” may happen, but never at any specific point in the spectrum, and the Lyapunov exponent is minorated in the region of the spectrum where it is positive. On the other hand, we do show that the number of phase transitions can be arbitrarily large.

In this paper, we propose a direct method to evaluate Hankel determinants for some generating functions satisfying a certain type of quadratic equations, which cover generating functions of Catalan numbers, Motzkin numbers and Schrder numbers. Additionally, four recent conjectures proposed by Cigler (2011) [3] are proved.

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products q_n d(q_n θ,Z) where the qn's are the denominators of the convergents associated with the real number θ by the ordinary continued fraction algorithm. Beside these two main results, we show that when d≥2, for almost all vectors θ∈R^d, lim inf_{n→∞} q_{n+k} d(q_n θ,Z^d)=0 for all positive integers k, where (q_n)_{n∈N} is the sequence of best approximation denominators of θ.