Dynamical generalizations of the Lagrange spectrum

Abstract

We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's ne_n, where e_n is the smallest measure of a cylinder of length n, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we get for the family of rotations with the upper limit of 1/ne_n

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

“…Also in this case one has a geometric definition as penetration spectra for the Teichmüller geodesic flow, as well as an interpretation motivated by Diophantine approximation for interval exchange maps, see [20]. A version of the latter already appears in the work by Boshernitzan, see [4]; different types of Lagrange spectra for interval exchange transformations, in particular in the case of 3 interval exchanges, are also studied by Ferenczi in [10]. For Lagrange spectra of strata of translation surfaces, the existence of Hall rays was in [20] and the Hurwitz constant was recently found by Boshernitzan and Delecroix [5].…”

“…These type of Lagrange spectra are also called dynamical Lagrange spectra in the literature. Dynamical spectra were in particular studied in the seminal works by [27,32,17] and have seen a recent surge of interest, see for example [1,5,7,10,19,20,26]. If the surface X has only one cusp at infinity, height(·) is an example of a proper function on X.…”

From a Packing Problem to Quantitative Recurrence in [0,1] and the Lagrange Spectrum of Interval Exchanges

“…But the situation changes dramatically when one goes to higher dimensional situations. For instance, for both the 2-dimensional rotations [AS13, Theorem 1] and 3-i.e.t.s [Fe12,Theorem 4.14] the Dirichlet spectrum is an interval. Nothing seems to be known about the structure of Dirichlet spectrum for rotations in R 3 or 4-i.e.t.s.…”

“…In[Fe12] the Dirichlet spectrum is called the upper Boshernitzan-Lagrange spectrum. The reason for this is that M. Boshernitzan proved that for i.e.t.…”

The Lagrange spectrum of a Veech surface has a Hall ray

“…We refer the interested reader to the book [2] by Cusick and Flahive. Moreover, several generalizations of the classical Lagrange spectrum have been studied by many authors, in particular in the context of Fuchsian groups and, more in general, negatively curved manifolds see [3,5,7,10,15,16,18,26]. For a very brief survey of these generalizations, we refer to the introduction of [8].…”