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

Abstract

Abstract:In this work, we use use a solution to a packing problem in the plane to study recurrence of maps on the interval [0,1]. First of all, we prove that 1/ √ 5 is the optiaml recurrence rate of measurable applications of the interval. Secondly, we analyze the bottom of the Lagrange spectrum of interval exchange transformations.

Estimating the Hausdorff measure using recurrence

“…And while it is possible that there exists a map with even slower recurrence, there does not seem to be any chance of improving up to the optimal lower bound. This is shown by a result of Boshernitzan and Delecroix, [2], which we will utilise below.…”

“…This shows that taking only the rotations, we have no chance of realising (4.2). And Boshernitzan and Delecroix generalise this to prove (in [2]) that inequality (4.3) is true for all maps preserving the Lebesgue measure. This shows that the method shown here has an irremovable obstacle in achieving the best bound.…”

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

“…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]. The authors proved in [1] that also for the particularly symmetric class of translation surfaces made of Veech surfaces, the Lagrange spectrum contained a Hall ray and the first values of the Lagrange spectrum a particular example of a square-tiled Veech surface are studied in detail in [19].…”

“…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.…”

“…It is well known that such Γ admits a fundamental domain for the action on D which is an ideal polygon, that is a hyperbolic polygon having finitely many vertices all lying on ∂D (see [39]). For technical reasons, we require some additional properties (in particular Condition (5) in the Lemma below) for the fundamental domain. We will hence construct a suitable fundamental domain F for the action of Γ on D. The following Lemma summarizes the choice of F. The reader can refer to Figure 3 for an example of a fundamental domain satisfying the requirements of the Lemma.…”