Prima Giornata DinAmica - 25 November 2016

The First DinAmicI Day to be held at Gran Sasso Science Institute in L’Aquila is approaching. For people arriving on the 24th, we are meeting at the restaurant “Il primo papavero” for dinner.

This is the program of the event (25 November 2016):

9:00-9:50   Mauro Artigiani, “Lagrange spectrum of Veech translation surfaces” 10:00-10:50  Anna Miriam Benini, “Singular values and periodic points” 11:00-11:30  Coffee Break 11:30-12:20  Sara Munday, “Slippery Devil’s Staircases”

12:45-14:30  Lunch Break

14:30-16:30  Assembly

Abstracts:

• Mauro Artigiani (SNS Pisa)

“Lagrange spectrum of Veech translation surfaces”

The Lagrange spectrum is a classical object in Diophantine approximation on the real line that has been generalised to many different settings. In particular, recently it has been generalised to a similar object for translation surfaces, which attracted quite some attention in the field. We study the Lagrange spectrum in the contest of Veech translation surfaces. These are particular translation surfaces with many symmetries, that can be thought as a dynamical equivalent of the torus in higher genera. Together with L. Marchese and C. Ulcigrai, we show that for such surfaces, similarly to what happens in the classical case, the Lagrange spectrum contains an infinite interval, called Hall ray. In our construction we use coding of hyperbolic geodesics and we deduce a formula that allows to describe high values in the spectrum as a sum of two Cantor sets.

• Anna Miriam Benini (U. Roma “Tor Vergata”)

“Singular values and periodic points”

We consider the iteration of entire transcendental functions on the complex plane. A special role is played by periodic points, that is points such that \( f^n(z)=z \)  for some \(n\).  Other special points are the singular values, that is points near which not all inverse branches of \(f\) are well defined and univalent (for example, critical values, or asymptotic values, like \(0\) for the map \(e^z\)).  We investigate the interplay between singular values, non-repelling periodic points and families of periodic curves called dynamic rays. We use topologial and geometric techniques. Our work also gives a new and  direct  proof of the famous  Fatou-Shishikura inequality for functions with finitely many singular values with bounded postsingular set, asserting that the number of non-repelling cycles is bounded by the number of singular values. This is joint work with N. Fagella.

• Sara Munday (U. Bologna)

“Slippery Devil’s Staircases”

In this talk, we consider various examples, classical and more recent, of Slippery Devil’s staircases, or, in other words, strictly increasing functions of the unit interval with derivative Lebesgue-almost everywhere equal to zero. Often, but not always, these functions appear as topological conjugacy maps between two dynamical systems, for example, then well-known and well-studied Minkowski question-mark function, which can be seen as the the topological conjugacy map between the Farey and tent systems. We consider the possible values, other than zero, that the derivative of such functions can take, the Hausdorff dimension of the exceptional sets and some perturbation results. We also offer some hopefully interesting open questions.