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Titres et résumésGiedrius Alkauskas First talk: The Minkowski question mark function and quasi-modular forms. In this talk, the Minkowski question mark-world is naturally integrated within a certain uniform construction with the classical world of modular forms for the full modular group. This construction leads to the new notion of mean-modular forms. The latter linear space of dimension 2k turns out to be canonically isomorphic to the space of quasi-modular forms of weight 2k, weight 2k subspace of the graded ring generated by E_2, E_4 and E_6. Second talk: The structure of the moments of the Minkowski question mark function. In this talk we present results on the moments of ?, with its relation to period functions for Maass wave forms, also structural constants from the theory of transfer operators. We also present few open problems. In particular, the fine arithmetic structure of these moments is still elusive. It is my deep conviction that these objects are much deeper than we know now. Pierre Arnoux, La fonction ? comme conjugaison de systèmes dynamiques The Minkowski question mark function is usually defined in an arithmetical way, by translating the continued fraction expansion of a real number to a dyadic expansion; there is however another natural définition, using the dynamical systems underlying the continued fraction (Gauss map) and the dyadic expansion. The additive version of the Gauss map, the Farey map, is a bimodal map, which is conjugate to a linear tent map; the conjugacy map can be written explicitly, using the fact that both maps are Markov maps with the same symbolic dynamics.It is even easier to define this map by using a modified (increasing) Farey map and the doubling map that underlies the binary system. This definition gives a symbolic interpretation to the Minkowski map which allows for nontrivial generalisations in higher dimension. Thomas Garrity, On Minkowski Question Mark Type Functions for a Family of Multidimensional Continued Fractions The overall goal is to explore the natural analogs of the Minkowski question mark function for triangle partition maps. We start with some motivations for studying multidimensional continued fractions in the first place. We will then look at Giovanni Panti's work on the question mark functions for the Monkemeyer map. We will then discuss triangle partition maps, which are a family that include many, if not most, well-known multidimensional continued fraction algorithms. In particular, we will start the process of seeing which triangle partition maps have singular associated question mark functions. This last part is joint work with Peter McDonald. Stefano Isola : Orderings of the rationals: dynamical systems, statistical mechanics and other issues We’ll discuss several topics orbiting around the Stern-Brocot ordering of positive rationals, such as codings and maps, spin chains and random walks, where the ? function plays a key role as a connecting tool. We shall also discuss how the latter enters in the construction of explicit correspondences between bifurcation sets of various families of dynamical systems, such as alpha-continued fraction transformations and unimodal maps. Thomas Jordan, Fourier transforms and the Minkowski question mark function In the first part of the talk I will describe the multifractal nature of the measure defined by the Minkowski ? mark function and large deviation results which can be obtained for the measure. In the second park I will talk about how this can be combined with ideas from Kaufman and Queffélec and Ramare to show that the Fourier transform for ? decays polynomially, answering a question of Salem. This is a joint work with Tuomas Sahlsten. Nikolay Moshchevitin, On the derivative of the Minkowski function We discuss some results related to the famous Minkowski question-mark function ?(x), defi ned as the limit distribution function for Stern-Brocot sequences. It is well known fact that if for a certain x in [0;1] the derivative ?'(x) exists, then either ?'(x) = 0 or ?'(x) = +infinity. We study the problem how to understand the behavior of the derivative ?'(x) from the continued fraction expansion of x. We consider similar problems for more general functions such as Denjoy's functions g_λ(x), and some others. Giovanni Panti : Moving rational points Rational points form a countable dense subset of the unit simplex, and any homeomorphism of the simplex sends them to another countable dense set. In the opposite direction, specifying a bijection between two such subsets does not usually induce a homeomorphism; when it does, it does so uniquely. If we are interested in homeos of a certain kind (e.g., induced by some algebraic structure), then the extension problem becomes quickly very delicate. We will discuss various instances of this phenomenon, the multidimensional Question Mark function being a remarkable one. Martine Queffélec : Around Rajchman measures Salem asked whether the measure defined by the Minkowski ? mark function is a Rajchman measure which means: its Fourier transform vanishes at infinity; this question has been solved by T. Jordan and T. Sahlsten. Actually, the decay rate of the Fourier transform of a Rajchman measure is a relevant parameter, providing informations on its support. We shall illustrate this fact with measures supported on continuous fraction Cantor sets, and may be also discuss the Erdös' conjecture on Bernoulli convolution measures. Muhammed Uludag : Jimm, a fundamental involution Dyer's outer automorphism of PGL(2,Z) induces an involution of the real line, which behaves very much like a kind of modular function. It has some striking properties: it preserves the set of quadratic irrationals sending them to each other in a non-trivial way and commutes with the Galois action on this set. It restricts to an highly non-trivial involution of the set unit of norm +1 of quadratic number fields. It conjugates the Gauss continued fraction map to the so-called Fibonacci map. It preserves harmonic pairs of numbers inducing a duality of Beatty partitions of N. It induces a subtle symmetry of Lebesgue's measure on the unit interval. On the other hand, it has jump discontinuities at rationals though its derivative exists almost everywhere and vanishes almost everywhere. In the talks, I plan to show how this involution arises from a special automorphism of the infinite trivalent tree and how it relates to the Minkowski question mark function.
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