Shigeki Akiyama (Tsukuba)
Title: Nearly recursive sequences
Abstract: We generalize the notion of linear recursive sequence to include the sequence which encodes discretized rotation dynamics and discuss fundamental properties. We study distribution of zeroes, the growth of the sequence, etc. We obtain rather surprizing behaviers of such sequences by number theoretical construction. This is a joint work with A.Petho and J.H Evertse.
Pierre Arnoux (Marseille)
Title: Geometric models for continued fractions of the Pisot type
Abstract: We first review the well-known connection between usual continued fraction, dynamics of circle rotations, sturmian sequences and the geodesic flow on the modular surface; we then show how this framework can be extended to higher dimensional continued fractions by using a special type of symbolic sequences generalizing sturmian sequences (S-adic sequences), provided that the continued fraction satisfies a generalization of the Pisot property. This talk is an introduction to the following lecture by Valérie Berthé.
Valérie Berthé (Paris 7)
Title: Beyond Sturmian dynamics: Brun renormalization and Kronecker sequences
Abstract: Sturmian dynamics is now completely understood via the deep connections that exist between word combinatorics of Sturmian words, regular continued fractions acting as a renormalization scheme, and discrepancy properties of Kronecker sequences $(n\alpha)_n$. We discuss the extension of this framework to the multidimensional case by focusing on the case of the Brun continued fraction algorithm. This is a joint work with P. Arnoux, M. Minervino, W. Steiner, J. Thuswaldner.
Yann Bugeaud (Strasbourg)
Title : On the representation of real numbers to distinct bases
Abstract : Let $b$ be an integer greater than or equal to $2$. A real number is called simply normal to base $b$ if each digit $0, \ldots , b-1$ occurs in its $b$-ary expansion with the same frequency $1/b$. It is called normal to base $b$ if it is simply normal to every base $b^k$, where $k$ is a positive integer (or, equivalently, if, for every positive integer $k$, each block of $k$ digits from $0, \ldots , b-1$ occurs in its $b$-ary expansion with the same frequency $1/b^k$). This notion was introduced in by 1909 \'Emile Borel, who established that almost every real number (in the sense of the Lebesgue measure) is normal to every integer base. We start with the existence of uncountably many numbers normal to base $2$ but not simply normal to base $3$, a result proved independently by Cassels and Schmidt more than fifty years ago. More generally, under some necessary conditions on a (finite or infinite) set $B$ of integers greater than or equal to $2$, we discuss the existence of uncountably many numbers which are simply normal to any base in $B$ and not simply normal to any base not in $B$. This is a joint work with Ver\’onica Becher and Ted Slaman. Finally, we show that no irrational real numbers $\xi$ have a `too simple’ expansion in base $2$ and in base $3$, in the following sense: if the sequence of binary digits of $\xi$ is Sturmian, then the sequence of its ternary digits cannot be Sturmian. This is a joint work with Dong Han Kim.
Yitwah Cheung (San Francisco)
Title: A Game Approach to Littlewood Conjecture
Abstract: In this talk I will describe a single player game in which the player chooses a nested sequence of rectangles with the goal of optimizing a certain parameter describing the state of the game. The player wins if the sequence of nested rectangles can be chosen to keep the parameter uniformly bounded. The state parameter is explicitly computable in terms of the rational endpoints determining the corners of the rectangles. The game is designed so that a counterexample to the Littlewood Conjecture would result if a winning strategy can be found.
Anish Ghosh (Tata Institute)
Title: Diophantine approximation on affine subspaces
Abstract: I will discuss Diophantine approximation on manifolds and connections to homogeneous dynamics. I will emphasize some recent work on the question of Diophantine approximation on affine subspaces.
Alan Haynes (Houston)
Title: Higher dimensional Steinhaus problems
Abstract: The Steinhaus theorem, known colloquially as the 3-distance theorem, states that for any positive integer N and for any real number x, the collection of points nx modulo 1, with 0<n<N, partitions R/Z into component intervals which each have one of at most 3 possible distinct lengths. Many authors have explored higher dimensional generalizations of this theorem. In this talk we will survey some of their results, and we will explore a two-dimensional version of the problem, which turns out to be closely related to the Littlewood conjecture. We will explain how tools from homogeneous dynamics can be employed to obtain new results about a problem of Erdos and Geelen and Simpson, proving the existence of parameters for which the number of distinct gaps in a higher dimensional version of the Steinhaus problem is unbounded. This is joint work with Jens Marklof.
Sanghoon Kwon (KlAS)
Title: Dynamics on the space of lattices in positive characteristic local fields
Abstract: We investigate the dynamics on the space of lattices SL(n,K)/SL(n,Z) of positive characteristic local fields K. We focus on the behavior of expanding translates of orbits under certain unipotent group and relate those to the properties of tiling system on affine buildings. In this talk, our goal is to discuss the Sobolev norm, the proof of effective equidistribution and the geometry of Bruhat-Tits buildings. This is a joint work with Seonhee Lim.
Lingmin Liao (Paris 12)
Title: Dynamical structure of some rational maps on the projective line of the field of $p$-adic numbers
Abstract : Rational maps of coefficients in the field $\mathbb{Q}_p$ of $p$-adic numbers are studied as dynamical systems on the projective line of $\mathbb{Q}_p$. For such a map, in general, we can find a minimal decomposition on its Fatou set, i.e., the Fatou set is a disjoint union of finite number of periodic orbits, finite or countably many minimal components (subsystems) and the attracting basins of periodic orbits and minimal components. While its Julia set usually contains some subsystems conjugate to subshift of finite type. The talk is based on some joint works with Ai-Hua Fan, Shi-Lei Fan and Yue-Fei Wang.
Seonhee Lim (Seoul National)
Title: Diophantine approximation and Hausdorff dimension
Abstract: In this talk, we will review some results on Diophantine approximation which are obtained by homogeneous dynamics on the space of lattices. One main question related to inhomogeneous Diophantine approximation will be explored : the Hausdorff dimension of bad grids. This is a joint work with U. Shapira and N. de Saxce.
Luca Marchese (Paris 13)
Title: Stable Hall’s ray for some generalized Lagrange spectra.
Abstract: The classical Lagrange spectrum is a subset of the positive real line, corresponding to a filtration of the set of badly approximable real numbers. Elements in the spectrum can be expressed also as maximal asymptotic excursion of bounded geodesic in the modular surface, which is the quotient of the upper half plane by SL(2,Z). Well-known features of such spectrum are its discrete lower part, whose elements are called Markoff numbers, and its upper part, which is an entire half-line and is know as Hall’s ray. In this talk we consider the same problem for any non co-compact Fuchsian group G with finite co-volume in SL(2,R). We prove the existence of Hall ray for the corresponding Legrange spectrum. We also prove that the Hall ray persists replacing the penetration in to cusp by a Lipschitz perturbation of it. If time will allow it, we will also discuss complementary results on spectra whose lower part is not discrete. This is a joint work with M. Artigiani and C. Ulcigrai. Complementary results come from a joint work with P. Hubert, S. Lelièvre and C. Ulcigrai.
Michał Rams (Polish Academy of Science)
Title: Lyapunov spectrum for random products of $SL(2,R)$ matrices.
Abstract: I will present our recent result (joint with Katrin Gelfert and Lorenzo Diaz) about the Lyapunov spectrum (in terms of topological entropy) for step one $SL(2,R)$ cocycles over the full shift on $k$ symbols. We investigate the nonuniformly hyperbolic (elliptic) situation, and we prove that generically (both in Baire sense and with respect to the Lebesgue measure) this spectrum is concave, has a unique maximum $\log k$ corresponding to a positive exponent, and the entropy of the zero exponent level set is positive but smaller than $\log k$. We are also able to present the exact (though not practically useful) formula for the spectrum in the form of Legendre transform of some pressure function.
Rene Ruhr (Tel Aviv)
Title: Effective Counting on Translation Surfaces
Abstract: I will report on an effective version of a result of Eskin and Masur: For almost all translation surfaces the number of saddle connections of length at most T, grows like cT2 + O(T2−κ). In this talk, we will focus on the dynamics of the SL_2(R) action that provides the method of counting. In fact, it is a somewhat simplified approach used by Eskin-Margulis-Mozes to give asymptotics on the Oppenheim conjecture. Joint with Amos Nevo and Barak Weiss.
Sanju Velani (York)
Title: Diophantine approximation in Kleinian groups: singular, extremal and bad limit points
Abstract: The aim is to initiate a "manifold'' theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural "manifold'' strengthening of Sullivan's logarithmic law for geodesics.
Short Presentations
Dong Han Kim (Dongguk)
Title: Examples of higher dimensional 3-Gap Problems
Sanghoon Kwon (KIAS)
Title: Littlewood conjecture in formal series fields: a combinatorial approach
Jungwon Lee (UNIST)
Title: Arithmetic statistics of modular symbols
Rene Ruhr (Tel Aviv)
Title: Expander graphs and property (τ)