Fractals and Numeration » Aan13Abstracts

Viviane Baladi: Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps
(joint with S. Marmi and D. Sauzin)
We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Psi(z), associated to precritical orbits. If the perturbation is horizontal, a generic condition (Birkho ff typicality of the postcritical orbit) implies that the nontangential limit of the Psi(z) as z tends to 1 exists and coincides with the derivative of the acim with respect to the map (linear response formula).

Frits Beukers: Geodesic continued fractions and LLL
In the lecture we briefly explain the idea of higher dimensional geodesic continued fractions and more briefly, the LLL-algorithm. The main goal of this lecture is to present a proposal to combine the best of these two ideas into a higher dimensional continued fraction algorithm.

Henk Bruin: On generalisations of Selmer's continued fraction algorithm
A higher-dimensional continued fraction algorithm is meant to find rational approximations of a higher-dimensional vectors: each component of the vector needs to be approximated by rationals with a common denominator.
Subtractive algorithms form a large class of such algorithms, including e.g. the Jacobi-Perron algorithm. Selmer's algorithm is another. In this joint paper with Fokkink & Kraaikamp we find invariant measures for generalisations of Selmer's, and apply this to estimate the quality of approximations.

Yann Bugeaud: Algebraic approximation to automatic continued fractions
We establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients can be generated by a finite automaton. The main new ingredient is an improvement of Liouville's inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we study the sets of values taken by Mahler's exponent $w_2$ and by Koksma's exponent $w_2^*$ at automatic continued fractions.

Vincent Delecroix: Lyapunov exponents and quality of diophantine approximation in dimension 2
To multidimensional continued fraction algorithms is naturally associated a matrix cocycle. The ratio of the second over the first Lyapunov exponents of this cocycle is a measure of quality of diophantine approximation of this algorithm. I will show that for many 2-dimensional multidimensional continued fraction algorithms we actually have strong convergence: the second Lyapunov exponent is negative. Our method applies to Brun, Selmer and Jacobi-Perron algorithms.

Michael Drmota: Subsequences and independence of generalized Halton sequences
(joint work with Lukas Spiegelhofer)
The classical Van-der-Corput sequence \phi_2(n) has the property that special subsequences like \phi_2(n^2) are not uniformly distributed modulo 1. This phenomenon is easy to explain and to characterize.
However, if we change to \beta-adic Van-der-Corput sequences \phi_Z(n) related to Zeckendorf enumeration system then we observe that several arithmetic subsequences like \phi_Z(P(n)), n \ge 0 and an integer polynomial P(x), or \ph_Z(P(p)), p prime, are still uniformly distributed. The reason for this nice property is that it is possible to use a proper Markov partition (in the unit square) to characterize the initial Zeckendorf expansion n by looking at (n/\gamma^k, n/\gamma^{k+1}) mod 1 and to relate the uniform distribution problem of \phi_Z(n) (or its subsequences) to proper exponential sums.
Furthermore, this Markov partition can be used, too, to provide explicit upper bounds for exponential sums of the form \sum_n e(\alpha s_2(n) + \beta s_Z(n)), where s_2(n) and s_Z(n) denote the binary and the Zeckendorf sum-of-digits function, and (re)prove in a quantitative way that s_2(n) and s_Z(n) behave independently.

Thomas Garrity: A generalized family of multidimensional continued fractions: TRIP maps
Most well-known multidimensional continued fractions, including the triangle map and the Mönkemeyer map, are generated by repeatedly subdividing triangles. We will construct a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field. (This is joint work with Krishna Dasaratha, Laure Flapan, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse and Matthew Stoffregen.)

Lingmin Liao: Dirichlet uniformly well-approximated numbers
For a fixed irrational $x$, we consider the numbers $y$ satisfying that for all large number $Q\gg1$, there exists an integer $1\leq n\leq Q$, such that the distance from $nx-y$ to the nearest integer is less than $Q^{-s}$, with $s>1$. These numbers are called Dirichlet uniformly well-approximated numbers by the reason of the classical uniform Dirichlet Theorem in Diophantine approximation. For any $s>1$, the Hausdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine approximation property of $x$. This is a joint work with Dong Han Kim.

Ali Messaoudi: Adding machine and fibered Julia sets
In 2000, Killeen and Taylor defined a stochastic adding machine in base 2. They proved that the spectrum of the transition operator associated to this machine is equal to the filled Julia set of a quadratic polynomial map. In this work, we extend Killeen-Taylor's construction to a class of adding machine and prove that the spectrum of its transition operator is connected to the fibered Julia set. We also study topological and combinatorial properties of this spectrum.

Arnaldo Nogueira (Aix-Marseille Université and Laboratoire Fibonacci, UMI 3483-CNRS): Ergodicity of multidimensional continued fraction algorithms
We will consider the homogeneous versions of the so called Brun and Selmer multidimensional continued fraction algorithms. Both maps are dissipative, we shall prove that they are ergodic with respect to Lebesgue measure.
(The talk is based on a joint work with Jon Chaika.)

Fritz Schweiger: Open problems for multidimensional continued fractions
Multidimensional continued fractions go back to Jacobi. Later Poincaré, Brun, and Selmer followed with different proposals. Now, even more algorithms have been considered. In my talk open problems in several areas will be presented:

  • Periodicity of algebraic numbers
  • Diophantine properties
  • Approximation by volume and generalization of Lagrange spectrum
  • Invariant subsets
  • Ergodicity and invariant measures

Sasha Skripchenko (Université Paris Diderot - Paris 7): On the Hausdorff dimension of the Rauzy gasket
(joint work with P. Hubert and A. Avila)
The Rauzy gasket was introduced by P. Arnoux and Š. Starosta as a subset of the standard 2-dimensional simplex associated with the letter frequencies of ternary episturmian words. The same fractal appears in connection with systems of isometries of thin type that are described by 2 independent parameters. In both cases, the Rauzy gasket plays the role of a parameter space endowed with a dynamics by piecewise projective maps. It is also homothetic to the invariant set of the fully subtractive algorithm.
It was proved independently by P. Arnoux and Š. Starosta and by I. Dynnikov and R. De Leo in different terms that this set has zero Lebesgue measure, but there were no theoretical estimations of its topological dimension. We prove that the Hausdoff dimension of the Rauzy gasket is strictly less than 2.

Robert Tichy: Discrepancy, Dynamical Systems and Limit Theorems
Using the so called von Neumann-Kakutani transformation we obtain a dynamical approach to low discrepancy sequences. By Grabner, Hellekalek, Liardet and others it is well-known that this approach leads to an analysis of van der Corput type sequences from an ergodic point of view. Recently, this approach was extended by Carbone, Iacó and Volčič to numeration systems with linear-recurring base sequences. This again reveals a dynamic system approach for describing the so-called odometer ("addition by 1 machine"). In the first part of the lecture we present an extension to the multidimensional setting. This explains the classical Halton sequences and various analogs for linear-recurring base sequences. The second part of the lecture is devoted to van der Corput sets and sets of recurrence. We give new constructions involving equidistribution of sequences of prime powers. This refines and unifies earlier results obtained by Sárkzőy, Furstenberg, Kamae and Mendès France and Bergelson and Lesigne. The proofs heavily depend on analytic machinery involving bounds for exponential sums. In the third part of the lecture we give some new sharp result concerning the probabilistic behaviour of lacunary sequences. In particular we show limit theorems for discrepancy functions and related quantities.

Giulio Tiozzo: Combinatorics and entropy of continued fractions with SL(2, Z) branches
We shall look at the dynamics of generalized Gauss maps whose branches are Moebius transformations in SL(2, Z). These maps, also known as (a, b)-continued fractions, have been introduced by S. Katok and I. Ugarcovici (following D. Zagier).
We provide an explicit construction of the bifurcation locus for this family, and show it is parametrized by Farey words and combinatorially isomorphic to the main cardioid in the Mandelbrot set. As a consequence, we prove that the entropy is a monotone function of the parameter.
This is joint work with C. Carminati and S. Isola.

Brigitte Vallée: Metrical versions of the Two-Distance Theorem
(joint work with Eda Cesaratto, Buenos Aires)
We perform a probabilistic study of the celebrated Kronecker sequence K(alpha) of the fractional parts of the multiples of a real alpha. When truncated to its first T terms, there are only two or three distinct distances between consecutive points. We focus here to the case when the truncation integer T gives rise to the two distances phenomenon. We choose at random the real alpha and wish to answer the question: Is such a random truncated Kronecker sequence actually pseudo-random? We choose five parameters of interest -- distances, covered space, discrepancy and Arnold measure--; each of them can be viewed as a measure of randomness of the sequence K(alpha), and is useful to study the pseudo-randomness of the sequence K(alpha) for a random input alpha.
We consider two main cases: the case where the "input" alpha is a random real, and the case when alpha is a random rational, and we exhibit strong similarities between the two situations. Moreover, as the size of the quotients in the continued fraction expansion of alpha plays an important rôle in the behaviour of Kronecker sequences, we also focus to the "constrained" case where all the quotients are bounded by a constant M, and consider the transition between the constrained case M<infty and the unconstrained case M = infty. This is also why we consider two types of truncation integers, where the rôle of the quotients is different, and we exhibit two distinct behaviours of pseudo-randomness, depending on the type of the truncation.

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