Fractals and Numeration » Abstracts

For the program, see this Page

Shigeki Akiyama Rotational beta expansion and self-similar tilings

Generalizing beta expansion to higher dimension, one obtains a systematic way to produce many self-similar tilings using polygonal tiles. We wish to discuss this class of substitutive tilings and their properties. This is a joint work with Jonathan Caalim.

Pierre Arnoux Natural extensions and suspensions for continued fraction algorithms (joint work with S. Labbé)

Sebastian Barbieri The domino problem for structures between Z and Z^d

We introduce a variant of the domino problem for subshifts where the symbols are restrained to subsets of Z^d. More specifically, subsets which have a self-similar structure defined by a family of substitutions. In this setting we exhibit non-trivial families of structures where the domino problem is decidable and undecidable. Amongst those we show that the Sierpinski triangle has decidable domino problem, while the Sierpinski carpet has undecidabe domino problem. This is joint work with Mathieu Sablik.

Olivier Carton Independence of normal numbers

Hiromi Ei and Rie Natsui On absolutely continuous invariant measures for complex continued fraction maps

The aim of our talk is to construct the natural extension of complex continued fraction maps to determine the density functions of the absolutely continuous invariant measures.

Christiane Frougny Beta-representations of 0 and Pisot numbers

Let beta be a number >1. We show that if beta is not a Pisot number, then the set of infinite sequences having value 0 in base beta is not recognizable by a finite Buchi automaton.

Maria Rita Iaco Multidimensional beta-adic sequences

The aim of this talk is to present a special class of multidimensional beta-adic sequences, the so-called beta-adic Halton sequences. They are uniformly distributed sequences in the multidimensional unit square and they are generated by dynamical systems.

Dong Han Kim Dirichlet uniformly well-approximated numbers

Fix an irrational alpha. For a positive real number tau, consider the numbers y satisfying that for all large number Q, there exists a positive integer n bounded by Q, such that the distance of n alpha -y to its nearest integer is smaller than 1/Q^tau. These numbers are called Dirichlet uniformly well-approximated numbers. For any positive tau, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the irrationality exponent of alpha. It is also proved that with respect to tau, the only possible discontinuous point of the Hausdorff dimension is where tau equals to 1. This is joint work with Lingmin Liao.

Luca Marchese The Lagrange spectrum of some square-tiled surfaces

Lagrange spectra have been defined for closed submanifolds of the moduli space of translation surfaces which are invariant under the action of SL(2,R). We consider the closed orbit generated by a specific covering of degree 7 of the standard torus, which is an element of the stratum H(2). We give an explicit formula for the values in the spectrum, in terms of a cocycle over the classical continued fraction. Differently from the classical case of the modular surface, where the lowest part of the Lagrange spectrum is discrete, we find an isolated minimum, and a set with a rich structure right above it. Joint work with Pascal Huber, Samuel Lelièvre, Corinna Ulcigrai.

Milton Minervino Non-stationary Markov partitions for Pisot cocycles

Makoto Mori On the discrepancy of random sequences generated by Dynamical systems

Hitoshi Nakada On a construction of normal series with respect to Artin continued fractions (Paris)

Hitoshi Nakada On equivalence relations of continued fractions (Lyon)

Fumihiko Nakano (Gakushuin University) Generalized carries process and riffle shuffles

This is a joint work with Taizo Sadahiro (Tsuda College). Carries process is a Markov chain of carries in adding n numbers. We consider a generalization of that, studied the transition probability matrix, and its relation to combinatorics. The results include : (1) the stationary distribution is proportional to the decent statistics of colored permutation group (2) left eigenvector matrix is equal to the Foulkes character table of G(p, n) (3) Stirling-Frobenius number appears in the right eigenvector matrix. (4) Discussion of the generalized riffle shuffles whose descent process is equally distributed to the carries process.

Svetlana Puzynina Ergodic infinite permutations

Infinite permutations can be defined as equivalence classes of real sequences with distinct elements, such that only the order of elements is taken into account. In other words, an infinite permutation can be defined as a linear ordering of the set of natural numbers. We investigate a new natural notion of ergodic infinite permutations, that is, infinite permutations which can be defined by equidistributed sequences. We study complexity and substitutions on ergodic permutations.

Asaki Saito (Future University Hakodate) Pseudorandom number generators using true orbits of the 2x modulo 1 map on algebraic integers

We introduce two methods for generating pseudorandom binary sequences using true orbits of the 2x modulo 1 map. The characteristic of these methods is that they exactly compute chaotic orbits of the 2x modulo 1 map on algebraic integers of degree 2 and 3. In addition, we develop ways to properly select initial points (seeds), which can distribute initial points almost uniformly (equidistantly) in the unit interval and which can avoid mergers of the orbits starting from them. This is a joint work with A. Yamaguchi

Wolfgang Steiner Natural extensions and entropy of alpha-continued fractions

Wolfgang Steiner Tilings with (non-unimodular) S-adic Rauzy fractals

Jun-ichi Tamura (Institute for Mathematics and Computer Sciences, Tsuda College) Calculi of substitutions over complex-powered symbols, Rauzy fractals, and multidimensional complex continued fractions (I and II).

Introducing fractional calculi of substitutions over complex powered symbols, we extend the meaning of Rauzy fractals to imaginary directions. Some new theorems concerning diophantine approximations and multidimensional complex continued fractions could be widely useful for simultaneous approximations of algebraic/transcendental numbers.

Reem Yassawi Dynamical systems over Z_p generated by constant length substitutions

Hisatoshi Yuasa On subshifts arising from non-primitive substitutions

Shin-ichi Yasutomi Generation of stepped surfaces via modified Jacobi-Perron algorithm and cubic numbers

This is collaborating with M.Furukado and S.Ito. We discuss that generation of stepped surfaces via modified Jacobi-Perron algorithm. We show that for a periodic point (alpha,beta) by modified Jacobi-Perron algorithm if Q(alpha) has complex embedding , then related tiling substitutions generate the whole stepped surface from a fundamental set.