The first aim of this work is to give information about the algebraic properties of alternate bases determining sofic systems. Émilie Charlier (Université de Liège): Spectrum, algebraicity and normalization in alternate bases ( video) ( slides) ( paper) This is joint work Nicolás Álvarez and Martín Mereb. We recently showed that there are computable Poisson generic real numbers and that all Martin-Löf real numbers are Poisson generic. They also showed that Poisson genericity implies Borel normality but the two notions do not coincide, witnessed by the famous Champernowne constant. Yuval Peres and Benjamin Weiss proved that almost all real numbers, with respect to Lebesgue measure, are Poisson generic. Years ago Zeev Rudnick defined the Poisson generic real numbers as those where the number of occurrences of the long strings in the initial segments of their fractional expansions in some base have the Poisson distribution. Verónica Becher (Universidad de Buenos Aires & CONICET Argentina): Poisson generic real numbers ( slides) ( paper) Finally I will also present further developments of the theory, leading to the notion of q-irrationals and q-unimodular matrices. In particular I will briefly mention connections with the combinatorics of posets, cluster algebras, Jones polynomials, homological algebra. I will explain the construction and give the main properties. The definition of q-rationals naturally extends the one of q-integers and leads to a ratio of polynomials with positive integer coefficients. With Valentin Ovsienko we recently suggested a notion of q-rationals based on combinatorial properties and continued fraction expansions. The most popular among them are certainly the q-integers and the q-binomial coefficients which both appear in various areas of mathematics and physics. Sophie Morier-Genoud (Université Reims Champagne Ardenne): q-analogues of real numbers ( video) ( paper1) ( paper2) ( paper3) ( paper4)Ĭlassical sequences of numbers often lead to interesting q-analogues. I will also discuss a new project whose tentative title is “ Can the flap of butterfly's wings shift a tornado out of Texas - without chaos? It has a dense set of periodic points that are 1 D unstable and another dense set of periodic points that are all 2 D unstable. Also, we have created a piecewise linear map on a 3D cube that is unstable in 2 dimensions in some places and unstable in 1 in others. Its dynamics can make us rethink climate models. Then I will discuss a couple related “very simple” maps that tell us a great deal about very complex models.
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To set the scene, I will discuss one large model, a whole-Earth model for predicting the weather, and how to initialize such a model and what aspects of chaos are essential. Yorke (University of Maryland): Large and Small Chaos Models ( video) ( slides) This talk is based on joint work with Marta Maggioni. Topics that we consider are periodic expansions, universal expansions, speed of convergence and approximation coefficients. In this talk we introduce a family of random dynamical systems that produce many Lüroth type expansions at once. A comparison between the two and other comparable number systems was then given by Barrionuevo, Burton, Dajani and Kraaikamp in 1996. In 1990 Kalpazidou, Knopfmacher and Knopfmacher introduced alternating Lüroth expansions and studied their properties. Since the introduction of Lüroth expansions by Lüroth in his paper from 1883 many results have appeared on their approximation properties. This is joint work with Bing Li and Yufeng Wu.Ĭharlene Kalle (Universiteit Leiden): Random Lüroth expansions ( video) ( slides) ( journal) ( arXiv) A quantitative result on the largest prime divisors of the denominators of rational numbers in A is also obtained. For any ×b-invariant, non-dense subset A of [0,1), we prove the finiteness of rational numbers in A whose denominators can only be divided by primes in S. Let b ≥ 2 be an integer and S be a finite non-empty set of primes not containing divisors of b. Ruofan Li (South China University of Technology): Rational numbers in ×b-invariant sets ( video) ( slides) ( paper)
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