Noticias en español
Acciones Por Difeomorfismos de Clase C¹ de los Grupos Baumslag-Solitar en Compactos Son Afines
RESUMEN: Los grupos Baumslag-Solitar BS(m,n) generan un interés dado la variedad en sus propiedades algebraicas y su comportamiento dinámico. De hecho, C. Bonatti, A. Navas, I. Monteverde y C. Rivas mostraron que ciertos grupos solubles (entre ellos BS(1,n)) solamente pueden actuar de manera afín en [0,1] cuando es por difeomorfismos de clase C¹. El motivo de esta charla es revisitar resultados de la dinamica 1-dimensional en acciones de grupos BS(m,n) y probar que en intervalos compactos las acciones por difeomorfismos de los grupos BS(m,n) (que no están incluidos en el trabajo de BMNR) son...
Read MorePartition Regularity for Quadratic Equations in Number Fields.
RESUMEN: An equation is partition regular over its domain if, for any finite coloring of that domain, there exists a monochromatic nontrivial solution. In this talk, we will review the background of this topic, focusing on the ergodic theoretic tools used to tackle such problems and present a recent joint work with A. Koutsogiannis, A. Ferré Moragues and W. Sun, concerning the partition regularity problem of quadratic equations over some number fields.
Read MoreA Normality Conjecture on Rational Base Number Systems
RESUMEN: The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called minimal and maximal words. We formulate the conjecture that every minimal and maximal word is normal over an appropriate subalphabet. The aim of the talk is to convince the audience that the conjecture seems true and of considerable difficulty. In particular, we shall discuss its connections with several older conjectures, including the existence of...
Read MoreSpectral Theory of Substitutions.
RESUMEN: Given a finite alphabet, a substitution is a rule that assigns to each letter a nontrivial word over the same alphabet. Although they are simple combinatorial objects, substitutions arise across a wide range of mathematical disciplines, including combinatorics on words, theoretical computer science (automata theory), number theory (Diophantine approximation, multiplicative functions), mathematical physics (quasicrystals), and ergodic theory (induced systems). In this talk, we will review recent work on the spectral properties of substitution dynamical systems, as well as other...
Read MoreRecent advances on multiple ergodic averages
RESUMEN In 1977, Furstenberg gave a dynamical proof of the theorem of Szemerédi on the existence of arithmetic progressions in dense subsets of integers. In doing so, he initiated the use of ergodic methods to solve problems originating from additive combinatorics and number theory. A central object of study in this field are multiple ergodic averages, a class of multilinear operators that generalize classical Birkhoff averages and can be used to count the number of arithmetic patterns in dense sets of integers. In this talk, I will outline the history of multiple ergodic averages, with...
Read MoreSingle orbits and Wiener-Wintner theorem
RESUMEN A single-orbit approach to dynamics links the global properties of a dynamical system with the behaviour of its orbits. During the talk, I shall discuss what can be deduced about the system from the existence of an orbit satisfying the conclusion of the Wiener-Wintner theorem (a Wiener-Wintner generic orbit). I will examine the spectrum of ergodic measures by examining the behaviour of their Wiener–Wintner generic points. Moreover, by investigating the properties of a “regular” subclass of such points, I shall characterise ergodic measures with discrete...
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