Noticias en español
A 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...
Read MoreCaracterización de relaciones regionalmente proximales mediante el semigrupo envolvente.
RESUMEN: El estudio de los sistemas de orden d ha despertado gran interés por sus aplicaciones en sistemas dinámicos, teoría de números y combinatoria. Un aspecto interesante es el estudio de las propiedades algebraicas de sus semigrupos envolventes. En esta charla se abordará la conexión entre el semigrupo envolvente y la relación regionalmente proximal, la cual define a los sistemas de orden d. En particular, se presentará una caracterización algebraica de estas relaciones. Luego mencionaré aplicaciones de estos resultados a la estructura de los cubos dinámicos y al estudio de los...
Read MoreComputer-assisted proof of robust transitivity.
RESUMEN: A smooth dynamical system is transitive if it has a dense orbit, loosely meaning that it has some chaos in a topological sense. If this property holds for all diffeomorphisms in a C¹-neighborhood, we say that systems in this neighborhood are robustly transitive. By Bonatti, Diaz and Pujals (2003), robustly transitive diffeomorphisms are volume hyperbolic, and thus they have positive topological entropy, being chaotic in a strict sense and in a robust way. Robust properties are key in classifying smooth dynamical systems, and they are also desirable to model applications. We develop...
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