Noticias en español
Complexities of words generated by a billiard in the hypercube.
RESUMEN: Sturmian words form a class of binary infinite words which sheds light, through its equivalent definitions, on remarkable interactions between combinatorics, dynamical systems, and number theory. They give rise to several generalizations over the d-letter alphabet, for d ≥ 3. A large program, initiated in the 80s, is to determine which characteristic properties of Sturmian words each of these generalizations still satisfy. My talk will focus on one dynamical representations of Sturmian words: as words generated by a billiard on a square table, which generalizes itself to a billiard...
Read MoreAperiodic Wang tiles associated with metallic means.
RESUMEN: A Penrose tiling consists of two polygonal tiles whose frequency ratio is equal to the golden ratio. Similarly, tilings by the aperiodic monotile discovered in 2023 by David Smith are such that the ratio of the frequencies of the two orientations of the monotile is equal to the fourth power of the golden ratio. The structure of Jeandel-Rao tilings discovered in 2015 is also explained using the golden ratio. We know of aperiodic tilings that are not related to the golden ratio. However, the characterization of possible numbers for such ratios is a question, posed as early as 1992 by...
Read MoreCondiciones para la inestabilidad estructural de contracciones a trozos del intervalo.
RESUMEN: En la presente charla mostraremos que, bajo condiciones razonables, los mapas contractivos a trozos del intervalo que admiten dinámicas asintóticas no-periódicas no pueden ser estructuralmente estables. Se mencionarán algunos obstáculos y preguntas que surgieron durante este trabajo, tales como: ¿qué tipo de perturbaciones o topología deberíamos considerar? ¿cómo controlamos que las perturbaciones de contracciones a trozos sean contractivas a trozos sin exigir hipótesis adicionales? Dado que varios resultados críticos para esta demostración no dependen de la “contracción a...
Read MoreSubstitutive structures on general countable groups.
RESUMEN: Symbolic dynamics has been largely used to represent dynamical systems through a coding system. This method was initially developed by M. Morse and G. A. Hedlund. One commonly used coding method involves infinite sequences of morphisms defined on finitely generated monoids, known as directive sequences or S-adic representations. Recent research has shown that understanding the underlying S-adic structures of some subshifts is valuable for studying their dynamical properties. Considering the previous studies and acknowledging the effectiveness of the S-adic framework, it is natural...
Read MoreMinimal configurations for Frenkel-Kontorova model on a quasicrystal.
RESUMEN: The Frenkel-Kontorova model is a physical model that is mathematically simple to describe and universal in the sense that it can be used to describe several underlying physical concepts. It was originally introduced in 1938 to represent the structure and dynamics of a crystal lattice in the vicinity of a dislocation core. It models a chain of classical particles coupled to their neighbors and subjected to an external potential. In this talk, I will present an overview of some known properties and open questions concerning equilibrium configurations when the external potential is...
Read MorePeriodic fractional Ambrosetti-Prodi for one-dimensional problem with drift.
Abstract: We prove Ambrosetti-Prodi type results for periodic solutions of some one-dimensional nonlinear problems that can have drift term whose principal operator is the fractional Laplacian of order s ∈ (0, 1). We establish conditions for the existence and nonexistence of solutions of those problems. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also obtain a priori bounds in order to get multiplicity results. We also prove that the solutions are C1,α under some regularity assumptions in the...
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