## Condiciones 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 More## Substitutive 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 More## Minimal 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 More## Periodic 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...

Read More## Characterising Retract Subshifts – Introducing The Notion of Contractible Subshift.

RESUMEN: A subshift X is a retract of another subshift Y ⊃ X if the embedding morphism emb: X → Y is split-monic in the category of subshifts (i.e. have a left inverse block-map, or “retract”). To characterise this notion, we introduce “contractibility”, a stregthening of the strong-irreducibility property which requires that the gluings are given by a block-map. After giving the characterisation of being a retract subshift, we will list out a few links between contractibility and other properties of subshifts (e.g. having dense periodic points, having the finite...

Read More## Cellular automata and percolation in groups.

RESUMEN: A famous theorem by Gilman shows that every cellular automaton over AZ satisfies an important dynamical dichotomy with respect to any Bernoulli measure: either almost every configuration is sensitive to initial conditions, or the system is equicontinuous. We show that there exists a fundamental relationship between the existence of a non-trivial percolation threshold on the Cayley graphs of a given group G and the failure of this dichotomy. We use this to give a characterization of the countable groups where Gilman’s dichotomy is satisfied, which correspond to the class of...

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