A pointwise cubic average for two commuting transformations

Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,\mu,S,T)$ with commuting transformations $S$ and $T$, the average $$\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)$$ converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(\mu)$

La escuela pública y la promoción de la mejora social : reflexiones a partir de Latinoamérica

El texto analiza la institución escolar en el marco de la temática señalada, revisando, a grandes líneas, su aporte a la mejora de la sociedad. Lo hace desde una visión centrada en los problemas más relevantes que nos aquejan. Se entiende que la institución educativa y los sistemas pedagógicos han de tener una funcionalidad muy estrecha con el presente y con el futuro de las sociedades y, en razón de ello, se puede valorar su impacto y su trascendencia, esencialmente desde esta perspectiva. Dichas materias se revisan desde la heterogeneidad que representa Latinoamérica, como un continente de grandes desigualdades sociales y, por ende, de grandes urgencias, muchas de las cuales llevan a pensar en la eficacia de la escuela, pero sin ligarla -en lo determinante- con la resolución de los problemas que afectan a la sociedad. Este es el desafío de magnitud para hoy: levantar la vista y comprender hacia dónde vamos y qué podría hacer la escuela en ese escenario para apoyar la construcción de una sociedad con verdaderas soluciones.El text analitza la institució escolar en el marc de la temàtica assenyalada, tot revisant, a grans trets, l'aportació que realitza a la millora de la societat. Ho fa des d'una visió centrada en els problemes més rellevants que ens afligeixen. S'entén que la institució educativa i els sistemes pedagògics han de tenir una funcionalitat molt estreta amb el present i el futur de les societats, i, quant a això, se'n pot valorar l'impacte i la transcendència, essencialment des d'aquesta perspectiva. Aquestes matèries es revisen des de l'heterogeneïtat que representa Llatinoamèrica, com un continent de grans desigualtats socials i, per tant, de grans urgències, moltes de les quals porten a pensar en l'eficàcia de l'escola, però sense lligar-la -en l'essencial- amb la resolució dels problemes que afecten la societat. Aquest és el desafiament de magnitud per avui: aixecar la vista i comprendre cap a on anem i què podria fer l'escola en aquest escenari per donar suport a la construcció d'una societat amb veritables solucions.This article examines educational institutions in Latin America through a broad review of their contribution to improving key problems facing society today. As key players in the present and future of societies, it is essential to assess the impact and importance of schools and educational systems from this perspective. These issues are reviewed from the viewpoint of the heterogeneity of Latin America; a continent with marked social inequalities that require urgent solutions. Many of these problems lead us to believe in the efficacy of schools, without linking them - in a determinant manner - to the resolution of problems affecting society. This is the greatest challenge of today: to look up and understand where we are headed and what schools can do in this scenario to contribute to building a society through real solutions

On automorphism groups of Toeplitz subshifts

In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. In the subquadratic complexity case, we show that the automorphism group modulo the torsion is generated by the roots of the shift map and that the result of the non superlinear case is optimal. Namely, for any $\varepsilon > 0$ we construct examples of minimal Toeplitz subshifts with complexity bounded by $C n^{1+\epsilon}$ whose automorphism groups are not finitely generated. Finally, we observe the coalescence and the automorphism group give no restriction on the complexity since we provide a family of coalescent Toeplitz subshifts with positive entropy such that their automorphism groups are arbitrary finitely generated infinite abelian groups with cyclic torsion subgroup (eventually restricted to powers of the shift)

On partial rigidity of S\mathcal{S}-adic subshifts

We develop combinatorial tools to study partial rigidity within the class of minimal $\mathcal{S}$-adic subshifts. By leveraging the combinatorial data of well-chosen Kakutani-Rokhlin partitions, we establish a necessary and sufficient condition for partial rigidity. Additionally, we provide an explicit expression to compute the partial rigidity rate and an associated partial rigidity sequence. As applications, we compute the partial rigidity rate for a variety of constant length substitution subshifts, such as the Thue-Morse subshift, where we determine a partial rigidity rate of 2/3. We also exhibit non-rigid substitution subshifts with partial rigidity rates arbitrarily close to 1 and as a consequence, using products of the aforementioned substitutions, we obtain that any number in $[0, 1]$ is the partial rigidity rate of a system.Comment: Comments welcome

Seminorms for multiple averages along polynomials and applications to joint ergodicity

Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study the criteria of joint ergodicity for sequences of the form $(T^{p_{1,j}(n)}_{1}\cdot\ldots\cdot T^{p_{d,j}(n)}_{d})_{n\in\mathbb{Z}},$ $1\leq j\leq k$, where $T_{1},\dots,T_{d}$ are commuting measure preserving transformations on a probability measure space and $p_{i,j}$ are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic. We also give a characterization for sequences of the form $(T^{p(n)}_{i})_{n\in\mathbb{Z}}, 1\leq i\leq d$ to be jointly ergodic, answering a question due to Bergelson.Comment: A. Ferr\'e Moragues and N. Frantzikinakis pointed out a mistake in the initial version of the article regarding the deduction of Theorems 1.3 and 1.4 from Proposition 5.1. Additionally, Proposition 6.1 was falsely stated for $k\geq 1$ (while in the stated form it is only true for $k\geq 2$). Some changes have been made correcting these issues. To appear in Journal d'Analyse Math\'ematiqu