arXiv subjects

Jaime Gómez

Publications and source records attributed to Jaime Gómez.

10 recordsLinked to original sources

arXiv2026

On the centralizer and normalizer groups of odometers and Toeplitz subshifts

For residually finite groups, we study the automorphism and normalizer groups of odometers and their symbolic extensions: Toeplitz subshifts. In relation to orbit equivalence theory, we show that two continuous orbit equivalent $G$-odometers have commensurable normalizers when $G=\mathbb{Z}^d$, but that this result fails for other nilpotent groups $G$. The automorphism and normalizer groups of any Toeplitz subshift embed into those of its underlying odometer; despite this constraint, they exhibit considerable flexibility. For instance, we exhibit examples of low-complexity $G$-Toeplitz subshifts whose centralizer is isomorphic to $G$ (or, at the other extreme, restricted to its center) yet whose normalizer group is as large as possible.

math.DS↗

arXiv2026

Sequence entropy and independence in free and minimal actions

For every countable infinite group that admits $\mathbb{Z}$ as a homomorphic image, we show that for each $m\in\mathbb{N}$, there exists a minimal action whose topological sequence entropy is $\log(m)$. Furthermore, for every countable infinite group $G$ that contains a finite index normal subgroup $G'$ isomorphic to $\mathbb{Z}^r$, and for every $m\in \mathbb{N}$, we found a free minimal action with topological sequence entropy $\log(n)$, where $m\leq n\leq m^{2^r[G:G']}$. In both cases, we also show that the aforementioned minimal actions admit non-trivial independence tuples of size $n$ but do not admit non-trivial independence tuples of size $n+1$ for some $n\geq m$.

math.DS↗

arXiv2026

Multisoliton solutions and blow up for the $L^2$-critical Hartree equation

We construct multisoliton solutions for the $L^2$-critical Hartree equation with trajectories asymptotically obeying a many-body law for an inverse square potential. Precisely, we consider the $m$-body hyperbolic and parabolic non-trapped dynamics. The pseudo-conformal symmetry then implies finite-time collision blow up in the latter case and a solution blowing up at $m$ distinct points in the former case. The approach we take is based on the ideas of [Krieger-Martel-Raphaël, 2009] and the third author's recent extension [Wu, 2026]. The approximation scheme requires new aspects in order to deal with a certain degeneracy for generalized root space elements.

math.AP↗

arXiv2025

Constructing Toeplitz arrays via cut and project schemes

We show a one-to-one correspondence between Toeplitz arrays over residually finite topological groups and model sets obtained via specific cut and project schemes, built from the odometer associated to the Toeplitz array. As an application, we construct irregular Toeplitz arrays which are extensions of maximal rank $k$ over their maximal equicontinuous factor (given by the associated odometer) and have exactly $k$ different ergodic measures. A modification of the construction also allows to obtain examples with the same measure-theoretic structure, but infinite maximal rank.

math.DS↗

arXiv2024

Uniform stability of concentration inequalities and applications

We prove a sharp quantitative version of recent Faber-Krahn inequalities for the continuous Wavelet transforms associated to a certain family of Cauchy wavelet windows . Our results are uniform on the parameters of the family of Cauchy wavelets, and asymptotically sharp in both directions. As a corollary of our results, we are able to recover not only the original result for the short-time Fourier transform as a limiting procedure, but also a new concentration result for functions in Hardy spaces. This is a completely novel result about optimal concentration of Poisson extensions, and our proof automatically comes with a sharp stability version of that inequality. Our techniques highlight the intertwining of geometric and complex-analytic arguments involved in the context of concentration inequalities. In particular, in the process of deriving uniform results, we obtain a refinement over the proof of a previous result by the first and fourth authors together with A. Guerra and P. Tilli, further improving the current understanding of the geometry of near extremals in all contexts under consideration.

math.FA↗

arXiv2024

Almost 1-1 extensions of Furstenberg-Weiss type and test for amenability

Let $G$ be an infinite residually finite group. We show that for every minimal equicontinuous Cantor system $(Z,G)$ with a free orbit, and for every minimal extension $(Y,G)$ of $(Z,G)$, there exist a minimal almost 1-1 extension $(X,G)$ of $(Z,G)$ and a Borel equivariant map $ψ:Y\to X$ that induces an affine bijection $ψ^*$ between $M(Y,G)$ and $M(X,G)$, the spaces of invariant probability measures of $(Y,G)$ and $(X,G)$, respectively. If $Y$ is a Cantor set, then $(Y,G)$ and $(X,G)$ are Borel isomorphic, i.e., $ψ^*$ is also a homeomorphism. As an application, we show that the family of Toeplitz subshifts is a test for amenability for residually finite groups, i.e., a residually finite group $G$ is amenable if and only if every Toeplitz $G$-subshift has invariant probability measures.

math.DS↗

arXiv2023

Almost automorphic systems have invariant measures

Let $G$ be a non-amenable countable group. We show that every almost automorphic $G$-action on a compact Hausdorff space, with a maximal equicontinuous factor whose phase space is a Cantor set, admits invariant probability measures (this partially answers a question posed by Veech). In particular, every Toeplitz $G$-subshift has a non-empty space of invariant measures, meaning that this family of subshifts is not a test for amenability for countable groups. We prove that almost one-to-one extensions without measures ensure the existence of symbolic almost one-to-one extensions with equal characteristics. As a consequence, we obtain the most general result of this paper. Finally, as a corollary of our results, we deduce that the class of Toeplitz subshifts is not dense in the space of infinite transitive subshifts of $Σ^G$, unlike $G=\mathbb{Z}$.

math.DS↗

arXiv2023

Topo-isomorphisms of irregular Toeplitz subshifts for residually finite groups

For each countable residually finite group $G$, we present examples of irregular Toeplitz subshifts in $\{0,1\}^G$ that are topo-isomorphic extensions of its maximal equicontinuous factor. To achieve this, we first establish sufficient conditions for Toeplitz subshifts to have invariant probability measures as limit points of periodic invariant measures of $\{0,1\}^G$. Next, we demonstrate that the set of Toeplitz subshifts satisfying these conditions is non-empty. When the acting group $G$ is amenable, this construction provides non-regular extensions of totally disconnected metric compactifications of $G$ that are (Weyl) mean-quicontinuous dynamical systems.

math.DS↗

arXiv2023

Stability of the Faber-Krahn inequality for the Short-time Fourier Transform

We prove a sharp quantitative version of the Faber--Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit $δ(f;Ω)$ which measures by how much the STFT of a function $f\in L^2(\mathbb R)$ fails to be optimally concentrated on an arbitrary set $Ω\subset \mathbb R^2$ of positive, finite measure. We then show that an optimal power of the deficit $δ(f;Ω)$ controls both the $L^2$-distance of $f$ to an appropriate class of Gaussians and the distance of $Ω$ to a ball, through the Fraenkel asymmetry of $Ω$. Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.

math.CA↗

arXiv2023

Invariant measures of Toeplitz subshifts on non-amenable groups

Let $G$ be a countable residually finite group (for instance $\mathbb{F}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\geq 1$ we construct a Toeplitz $G$-subshift $(X,σ,G)$, which is an almost one-to-one extension of $\overleftarrow{G}$, having $r$ ergodic measures $ν_1, \cdots,ν_r$ such that for every $1\leq i\leq r$ the measure-theoretic dynamical system $(X,σ,G,ν_i)$ is isomorphic to $\overleftarrow{G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups), however, we point out the differences and obstructions that could appear when the acting group is not amenable.

math.DS↗