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Néstor Jara

Publications and source records attributed to Néstor Jara.

9 recordsLinked to original sources
arXiv2026

Dichotomies Faster or Slower than exponential are Irrelevant for Skew-Product Flows

We prove that dichotomies given by growth rates that are either faster or slower than exponential either do not occur or are inconsequential in the setting of skew-products with compact base. A similar conclusion is obtained for the nonuniform exponential behavior. To achieve this, we study families of translated linear nonautonomous differential equations for which we prove propagation results. We also study translations of growth rates under a comparison criteria.

math.DS↗
arXiv2025

The interplay of $μ$-dichotomy, bounded growth, and spectral properties via growth rate comparisons

We investigate the behavior of the dichotomy spectrum of nonautonomous linear systems under general growth rates. By introducing comparison criteria we clarify how $μ$-dichotomy and $μ$-bounded growth interact. We also study the evolution of the dichotomy spectrum under these comparisons, revealing that faster growth rates compress the spectrum, while slower growth rates expand it. Moreover, we introduce equivalence relations on the set of growth rates, which enable us to establish that, for any given system, there exists -- up to equivalence -- at most one growth rate under which both properties, bounded growth and dichotomy, hold. Finally, we show that these equivalence relations lead to a classification of dichotomy spectra. Our results are valid in both discrete and continuous time settings.

math.DS↗
arXiv2025

Discrete $μ$-dichotomy spectrum: beyond uniformity and new insights

We develop spectral theorems for nonautonomous linear difference systems, considering different types of $μ$-dichotomies, both uniform and nonuniform. In the nonuniform case, intriguing scenarios emerge -- that have been employed but whose consequences have not been thoroughly explored -- which surprisingly exhibit unconventional behavior. These particular cases motivate us to introduce two novel properties of nonautonomous systems (even in the continuous-time framework), which appear to have been overlooked in the existing literature. Additionally, we introduce a new conceptualization of a nonuniform $μ$-dichotomy spectrum, which lies between the traditional nonuniform $μ$-dichotomy spectrum and the slow nonuniform $μ$-dichotomy spectrum. Moreover, and this is particularly noteworthy, we propose a conjecture that enables the derivation of spectral theorems in this new setting. Finally, contrary to what has been believed in recent years, through the lens of optimal ratio maps, we show that the nonuniform exponential dichotomy spectrum is not preserved between systems that are weakly kinematically similar.

math.DS↗
arXiv2024

Generalized nonautonomous dynamics through groupoid morphisms

We extend the notions of nonautonomous dynamics to arbitrary groups, through groupoid morphisms. This also presents a generalization of classic dynamical systems and group actions. We introduce the structure of cotranslations, as a specific kind of groupoid morphism, and establish a correspondence between cotranslations and skew-products. We give applications of cotranslations to nonautonomous equations, both in differences and differential. Our results delve into the differentiability of cotranslations, along with dimension invariance and diagonalization, utilizing a generalized notion of kinematic similarity.

math.DS↗
arXiv2024

Spectrum invariance dilemma for nonuniformly kinematically similar systems

We unveil instances where nonautonomous linear systems manifest distinct nonuniform $μ$-dichotomy spectra despite admitting nonuniform $(μ, \varepsilon)$-kinematic similarity. Exploring the theoretical foundations of this lack of invariance, we discern the pivotal influence of the parameters involved in the property of nonuniform $μ$-dichotomy such as in the notion of nonuniform $(μ, \varepsilon)$-kinematic similarity. To effectively comprehend these dynamics, we introduce the stable and unstable optimal ratio maps, along with the $\varepsilon$-neighborhood of the nonuniform $μ$-dichotomy spectrum. These concepts provide a framework for understanding scenarios governed by the noninvariance of the nonuniform $μ$-dichotomy spectrum.

nlin.CD↗
arXiv2023

Groupoid morphisms as an algebraic structure for nonautonomous dynamics

We present groupoid morphisms as an algebraic structure for nonautonomous dynamics, as well as a generalization of group morphisms, which describe classic dynamical systems. We introduce the structure of cotranslations, as a specific kind of groupoid morphism, and establish a correspondence between cotranslations and skew-products. We give applications of cotranslations to nonautonomous equations, both in differences and differential. We obtain results about the differentiability of cotranslations, as well as dimension invariance and diagonalization (through a generalized notion of kinematic similarity) for a partial version of them, admitting noninvertible transformations.

math.DS↗
arXiv2023

Gohberg Lemma and Spectral Results for Pseudodifferential Operators on Locally Compact Abelian Groups

We provide a new type of proof for known or new Gohberg lemmas for pseudodifferential operators on Abelian locally compact groups $\mathrm{X}$. We use $C^*$-algebraic techniques, which also give spectral results to which the Gohberg lemma is just a corollary. These results extend most of those appearing in the literature in various directions. In particular, compactness or a a Lie structure are not needed. The ideal of all the compact operators in $\mathrm{L}^2(\mathrm{X})$ is replaced by all the ideals having a crossed product structure, which is a consistent generalization. We also indicate several new examples, mostly connected to specific behaviors of functions on the dual $Ξ$ of $\mathrm{X}$.

math.FA↗
arXiv2021

A note on the differentiability of Palmer's topological equivalence for discrete systems

A linear system of difference equations and a nonlinear perturbation are considered. We propose sufficient conditions to ensure that the homeomorphism of topological equivalence between them is actually a $C^1$ diffeomorphism. These conditions consider that the linear part satisfies a dichotomy on the positive half-line, while the perturbation satisfies boundness and Lipschitzness conditions, and anothers that are tailored for our goal. The family of dichotomies that we study consider (but are not limited to) the exponential and nonuniform exponential dichotomies. We discuss the possibility of extending this result to a higher class of differentiability.

math.DS↗