arXiv subjects

Lucas Backes

Publications and source records attributed to Lucas Backes.

At least 19 recordsLinked to original sources

arXiv2026

Periodic approximation of Lyapunov exponents for cocycles admitting invariant holonomies

Classical results establish that the Lyapunov exponents of an ergodic measure for linear cocycles over hyperbolic systems can be approximated by the Lyapunov exponents of periodic orbits, provided the cocycle is H\"older continuous. A recent counterexample by Bochi demonstrates that this approximation property fails in general if the H\"older assumption is relaxed to mere continuity. In this paper, we introduce a geometric condition that successfully substitutes this analytical regularity hypothesis. More precisely, we prove that if a cocycle - even a discontinuous one - admits a continuous family of invariant holonomies, the periodic approximation of Lyapunov exponents still holds. Our geometric approach yields a proof that is substantially simpler and more direct than existing arguments in the literature, even when applied to classical settings such as fiber-bunched cocycles for which previous results were already available.

math.DS↗

arXiv2026

Smooth linearization of nonautonomous dynamics under general dichotomic behaviour

The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a very general form of dichotomy known as $\mu$-dichotomy and that the associated $\mu$-dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. We observe that our notion of $\mu$-dichotomy encompasses the classical notions of exponential, polynomial and logarithmic dichotomies as very particular cases. In particular, our result is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between $\mu$-dichotomy and exponential dichotomy via a suitable reparametrization of time.

math.DS↗

arXiv2025

Dominated splittings and periodic data for quasi-compact operator cocycles

For infinite-dimensional quasi-compact cocycles over a map satisfying a certain closing condition, we show that periodic orbits carry enough information to guarantee the existence of a dominated splitting. More precisely, we establish that if the moduli of the $(k+1)$-largest eigenvalues of the cocycle are $e^{\lambda_1n}\geq e^{\lambda_2n}\geq \ldots\geq e^{\lambda_kn}\geq e^{\lambda_{k+1}n}$ at every periodic point of period $n$, and $\lambda_k>\lambda_{k+1}$, then the cocycle admits a dominated splitting of index $k$. As a consequence, if $\lambda_k>0>\lambda_{k+1}$ then the cocycle is uniformly hyperbolic. Furthermore, we are able to obtain these same conclusions even when the eigenvalues are only close to constant, not strictly constant.

math.DS↗

arXiv2025

Liv\v{s}ic regularity for random and sequential dynamics through transfer operators

We prove Liv\v{s}ic-type regularity results of coboundary representations for non-autonomous dynamical systems. Our results have an abstract nature and apply to several important specific situations, such as (higher-dimensional) random or sequential piecewise expanding maps and subshifts of finite type, which have applications to Markov interval maps and to finite state inhomogeneous elliptic Markov shifts, via symbolic representations. We also obtain results for some classes of non-autonomous hyperbolic systems. Our results can be seen as non-autonomous versions of a recent result obtained by Morris. However, we emphasize that our proof differs from the one mentioned previously even in the deterministic case. Finally, we show that our results provide a more relaxed characterization for having variance growth of Birkhoff sums on random and sequential dynamical systems; we show that such growth can fail only when the underlying functions are a coboundary without special restrictions on the regularity of the coboundary. For random systems, we show that this is equivalent to having a coboundary with bounded ``variation", but for sequential systems it turns out that this is no longer true, as demonstrated by examples.

math.DS↗

arXiv2024

Ion manipulation from liquid Xe to vacuum: Ba-tagging for a nEXO upgrade and future $0 \nu \beta \beta$ experiments

Neutrinoless double beta decay {($0\nu\beta\beta$)} provides a way to probe physics beyond the Standard Model of particle physics. The upcoming nEXO experiment will search for $0\nu\beta\beta$ decay in $^{136}$Xe with a projected half-life sensitivity exceeding $10^{28}$ years at the 90\% confidence level using a liquid xenon (LXe) Time Projection Chamber (TPC) filled with 5 tonnes of Xe enriched to $\sim$90\% in the {$\beta \beta$}-decaying isotope $^{136}$Xe. In parallel, a potential future upgrade to nEXO is being investigated with the aim to further suppress radioactive backgrounds and to confirm $\beta \beta$-decay events. This technique, known as Ba-tagging, comprises extracting and identifying the $\beta \beta$-decay daughter $^{136}$Ba ion. One tagging approach being pursued involves extracting a small volume of LXe in the vicinity of a potential $\beta \beta$-decay using a capillary tube and facilitating a liquid-to-gas phase transition by heating the capillary exit. The Ba ion is then separated from the accompanying Xe gas using a radio-frequency (RF) carpet and RF funnel, conclusively identifying the ion as $^{136}$Ba via laser-fluorescence spectroscopy and mass spectrometry. Simultaneously, an accelerator-driven Ba ion source is being developed to validate and optimize this technique. The motivation for the project, the development of the different aspects, along with the current status and results, are discussed here.

physics.ins-det↗

arXiv2024

Variational principles for metric mean dimension with potential of level sets

We establish three variational principles for the upper metric mean dimension with potential of level sets of continuous maps in terms of the entropy of partitions and Katok's entropy of the underlying system. Our results hold for dynamical systems exhibiting the specification property. Moreover, we apply our results to study the metric mean dimension of suspension flows.

math.DS↗

arXiv2024

A characterization of $(\mu,\nu)$-dichotomies via admissibility

We present a characterization of $(\mu,\nu)$-dichotomies in terms of the admissibility of certain pairs of weighted spaces for nonautonomous discrete time dynamics acting on Banach spaces. Our general framework enables us to treat various settings in which no similar result has been previously obtained as well as to recover and refine several known results. We emphasize that our results hold without any bounded growth assumption and the statements make no use of Lyapunov norms. Moreover, as a consequence of our characterization, we study the robustness of $(\mu, \nu)$-dichotomies, i.e. we show that this notion persists under small but very general linear perturbations.

math.DS↗

arXiv2024

Shadowing and hyperbolicity for linear delay difference equations

It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this paper, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two classes of linear delay difference equations: (a)~for nonautonomous equations with finite delays and uniformly bounded compact coefficient operators in (possibly infinite-dimensional) Banach spaces, (b)~for Volterra difference equations with infinite delay in finite dimensional spaces.

math.DS↗

arXiv2024

Shadowing, Hyers--Ulam stability and hyperbolicity for nonautonomous linear delay differential equations

It is known that hyperbolic non\-autonomous linear delay differential equations in a finite dimensional space are Hyers--Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.

math.DS↗

arXiv2023

Parameterized shadowing for nonautonomous dynamics

For nonautonomous and nonlinear differential and difference equations depending on a parameter, we formulate sufficient conditions under which they exhibit $C^k$, $k\in \N$ shadowing with respect to a parameter. Our results are applicable to situations when the linear part is not hyperbolic. In the case when the linear part is hyperbolic, we obtain results dealing with parameterized Hyers-Ulam stability.

math.DS↗

arXiv2022

Conditional Lipschitz shadowing for ordinary differential equations

We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain non\-auto\-nomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to the state variable is uniformly negative in a neighborhood of the prescribed set. The results are applicable to important classes of model equations including the logistic equation, whose conditional shadowing has recently been studied. Several examples are constructed showing that the obtained conditions are optimal.

math.DS↗

arXiv2022

Smooth linearization of nonautonomous dynamics under polynomial behaviour

The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a nonuniform polynomial dichotomy and that the associated polynomial dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. This is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between polynomial and exponential dichotomies via a suitable reparametrization of time.

math.DS↗

arXiv2022

A variational principle for the metric mean dimension of level sets

We prove a variational principle for the upper and lower metric mean dimension of level sets \[ \left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\right\} \] associated to continuous potentials $\varphi:X\to \mathbb R$ and continuous dynamics $f:X\to X$ defined on compact metric spaces and exhibiting the specification property. This result relates the upper and lower metric mean dimension of the above mentioned sets with growth rates of measure-theoretic entropy of partitions decreasing in diameter associated to some special measures. Moreover, we present several examples to which our result may be applied to. Similar results were previously known for the topological entropy and for the topological pressure.

math.DS↗

arXiv2022

Multiscale linearization of nonautonomous systems

We present sufficient conditions under which a given linear nonautonomous system and its nonlinear perturbation are topologically conjugated. Our conditions are of a very general form and provided that the nonlinear perturbations are well-behaved, we do not assume any asymptotic behaviour of the linear system. Moreover, the control on the nonlinear perturbations may differ along finitely many mutually complementary directions. We consider both the cases of one-sided discrete and continuous dynamics.

math.CA↗

arXiv2022

Smooth Linearization of Nonautonomous Coupled Systems

In a joint work with Palmer we have formulated sufficient conditions under which there exist continuous and invertible transformations of the form $H_n(x,y)$ taking solutions of a coupled system \begin{equation*} x_{n+1} =A_nx_n+f_n(x_n, y_n), \quad y_{n+1}=g_n( y_n), \end{equation*} onto the solutions of the associated partially linearized uncoupled system \begin{equation*} x_{n+1} =A_nx_n, \quad y_{n+1}=g_n( y_n). \end{equation*} In the present work we go one step further and provide conditions under which $H_n$ and $H_n^{-1}$ are smooth in one of the variables $x$ and $y$. We emphasise that our conditions are of a general form and do not involve any kind of dichotomy, nonresonance or spectral gap assumptions for the linear part which are present on most of the related works.

math.DS↗

arXiv2021

On the linearization of infinite-dimensional random dynamical systems

We present a new version of the Grobman-Hartman's linearization theorem for random dynamics. Our result holds for infinite dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on the non-linear perturbations, we don't require for the linear part to be nonuniformly hyperbolic in the sense of Pesin but rather (besides requiring the existence of stable and unstable directions) allow for the existence of a third (central) direction on which we don't prescribe any behaviour for the dynamics. Moreover, under some additional nonuniform growth condition, we prove that the conjugacies given by the linearization procedure are H\"older continuous when restricted to bounded subsets of the space.

math.DS↗

arXiv2020

A general approach to nonautonomous shadowing for nonlinear dynamics

Given a nonautonomous and nonlinear differential equation \begin{equation}\label{DE} x'=A(t)x+f(t,x) \quad t\geq 0, \end{equation} on an arbitrary Banach space $X$, we formulate very general conditions for the associated linear equation $x'=A(t)x$ and for the nonlinear term $f:[0,+\infty)\times X\to X$ under which the above system satisfies an appropriate version of the shadowing property. More precisely, we require that $x'=A(t)x$ admits a very general type of dichotomy, which includes the classical hyperbolic behaviour as a very particular case. In addition, we require that $f$ is Lipschitz in the second variable with a sufficiently small Lipschitz constant. Our general framework enables us to treat various settings in which no shadowing result has been previously obtained. Moreover, we are able to recover and refine several known results. We also show how our main results can be applied to the study of the shadowing property for higher order differential equations. Finally, we conclude the paper by presenting a discrete time versions of our results.

math.DS↗

arXiv2020

Linearization and H\" older Continuity for Nonautonomous Systems

We consider a nonautonomous system \[ \dot x=A(t)x+f(t,x,y),\quad \dot y = g(t,y)\] and give conditions under which there is a transformation of the form $H(t,x,y)=(x+h(t,x,y),y)$ taking its solutions onto the solutions of the partially linearized system \[ \dot x=A(t)x,\quad \dot y = g(t,y).\] Shi and Xiong \cite{SX} proved a special case where $g(t,y)$ was a linear function of $y$ and $\dot x=A(t)x$ had an exponential dichotomy. Our assumptions on $A$ and $f$ are of the general form considered by Reinfelds and Steinberga \cite{RS}, which include many of the generalizations of Palmer's theorem proved by other authors. Inspired by the work of Shi and Xiong, we also prove H\" older continuity of $H$ and its inverse in $x$ and $y$. Again the proofs are given in the context of Reinfelds and Steinberga but we show what the results reduce to when $\dot x=A(t)x$ is assumed to have an exponential dichotomy. The paper is concluded with the discrete version of the results.

math.DS↗