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This study examines the synchronization of three identical oscillators arranged in an array and coupled by small impacts, wherein each oscillator interacts solely with its nearest neighbor. The synchronized state, which is asymptotically stable, is characterized by phase opposition among alternating oscillators. We analyze the system using a non-linear discrete dynamical system based on a difference equation derived from the iteration of a plane diffeomorphism. We illustrate these results with the application to a system of three aligned Andronov clocks, showcasing their applicability to a broad range of oscillator systems., Comment: 19 Pages

The concept of $\mathbb C$-supercyclic operators was introduced by Hilden and Wallen in \cite{hilden} and, since then, a multitude of variants have been studied. In \cite{herzog1992linear}, Herzog proved that every real or complex, separable, infinite dimensional Banach space supports a $\mathbb C$-supercyclic operator. In the present paper we investigate different variants of supercyclicity, precisely $\mathbb R^+$-, $\mathbb R$- and $\mathbb C$-supercyclicity in the context of composition operators. We characterize $\mathbb R$-supercyclic composition operators on $L^p$, $1 \leq p < \infty$. Then, we turn our attention to dissipative composition operators, and we show that $\mathbb R$- and $\mathbb C$-supercyclicity are equivalent notions in this setting and they have a ``shift-like'' characterization.

We prove the spaceability of the set of hypercyclic vectors for {\em shifts-like operators}. Shift-like operators appear naturally as composition operators on $L^p(X)$, when the underlying space $X$ is dissipative. In the process of proving the main theorem, we provide, among other results of independent interest, a characterization of weakly mixing dissipative composition operators of bounded distortion.

It is well-known that, in Linear Dynamics, the most studied class of linear operators is certainly that of weighted shifts, on the separable Banach spaces $c_0$ and $\ell^p$, $1 \leq p< \infty$. Over the last decades, the intensive study of such operators has produced an incredible number of versatile, deep and beautiful results, applicable in various areas of Mathematics and the relationships between various important notions, especially concerning chaos and hyperbolic properties, and the spectrum of weighted shifts are investigated. In this paper, we investigate the point spectrum of weighted shifts and, under some regularity hypotheses on the weight sequence, we deduce the spectrum.

In this paper, we study Lebesgue differentiation processes along rectangles $R_k$ shrinking to the origin in the Euclidean plane, and the question of their almost everywhere convergence in $L^p$ spaces. In particular, classes of examples of such processes failing to converge a.e. in $L^\infty$ are provided, for which $R_k$ is known to be oriented along the slope $k^{-s}$ for $s>0$, yielding an interesting counterpart to the fact that the directional maximal operator associated to the set $\{k^{-s}:k\in\mathbb{N}^*\}$ fails to be bounded in $L^p$ for any $1\leq p<\infty$.

We introduce the notion of \textit{Perron capacity} of a set of slopes $\Omega \subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $\Omega$ is finite then the directional maximal operator associated $M_\Omega$ is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set $$\Omega_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.

In this article we develop a general technique which takes a known characterization of a property for weighted backward shifts and lifts it up to a characterization of that property for a large class of operators on $L^p(X)$. We call these operators ``shift-like''. The properties of interest include chaotic properties such as Li-Yorke chaos, hypercyclicity, frequent hypercyclicity as well as properties related to hyperbolic dynamics such as shadowing, expansivity and generalized hyperbolicity. Shift-like operators appear naturally as composition operators on $L^p(X)$ when the underlying space is a dissipative measure system. In the process of proving the main theorem, we provide some results concerning when a property is shared by a linear dynamical system and its factors., Comment: arXiv admin note: text overlap with arXiv:2009.11526

It is rather well-known that hyperbolic operators have the shadowing property. In the setting of finite dimensional Banach spaces, having the shadowing property is equivalent to being hyperbolic. In 2018, Bernardes et al. constructed an operator with the shadowing property which is not hyperbolic, settling an open question. In the process, they introduced a class of operators which has come to be known as generalized hyperbolic operators. This class of operators seems to be an important bridge between hyperbolicity and the shadowing property. In this article, we show that for a large natural class of operators on $L^p(X)$ the notion of generalized hyperbolicity and the shadowing property coincide. We do this by giving sufficient and necessary conditions for a certain class of operators to have the shadowing property. We also introduce computational tools which allow construction of operators with and without the shadowing property. Utilizing these tools, we show how some natural probability distributions, such as the Laplace distribution and the Cauchy distribution, lead to operators with and without the shadowing property on $L^p(X)$.

We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties., Comment: Accepted for publication in Aequationes Mathematicae (AEQM)

Weighted shifts are an important concrete class of operators in linear dynamics. In particular, they are an essential tool in distinguishing variety dynamical properties. Recently, a systematic study of dynamical properties of composition operators on $L^p$ spaces has been initiated. This class of operators includes weighted shifts and also allows flexibility in construction of other concrete examples. In this article, we study one such concrete class of operators, namely composition operators induced by measures on odometers. In particular, we study measures on odometers which induce mixing and transitive linear operators on $L^p$ spaces., Comment: 15 pages, keywords: linear dynamics, composition operators, topological mixing, topological transitivity, odometers

Let $X = [0,1]^{n}$, $n \geq1$. We show that the typical (in the sense of Baire category) compact subset of $X$ is not only a zero dimensional Cantor space but it satisfies the property of being strongly microscopic, which is stronger than being of dimension zero.

This paper investigates the synchronization of three identical oscillators, or clocks, suspended from a common rigid support. We consider scenarios where each clock interacts with the other two, achieving synchronization through small impacts exchanged between oscillator pairs. The fundamental outcome of our study reveals that the ultimate synchronized state maintains a phase difference of $\frac{2\pi}{3}$ between successive clocks, either clockwise or counter-clockwise. Furthermore, these locked states exhibit an attracting set, which closure encompasses the entire initial conditions space. Our analytical approach involves constructing a nonlinear discrete dynamical system in dimension two. These findings hold significance for sets of three weakly coupled periodic oscillators engaged in mutual symmetric impact periodic interaction, irrespective of the specific oscillator models employed. Lastly, we explore the amplitude of oscillations at the final locked state in the context of two and three interacting Andronov pendulum clocks. Our analysis reveals a precise small increase in the amplitude of the locked-state oscillations, as quantified in this paper.

In the present note, we examine the behavior of some homo\-thecy-invariant differentiation basis of rectangles in the plane satisfying the following requirement: for a given rectangle to belong to the basis, the ratio of the largest of its side-lengths by the smallest one (which one calls its \emph{shape}) has to be a fixed real number depending on the angle between its longest side and the horizontal line (yielding a \emph{shape-function}). Depending on the allowed angles and the corresponding shape-function, a basis may differentiate various Orlicz spaces. We here give some examples of shape-functions so that the corresponding basis differentiates $L\log L(\R^2)$, and show that in some `model' situations, a fast-growing shape function (whose speed of growth depends on $\alpha>0$) does not allow the differentiation of $L\log^\alpha L(\R^2)$.

In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space.

In this work we investigate families of translation invariant differentiation bases $B$ of rectangles in $R^n$, for which $L\log^{n-1}L(R^n)$ is the largest Orlicz space that $B$ differentiates. In particular, we improve on techniques developed by A.~Stokolos 1988 and 2008., Comment: 17 pages, 2 figures