arXiv subjects

Nilson C. Bernardes Jr.

Publications and source records attributed to Nilson C. Bernardes Jr..

12 recordsLinked to original sources

arXiv2026

The specification property for composition operators

It is known that the operator specification property does not coincide, in general, with other notions of chaos in linear dynamics. In this paper, we show that, for a natural and widely studied class of linear operators, namely the dissipative composition operators of bounded distortion (including the class of weighted shifts), specification-type properties do not introduce new dynamical behaviors: they coincide with classical notions of chaos. We also extend these equivalences with conditions on periodic points and examine in detail the special case of the translation operator $T_α$ acting on $L^p(\mathbb R,\mathcal B,μ)$, where the measure $μ$ is induced by a density function. Moreover, we show that, in general, there is no relationship between the operator specification property and the shadowing property in the linear context. Finally, we analyse the important role played by the bounded distortion hypothesis.

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arXiv2026

On the Dynamics of Weighted Composition Operators II

We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces $C_0(X)$ of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space $X$, endowed with the sup norm, and the locally convex spaces $C(X)_c$ of continuous scalar-valued functions on a completely regular space $X$, endowed with the compact-open topology. We also obtain complete characterizations of various notions of expansivity for weighted composition operators on $L^p(μ)$ spaces, thereby complementing and extending previously known results in the unweighted case. Finally, we establish a conjugation between weighted and unweighted composition operators in the case of dissipative systems on $L^p(μ)$ spaces and apply it to the study of several dynamical properties.

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arXiv2025

On Notions of Expansivity for Operators on Locally Convex Spaces

We extend the concept of average expansivity for operators on Banach spaces to operators on arbitrary locally convex spaces. We obtain complete characterizations of the average expansive weighted shifts on Fréchet sequence spaces. Moreover, we give a partial answer to a problem proposed by Bernardes et al. in J. Funct. Anal. 288 (2025), Paper No. 110696, by obtaining complete characterizations of the uniformly expansive weighted shifts on Köthe sequence spaces. Some general properties of the various concepts of expansivity in linear dynamics and several concrete examples are also presented.

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arXiv2025

On the Dynamics of Weighted Composition Operators

We study the properties of power-boundedness, Li-Yorke chaos, distributional chaos, absolutely Cesàro boundedness and mean Li-Yorke chaos for weighted composition operators on $L^p(μ)$ spaces and on $C_0(Ω)$ spaces. We illustrate the general results by presenting several applications to weighted shifts on the classical sequence spaces $c_0(\mathbb{N})$, $c_0(\mathbb{Z})$, $\ell^p(\mathbb{N})$ and $\ell^p(\mathbb{Z})$ ($1 \leq p < \infty$) and to weighted translation operators on the classical function spaces $C_0[1,\infty)$, $C_0(\mathbb{R})$, $L^p[1,\infty)$ and $L^p(\mathbb{R})$ ($1 \leq p < \infty$).

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arXiv2024

Li-Yorke Chaotic Weighted Composition Operators

We establish complete characterizations of the notion of Li-Yorke chaos for weighted composition operators on $C_0(X)$ spaces and on $L^p(μ)$ spaces. As a consequence, we obtain simple characterizations of the Li-Yorke chaotic weighted shifts on $c_0$ and on $\ell^p$ ($1 \leq p < \infty$) that complement previously known results. We also investigate the notion of Li-Yorke chaos for weighted shifts on Fréchet sequence spaces. As applications, we obtain characterizations of the Li-Yorke chaotic weighted shifts on Köthe sequence spaces depending only on the entries of the Köthe matrix and the weights of the shift.

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arXiv2024

Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex Spaces

We introduce and study the notions of (generalized) hyperbolicity, topological stability and (uniform) topological expansivity for operators on locally convex spaces. We prove that every generalized hyperbolic operator on a locally convex space has the finite shadowing property. Contrary to what happens in the Banach space setting, hyperbolic operators on Fréchet spaces may fail to have the shadowing property, but we find additional conditions that ensure the validity of the shadowing property. Assuming that the space is sequentially complete, we prove that generalized hyperbolicity implies the strict periodic shadowing property, but we also show that the hypothesis of sequential completeness is essential. We show that operators with the periodic shadowing property on topological vector spaces have other interesting dynamical behaviors, including the fact that the restriction of such an operator to its chain recurrent set is topologically mixing and Devaney chaotic. We prove that topologically stable operators on locally convex spaces have the finite shadowing property and the strict periodic shadowing property. As a consequence, topologically stable operators on Banach spaces have the shadowing property. Moreover, we prove that generalized hyperbolicity implies topological stability for operators on Banach spaces. We prove that uniformly topologically expansive operators on locally convex spaces are neither Li-Yorke chaotic nor topologically transitive. Finally, we characterize the notion of topological expansivity for weighted shifts on Fréchet sequence spaces. Several examples are provided.

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arXiv2024

On Shadowing and Chain Recurrence in Linear Dynamics

In the present work we study the concepts of shadowing and chain recurrence in the setting of linear dynamics. We prove that shadowing and finite shadowing always coincide for operators on Banach spaces, but we exhibit operators on the Fréchet space $H(\mathbb{C})$ of entire functions that have the finite shadowing property but do not have the shadowing property. We establish a characterization of mixing for continuous maps with the finite shadowing property in the setting of uniform spaces, which implies that chain recurrence and mixing coincide for operators with the finite shadowing property on any topological vector space. We establish a characterization of dense distributional chaos for operators with the finite shadowing property on Fréchet spaces. As a consequence, we prove that if a Devaney chaotic (resp. a chain recurrent) operator on a Fréchet space (resp. on a Banach space) has the finite shadowing property, then it is densely distributionally chaotic. We obtain complete characterizations of chain recurrence for weighted shifts on Fréchet sequence spaces. We prove that generalized hyperbolicity implies periodic shadowing for operators on Banach spaces. Moreover, the concepts of shadowing and periodic shadowing coincide for unilateral weighted backward shifts, but these notions do not coincide in general, even for bilateral weighted shifts.

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arXiv2020

Uniformly Positive Entropy of Induced Transformations

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space $X$ and a continuous surjective map $T : X \to X$. By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $(\cM(X),\wt{T})$ on the space of Borel probability measures endowed with the weak$^*$ topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

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arXiv2019

Shadowing and structural stability in linear dynamical systems

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by P. Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by J. Palis and C. Pugh around 1968. We will exhibit examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_0(\mathbb{Z})$ and $\ell_p(\mathbb{Z})$ ($1 \leq p < \infty$) that satisfy the shadowing property.

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arXiv2016

Expansivity and Shadowing in Linear Dynamics

In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators.

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arXiv2014

On the Dynamics of Induced Maps on the Space of Probability Measures

For the generic continuous map and for the generic homeomorphism of the Cantor space, we study the dynamics of the induced map on the space of probability measures, with emphasis on the notions of Li-Yorke chaos, topological entropy, equicontinuity, chain continuity, chain mixing, shadowing and recurrence. We also establish some results concerning induced maps that hold on arbitrary compact metric spaces.

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arXiv2012

Graph Theoretic Structure of Maps of the Cantor Space

In this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of the space of homeomorphisms and the space of self-maps of the Cantor space. Using our methods, we give characterizations which determine when two homeomorphisms of the Cantor space are conjugate to each other. We also give a new characterization of the comeager conjugacy class of the space of homeomorphisms of the Cantor space. The existence of this class was established by Kechris and Rosendal and a specific element of this class was described concretely by Akin, Glasner and Weiss. Our characterization readily implies many old and new dynamical properties of elements of this class. For example, we show that no element of this class has a Li-Yorke pair, implying the well known Glasner-Weiss result that there is a comeager subset of homeomorphism space of the Cantor space each element of which has topological entropy zero. Our analogous investigation in the space of continuous self-maps of the Cantor space yields a surprising result: there is a comeager subset of the space of self-maps of the Cantor space such that any two elements of this set are conjugate to each other by an homeomorphism. Our description of this class also yields many old and new results concerning dynamics of a comeager subset of the space of continuous self-maps of the Cantor space.

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