Nilson C. Bernardes
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View article: On Notions of Expansivity for Operators on Locally Convex Spaces
On Notions of Expansivity for Operators on Locally Convex Spaces Open
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…
View article: On the Dynamics of Weighted Composition Operators
On the Dynamics of Weighted Composition Operators Open
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 gener…
View article: Generalized hyperbolicity, stability and expansivity for operators on locally convex spaces
Generalized hyperbolicity, stability and expansivity for operators on locally convex spaces Open
View article: Li-Yorke Chaotic Weighted Composition Operators
Li-Yorke Chaotic Weighted Composition Operators Open
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 shi…
View article: Time-Dependent Stable Operators
Time-Dependent Stable Operators Open
We prove that every invertible generalized hyperbolic operator on a Banach space is time-dependent stable.
View article: Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex Spaces
Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex Spaces Open
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 …
View article: On shadowing and chain recurrence in linear dynamics
On shadowing and chain recurrence in linear dynamics Open
[EN] 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 th…
View article: On Shadowing and Chain Recurrence in Linear Dynamics
On Shadowing and Chain Recurrence in Linear Dynamics Open
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 Fré…
View article: Chain recurrence and average shadowing in dynamics
Chain recurrence and average shadowing in dynamics Open
View article: Uniformly positive entropy of induced transformations
Uniformly positive entropy of induced transformations Open
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…
View article: A generalized Grobman-Hartman theorem
A generalized Grobman-Hartman theorem Open
We prove that any generalized hyperbolic operator on any Banach space is structurally stable. As a consequence, we obtain a generalization of the classical Grobman-Hartman theorem.
View article: Shadowing and structural stability for operators
Shadowing and structural stability for operators Open
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 Hartman in 1960 for operators on finite-dimensional spa…
View article: Mean Li-Yorke chaos in Banach spaces
Mean Li-Yorke chaos in Banach spaces Open
View article: Shadowing and structural stability in linear dynamical systems
Shadowing and structural stability in linear dynamical systems Open
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 …
View article: Li-Yorke Chaos for Composition Operators on $L^p$-Spaces
Li-Yorke Chaos for Composition Operators on $L^p$-Spaces Open
Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more use…
View article: Mean Li-Yorke chaos in Banach spaces
Mean Li-Yorke chaos in Banach spaces Open
We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric s…
View article: Expansivity and shadowing in linear dynamics
Expansivity and shadowing in linear dynamics Open
View article: Distributional chaos for operators on Banach spaces
Distributional chaos for operators on Banach spaces Open
View article: Set-Valued Chaos in Linear Dynamics
Set-Valued Chaos in Linear Dynamics Open
View article: On the dynamics of induced maps on the space of probability measures
On the dynamics of induced maps on the space of probability measures Open
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, equic…