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Welington Cordeiro

Publications and source records attributed to Welington Cordeiro.

12 recordsLinked to original sources
arXiv2026

Mean dimension of general iterated function systems

In this paper, we introduce and investigate the notions of Mean Dimension and Metric Mean Dimension for generalized iterated function systems (IFS). We establish basic properties of these invariants and prove that Mean Dimension is always bounded above by the lower Metric Mean Dimension and the upper Metric Mean Dimension in this setting. We further show that generalized iterated function systems with the Small Boundary Property have zero Mean Dimension. Finally, we introduce a Gluing Orbit Property for generalized iterated function systems and prove that, under suitable transitivity and non-rigidity assumptions, it guarantees positive topological entropy.

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arXiv2025

L-shadowing for the induced hyperspace homeomorphism

We prove that a homeomorphism f of a compact metric space X satisfies the L-shadowing property if and only if its induced hyperspace homeomorphism also satisfies the L-shadowing property. In the proof, assuming only the L-shadowing property, we obtain the existence of points in the asymptotic local-product-structure with iterates approaching in a uniform rate of convergence to zero. This contrasts with the lack of uniformity of contraction on local stable/unstable sets on many homeomorphisms with the L-shadowing property.

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arXiv2024

Gluing-orbit property, local stable/unstable sets, and induced dynamics on hyperspace

We prove that local stable/unstable sets of homeomorphisms of an infinite compact metric space satisfying the gluing-orbit property always contain compact and perfect subsets of the space. As a consequence, we prove that if a positively countably expansive homeomorphism satisfies the gluing-orbit property, then the space is a single periodic orbit. We also prove that there are homeomorphisms with gluing-orbit such that its induced homeomorphism on the hyperspace of compact subsets does not have gluing-orbit, contrasting with the case of the shadowing and specification properties, proving that if the induced map has gluing-orbit, then the base map has gluing-orbit and is topologically mixing.

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arXiv2021

N-expansive Flows

We define the concept of $N$-expansivity for flows and extend some of the results already established for discrete dynamics and for $CW$-expansive flows. We show examples of $N$-expansive flows but not expansive, and examples of $CW$-expansive flows but not $N$-expansive for any natural number $N$.} We also define Komuro $N$-expansivity and prove that on compact surfaces it implies Komuro expansivity.

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arXiv2020

Continuum-wise hyperbolicity

We introduce continuum-wise hyperbolicity, a generalization of hyperbolicity with respect to the continuum theory. We discuss similarities and differences between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing lemma for cw-hyperbolic homeomorphisms is proved in the form of the L-shadowing property and a Spectral Decomposition is obtained in this scenario. In the proof we generalize the construction of Fathi \cite{Fat89} of a hyperbolic metric using only cw-expansivity, obtaining a hyperbolic cw-metric. We also introduce cwN-hyperbolicity, exhibit examples of these systems for arbitrarily large $N\in\mathbb{N}$ and obtain further dynamical properties of these systems such as finiteness of periodic points with the same period. We prove that homeomorphisms of $\mathbb{S}^2$ that are induced by topologically hyperbolic homeomorphisms of $\mathbb{T}^2$ are continuum-wise-hyperbolic and topologically conjugate to linear cw-Anosov diffeomorphisms of $\mathbb{S}^2$, being in particular cw2-hyperbolic.

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arXiv2019

Suspensions of homeomorphisms with the two-sided limit shadowing property

In this paper we discuss the two-sided limit shadowing property for continuous flows defined in compact metric spaces. We analyze some of the results known for the case of homeomorphisms in the case of continuous flows and observe that some differences appear in this scenario. We prove that the suspension flow of a homeomorphism satisfying the two-sided limit shadowing property also satisfies it. This gives a lot of examples of flows satisfying this property, however it enlighten an important difference between the case of flows and homeomorphisms: there are flows satisfying the two-sided limit shadowing property that are not topologically mixing, while homeomorphisms satifying the two-sided limit shadowing property satisfy even the specification property. There are no homeomorphisms on the circle satisfying the two-sided limit shadowing property but we exhibit examples of flows on the circle satisfying it. It can happen that a suspension flow has the two-sided limit shadowing property but the base homeomorphism does not, though it is proved that it must satisfy a strictly weaker property called two-sided limit shadowing with a gap (as in \cite{CK}). We define a similar notion of two-sided limit shadowing with a gap for flows and prove that these notions are actually equivalent in the case of flows. Finally, we prove that singular suspension flows (in the sense of Komuro \cite{K}) do not satisfy the two-sided limit shadowing property.

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arXiv2019

Countably and entropy expansive homeomorphisms with the shadowing property

We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomorphisms satisfying the shadowing property are expansive in the set of transitive points. This is in contrast with pseudo-Anosov diffeomorphisms of the two-dimensional sphere that are transitive, cw-expansive, satisfy the shadowing property but the dynamical ball in each transitive point contains a Cantor subset. We exhibit examples of countably expansive homeomorphisms that are not finite expansive, satisfy the shadowing property and admits an infinite number of chain-recurrent classes. We further explore the relation between countable and entropy expansivity and prove that for surface homeomorphisms $f:S\to S$ satisfying the shadowing property and $Ω(f)=S$, both countably expansive and entropy cw-expansive are equivalent to being topologically conjugate to an Anosov diffeomorphism.

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arXiv2018

On specification and measure expansiveness

We relate the local specification and periodic shadowing properties. We also clarify the relation between local weak specification and local specification if the system is measure expansive. The notion of strong measure expansiveness is introduced, and an example of a non-expansive systems with the strong measure expansive property is given. Moreover, we find a family of examples with the $N$-expansive property, which are not strong measure expansive. We finally show a spectral decomposition theorem for strong measure expansive dynamical systems with shadowing.

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arXiv2017

Positively n-expansive homeomorphisms and the L-shadowing property

We discuss further the dynamics of n-expansive homeomorphisms with the shadowing property, started in [7]. The L-shadowing property is defined and the dynamics of n-expansive homeomorphisms with such property is explored. In particular, we prove that positively n-expansive homeomorphisms with the L-shadowing property can only be defined in finite metric spaces.

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arXiv2016

N-expansive homeomorphisms with the shadowing property

We discuss the dynamics of $n$-expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every $n\in\mathbb{N}$, we exhibit an $n$-expansive homeomorphism, which is not $(n-1)$-expansive, has the shadowing property and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of $n$-expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. This deals some direction to the problem of non-existence of topologically mixing $n$-expansive homeomorphisms that are not expansive.

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arXiv2015

CW-expansivity and entropy for flows

We define the concept of continuum wise expansive for flows, and we prove that continuum wise expansive flows on compact metric spaces with topological dimension greater than one have positive entropy.

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