Bernardo Carvalho

Bernardo Carvalho
Federal University of Minas Gerais | UFMG · Departamento de Matemática

PhD

About

16
Publications
2,064
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183
Citations
Introduction
I am a pure mathematician working in the area of dynamical systems, with emphasis in hyperbolic and topological dynamics.
Additional affiliations
December 2018 - December 2020
Friedrich Schiller University Jena
Position
  • PostDoc Position
Description
  • Capes/Humboldt Fellowship
February 2017 - present
Federal University of Minas Gerais
Position
  • Professor (Assistant)
January 2016 - February 2017
Universidade Federal de Viçosa (UFV)
Position
  • Professor (Associate)
Education
March 2012 - December 2015
Federal University of Rio de Janeiro
Field of study
  • Mathematics

Publications

Publications (16)
Article
We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomor-phisms satisfying the shadowing property are expansive in the set of transitive points. This is in contrast with pseudo-Anosov diffeomorph...
Preprint
In this paper we study local stable/unstable sets of sensitive homeomorphisms with the shadowing property defined on compact metric spaces. We prove that local stable/unstable sets always contain a compact and perfect subset of the space. As a corollary we generalize results in [6] and [11] proving that positively countably expansive homeomorphisms...
Preprint
Full-text available
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 Decompositi...
Article
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-...
Preprint
Full-text available
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-...
Preprint
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-...
Article
In this paper we further explore the L-shadowing property defined in [17] for dynamical systems on compact spaces. We prove that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere satisfy this property. Homeomorphisms satisfying the L-shadowing property have a spectral decomposition where the ba...
Preprint
We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomor-phisms satisfying the shadowing property are expansive in the set of transitive points. This is in contrast with pseudo-Anosov diffeomorph...
Article
Full-text available
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 m...
Preprint
In this paper we further explore the L-shadowing property defined in [17] for dynamical systems on compact spaces. We prove that structurally stable diffeomorphisms and some pseudo-Anosov diffeomorphisms of the two-dimensional sphere satisfy this property. Homeomorphisms satisfying the L-shadowing property have a spectral decomposition where the ba...
Article
We characterize product Anosov diffeomorphisms in terms of the two-sided limit shadowing property. It is proved that an Anosov diffeomorphism is a product Anosov diffeomorphism if and only if any lift to the universal covering has the unique two-sided limit shadowing property. Then we introduce two maps in a suitable Banach space such that fixed po...
Article
We discuss the dynamics of n-expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every natural 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)...
Thesis
Full-text available
In this thesis we study in detail the two-sided limit shadowing property in the theory of dynamical systems. We discuss some of its consequences and its role in the hyperbolic theory. This property is among the strongest known pseudo-orbit tracing properties. It implies shadowing, average shadowing, asymptotic average shadowing and even the specifi...
Article
We prove that the two-sided limit shadowing property is among the strongest known notions of pseudo-orbit tracing. It implies shadowing, average shadowing, asymptotic average shadowing and specification properties. We also introduce a weaker notion that is called two-sided limit shadowing with a gap and prove that it implies shadowing and transitiv...
Article
For a C 1 generic diffeomorphism if a bi-Lyapunov stable homoclinic class is homogeneous then it does not have weak eigenvalues. Using this, we show that such homoclinic classes are hyperbolic if it has one of the following properties: shadowing, specification or limit shadowing.
Article
Full-text available
We explore the notion of two-sided limit shadowing property introduced by Pilyugin \cite{P1}. Indeed, we characterize the $C^1$-interior of the set of diffeomorphisms with such a property on closed manifolds as the set of transitive Anosov diffeomorphisms. As a consequence we obtain that all codimention-one Anosov diffeomorphisms have the two-sided...

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Projects

Project (1)
Project
To understand the dynamics of homeomorphisms satisfying generalizations of topological hyperbolicity.