Searcharxiv⌕ Search

arXiv subjects

Bernardo Carvalho

Publications and source records attributed to Bernardo Carvalho.

At least 19 recordsLinked to original sources
arXiv2026

Box dimension of stable sub-slices of fractal graphs over Anosov automorphisms

We consider fractal graphs invariant by a skew product $F:\mathbb{T}^k\times \mathbb{R}\rightarrow \mathbb{T}^k\times \mathbb{R}$ of the form $F(x,y)=(Ax, λy+p(x))$ where $0<λ<1$, $p\colon\mathbb{T}^k\to\mathbb{R}$ is a $C^{k+1}$ function, and $A$ is an Anosov diffeomorphism of $\mathbb{T}^k$ admitting $k$ distinct eigenvalues with respective eigenvectors forming a basis of $\mathbb{R}^k$. We note that the stable sub-slices can give information of the fractal structure of the graph that is not captured by the box dimension of the graph. Using the results of Kaplan,Mallet-Paret, and York [6], we exhibit conditions on the skew product that ensure the box dimension of the graph is smaller than the sum of the box dimensions of its stable/unstable sub-slices. We prove that these conditions hold for generic functions $p\in C^{k+1}$.

math.DS↗
arXiv2026

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities

We establish a complete structural classification for continuum-wise hyperbolic surface homeomorphisms. Specifically, we prove that a surface homeomorphism is cw$_F$-hyperbolic if, and only if, it is a pseudo-Anosov homeomorphism whose singularities consist exclusively of spines (1-prongs). Furthermore, we classify these systems up to topological conjugacy, showing that every such homeomorphism is conjugate to either an Anosov automorphism on the torus $\mathbb{T}^2$ or to its standard hyperelliptic quotient on the sphere $\mathbb{S}^2$. As a rigid consequence of this classification, we show that such dynamics are strictly obstructed on surfaces of genus greater than one, the Klein bottle, and the projective plane.

math.DS↗
arXiv2026

Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy

We prove that cw-hyperbolic homeomorphisms with jointly continuous stable/unstable holonomies satisfy the periodic shadowing property and, if they are topologically mixing, the periodic specification property. We discuss difficulties to adapt Bowen's techniques to obtain a measure of maximal entropy for cw-hyperbolic homeomorphisms, exhibit the unique measure of maximal entropy for Walter's pseudo-Anosov diffeomorphism of $\mathbb{S}^2$, and prove it can be obtained, as in the expansive case, as the weak* limit of an average of Dirac measures on periodic orbits. As an application, we exhibit the unique measure of maximal entropy for the homeomorphism on the Sierpiński Carpet defined in [12], which does not satisfy the specification property.

math.DS↗
arXiv2025

Transitivity of real Anosov diffeomorphisms

We prove the transitivity of real Anosov diffeomorphisms, which are Anosov diffeomorphisms where stable and unstable spaces decompose into a continuous sum of invariant one-dimensional sub-spaces with uniform contraction/expansion over the ambient manifold. We prove that if a stable/unstable curve has a well-defined length in a conformal hyperbolic distance, then it has a globally defined holonomy. We exhibit a conformal hyperbolic distance with well-defined length of stable/unstable curves for each real Anosov diffeomorphism.

math.DS↗
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.

math.DS↗
arXiv2025

Shadowing in the hyperspace of continua

We discuss whether classical examples of dynamical systems satisfying the shadowing property also satisfy the shadowing property for the induced map on the hyperspace of continua, obtaining both positive and negative results. We prove that transitive Anosov diffeomorphisms, or more generally continuum-wise hyperbolic homeomorphisms, do not satisfy the shadowing property for the induced map on the hyperspace of continua. We prove that dendrite monotone maps satisfy the shadowing property if, and only if, their induced map on the hyperspace of continua also satisfies it. We give some algorithms that show that there are abundant dynamical systems satisfying the shadowing property with dendrites (compact metric trees) as the underlying space. As a consequence, we show that the universal dendrite of order n admits a homeomorphism with the shadowing property.

math.DS↗
arXiv2024

Stable/unstable continua of cw-expansive flows

We introduce distinct definitions of local stable/unstable sets for flows without fixed points, namely, kinematic, geometric, and sectionally geometric, and discuss relations between them. We prove the existence of continua with a uniform diameter within each sectionally geometric local stable/unstable set for cw-expansive flows defined on Peano continua.

math.DS↗
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.

math.DS↗
arXiv2024

Extending 3D body pose estimation for robotic-assistive therapies of autistic children

Robotic-assistive therapy has demonstrated very encouraging results for children with Autism. Accurate estimation of the child's pose is essential both for human-robot interaction and for therapy assessment purposes. Non-intrusive methods are the sole viable option since these children are sensitive to touch. While depth cameras have been used extensively, existing methods face two major limitations: (i) they are usually trained with adult-only data and do not correctly estimate a child's pose, and (ii) they fail in scenarios with a high number of occlusions. Therefore, our goal was to develop a 3D pose estimator for children, by adapting an existing state-of-the-art 3D body modelling method and incorporating a linear regression model to fine-tune one of its inputs, thereby correcting the pose of children's 3D meshes. In controlled settings, our method has an error below $0.3m$, which is considered acceptable for this kind of application and lower than current state-of-the-art methods. In real-world settings, the proposed model performs similarly to a Kinect depth camera and manages to successfully estimate the 3D body poses in a much higher number of frames.

cs.RO↗
arXiv2023

Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms

We discuss different regularities on stable/unstable holonomies of cw-hyperbolic homeomorphisms and prove that if a cw-hyperbolic homeomorphism has continuous joint stable/unstable holonomies, then it has a dense set of periodic points in its non-wandering set. For that, we prove that the hyperbolic cw-metric (introduced in [9]) can be adapted to be self-similar (as in [6]) and, in this case, continuous joint stable/unstable holonomies are pseudo-isometric. We also prove transitivity of cw-hyperbolic homeomorphisms assuming that the stable/unstable holonomies are isometric. In the case the ambient space is a surface, we prove that a cw$_F$-hyperbolic homeomorphism has continuous joint stable/unstable holonomies when every bi-asymptotic sector is regular.

math.DS↗
arXiv2023

Continuum-wise hyperbolic homeomorphisms on surfaces

This paper discusses the dynamics of continuum-wise hyperbolic surface homeomorphisms. We prove that $cw_F$-hyperbolic surface homeomorphisms containing only a finite set of spines are $cw_2$-hyperbolic. In the case of $cw_3$-hyperbolic homeomorphisms we prove the finiteness of spines and, hence, that $cw_3$-hyperbolicity implies $cw_2$-hyperbolicity. In the proof, we adapt techniques of Hiraide [11] and Lewowicz [15] in the case of expansive surface homeomorphisms to prove that local stable/unstable continua of $cw_F$-hyperbolic homeomorphisms are continuous arcs. We also adapt techniques of Artigue, Pacífico and Vieitez [6] in the case of N-expansive surface homeomorphisms to prove that the existence of spines is strongly related to the existence of bi-asymptotic sectors and conclude that spines are necessarily isolated from other spines.

math.DS↗
arXiv2023

First-time sensitive homeomorphisms

We introduce first-time sensitivity for a homeomorphism of a compact metric space, that is a condition on the first increasing times of open balls of the space. Continuum-wise expansive homeomorphisms, the shift map on the Hilbert cube, and also some partially hyperbolic diffeomorphisms satisfy this condition. We prove the existence of local unstable continua satisfying similar properties with the local unstable continua of cw-expansive homeomorphisms, but assuming first-time sensitivity. As a consequence we prove that first-time sensitivity (with some additional technical assumptions) implies positive topological entropy.

math.DS↗
arXiv2021

Sensitivity, local stable/unstable sets and shadowing

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 defined on compact metric spaces satisfying either transitivity and the shadowing property, or the L-shadowing property, can only be defined in countably spaces.

math.DS↗
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.

math.DS↗
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.

math.DS↗
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.

math.DS↗
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.

math.DS↗