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Krerley Oliveira

Publications and source records attributed to Krerley Oliveira.

17 recordsLinked to original sources
arXiv2025

Uniqueness of Equilibrium Measure for a Family of Partially Hyperbolic Horseshoes

In this paper, we show the uniqueness of equilibrium state for a family of partially hyperbolic horseshoes, introduced in [12] for some classes of continuous potentials. For the first class, the method used here is making use of the Sarig's theory for countable shifts. For this purpose, we study the dynamics of an induced map associated to the horseshoe map, we build a symbolic system with infinitely many symbols that is topologically conjugated to this induced map and we show that the induced potential is locally Hölder and recurrent. For the second class, by following ideas of [29] and [31], we prove that uniqueness for the horseshoe is equivalent to uniqueness for the restriction to a non-uniformly expanding map, which is a hyperbolic potential and then has uniqueness. Both classes include potentials with high variation, differently from previous results for potentials with low variation. We also prove uniqueness when the potential presents its supremum at a special fixed point and less than the pressure.

math.DS↗
arXiv2024

Equilibrium states for hyperbolic potentials via inducing schemes

In a context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. This generalizes a result due to Ramos and Viana, where analytical methods are used for maps with no critical sets. The strategy here consists in using a finite number of inducing schemes with a Markov structure in infinitely many symbols to code the dynamics, to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. We apply our results to the important class of multidimensional Viana maps.

math.DS↗
arXiv2022

Quantifying protocols for safe school activities

By the peak of COVID-19 restrictions on April 8, 2020, up to 1.5 billion students across 188 countries were by the suspension of physical attendance in schools. Schools were among the first services to reopen as vaccination campaigns advanced. With the emergence of new variants and infection waves, the question now is to find safe protocols for the continuation of school activities. We need to understand how reliable these protocols are under different levels of vaccination coverage, as many countries have a meager fraction of their population vaccinated, including Uganda where the coverage is about 8\%. We investigate the impact of face-to-face classes under different protocols and quantify the surplus number of infected individuals in a city. Using the infection transmission when schools were closed as a baseline, we assess the impact of physical school attendance in classrooms with poor air circulation. We find that (i) resuming school activities with people only wearing low-quality masks leads to a near fivefold city-wide increase in the number of cases even if all staff is vaccinated, (ii) resuming activities with students wearing good-quality masks and staff wearing N95s leads to about a threefold increase, (iii) combining high-quality masks and active monitoring, activities may be carried out safely even with low vaccination coverage. These results highlight the effectiveness of good mask-wearing. Compared to ICU costs, high-quality masks are inexpensive and can help curb the spreading. Classes can be carried out safely, provided the correct set of measures are implemented.

physics.soc-ph↗
arXiv2020

LRCN-RetailNet: A recurrent neural network architecture for accurate people counting

Measuring and analyzing the flow of customers in retail stores is essential for a retailer to better comprehend customers' behavior and support decision-making. Nevertheless, not much attention has been given to the development of novel technologies for automatic people counting. We introduce LRCN-RetailNet: a recurrent neural network architecture capable of learning a non-linear regression model and accurately predicting the people count from videos captured by low-cost surveillance cameras. The input video format follows the recently proposed RGBP image format, which is comprised of color and people (foreground) information. Our architecture is capable of considering two relevant aspects: spatial features extracted through convolutional layers from the RGBP images; and the temporal coherence of the problem, which is exploited by recurrent layers. We show that, through a supervised learning approach, the trained models are capable of predicting the people count with high accuracy. Additionally, we present and demonstrate that a straightforward modification of the methodology is effective to exclude salespeople from the people count. Comprehensive experiments were conducted to validate, evaluate and compare the proposed architecture. Results corroborated that LRCN-RetailNet remarkably outperforms both the previous RetailNet architecture, which was limited to evaluating a single image per iteration; and a state-of-the-art neural network for object detection. Finally, computational performance experiments confirmed that the entire methodology is effective to estimate people count in real-time.

cs.CV↗
arXiv2020

Equilibrium states for natural extensions of non-uniformly expanding local homeomorphisms

We examine uniqueness of equilibrium states for the natural extension of a topologically exact, non-uniformly expanding, local homeomorphism with a Hölder continuous potential function. We do this by applying general techniques developed by Climenhaga and Thompson, and show there is a natural condition on decompositions that guarantees that a unique equilibrium state exists. We then show how to apply these results to partially hyperbolic attractors.

math.DS↗
arXiv2019

Equilibrium states for certain partially hyperbolic attractors

We prove that a class of partially hyperbolic attractors introduced by Castro and Nascimento have unique equilibrium states for natural classes of potentials. We also show if the attractors are $C^2$ and have invariant stable and centerunstable foliations, then there is a unique equilibrium state for the geometric potential and its 1-parameter family. We do this by applying general techniques developed by Climenhaga and Thompson.

math.DS↗
arXiv2018

Sequential Gibbs Measures and Factor Maps

We define the notion of sequential Gibbs measures, inspired by on the classical notion of Gibbs measures and recent examples from the study of non-uniform hyperbolic dynamics. Extending previous results of Kempton-Pollicott and Ugalde-Chazottes, we show that the images of one block factor maps of a sequential Gibbs measure are also a sequential Gibbs measure, with the same sequence of Gibbs times. We obtain some estimates on the regularity of the potential of the image measure at almost every point.

math.DS↗
arXiv2011

Non-uniform Hyperbolicity and Non-uniform Specification

In this paper we deal with an invariant ergodic hyperbolic measure $μ$ for a diffeomorphism $f,$ assuming that $f$ it is either $C^{1+α}$ or $f$ is $C^1$ and the Oseledec splitting of $μ$ is dominated. We show that this system $(f,μ)$ satisfies a weaker and non-uniform version of specification, related with notions studied in several recent papers, including \cite{STV,Y, PS, T,Var, Oli}. Our main results have several consequences: as corollaries, we are able to improve the results about quantitative Poincaré recurrence, removing the assumption of the non-uniform specification property in the main Theorem of \cite{STV} that establishes an inequality between Lyapunov exponents and local recurrence properties. Another consequence is the fact that any of such measure is the weak limit of averages of Dirac measures at periodic points, as in \cite{Sigmund}. Following \cite{Y} and \cite{PS}, one can show that the topological pressure can be calculated by considering the convenient weighted sums on periodic points, whenever the dynamics is positive expansive and every measure with pressure close to the topological pressure is hyperbolic.

math.DS↗
arXiv2010

Every expanding measure has the nonuniform specification property

Exploring abundance and non lacunarity of hyperbolic times for endomorphisms preserving an ergodic probability with positive Lyapunov exponents, we obtain that there are periodic points of period growing sublinearly with respect to the lenght of almost every dynamical ball. In particular, we conclude that any ergodic measure with positive Lyapunov exponents satisfy the nonuniform specification property. As consequences, we (re)-obtain estimates on the recurrence to a ball in terms of the Lyapunov exponents and we prove that any expanding measure is limit of Dirac measures on periodic points.

math.DS↗
arXiv2008

Equilibrium States for Partially Hyperbolic Horseshoes

In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in \cite{DHRS07}. Despite the fact that the non-wandering set is a horseshoe, it contains intervals. We prove that every recurrent point has non-zero Lyapunov exponents and all ergodic invariant measures are hyperbolic. As a consequence, we obtain the existence of equilibrium measures for any continuous potential. We also obtain an example of a family of $C^\infty$ potentials with phase transition.

math.DS↗
arXiv2006

Equilibrium States for Non-uniformly expanding maps

We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties.

math.DS↗
arXiv2006

Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms

We prove existence of maximal entropy measures for an open set of non-uniformly expanding local diffeomorphisms on a compact Riemannian manifold. In this context the topological entropy coincides with the logarithm of the degree, and these maximizing measures are eigenmeasures of the transfer operator. When the map is topologically mixing, the maximizing measure is unique and positive on every open set.

math.DS↗
arXiv2006

Shadowing by non uniformly hyperbolic periodic points and uniform hyperbolicity

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map $g$ which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context.

math.DS↗
arXiv2005

Geometrical versus Topological Properties of Manifolds

Given a compact $n$-dimensional immersed Riemannian manifold $M^n$ in some Euclidean space we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then $M^n$ is homeomorphic to the sphere $S^n$. Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with small set of points of zero Gauss-Kronecker curvature are topologically the sphere minus a finite number of points. A characterization of the $2n$-catenoid is obtained.

math.DG↗
arXiv2004

On the continuity of the SRB entropy for endomorphisms

We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of non-uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the SRB measure. In particular, we show that the SRB entropy of Viana maps varies continuously with the map.

math.DS↗
arXiv2003

Equilibrium States for Random Non-uniformly Expanding Maps

We show that, for a robust ($C^2$-open) class of random non-uniformly expanding maps, there exists equilibrium states for a large class of potentials.In particular, these sytems have measures of maximal entropy. These results also give a partial answer to a question posed by Liu-Zhao. The proof of the main result uses an extension of techniques in recent works by Alves-Araújo, Alves-Bonatti-Viana and Oliveira.

math.DS↗