Research

Abstract

My research is in the area of Dynamical Systems, focusing on piecewise smooth dynamics, entropy, and symbolic dynamics. On this page I list my published articles and preprints, my research projects, the research groups I participate in, and a list of collaborators and colleagues.

Publications

Published articles

N. Martins, P. G. Mattos, R. Varão. Folding and Metric Entropies for Extended Shifts. Journal of Dynamics and Differential Equations, 2026.

In this paper we calculate the metric and folding entropies for a family of non-invertible symbolic dynamical systems $(\Sigma_{m_-,m_+}, \sigma_\phi)$ which generalizes the standard bilateral Bernoulli shifts. The space $\Sigma_{m_-,m_+}$ consists of symbolic sequences over two distinct finite alphabets, with dynamics governed by a shift map $\sigma_\phi$ incorporating a non-invertible function $\phi$ that maps one of the alphabets to the other one. These systems are, for instance, particularly useful for encoding the many-to-one baker's transformation endomorphisms, and they can also be seen as a skew product with a unilateral Bernoulli shift on the base.
R. D. Euzébio, P. G. Mattos, R. Varão. Characterization of Non-deterministic Chaos in Two-dimensional Non-smooth Vector Fields. Differential Equations and Dynamical Systems, 2025.

Our context is Filippov systems defined on two-dimensional manifolds having a finite number of tangency points. We prove that topological transitivity is a necessary and sufficient condition for the occurrence of non-deterministic chaos when the Filippov system has non-empty sliding or escaping regions. A fundamental result for continuous flows is the equivalence of topological transitivity and existence of a dense orbit. We prove in our setting that topological transitivity for Filippov systems is indeed equivalent to the existence of a dense Filippov orbit, although, in contrast to the continuous case, we are not able to guarantee that the dense orbit implies the existence of a residual set of dense orbits. Finally, we prove that, in this context, topological transitivity implies strictly positive topological entropy for the Filippov system. This calculation is made using techniques similar to those from symbolic dynamics.

Preprints

O. M. L. Gomide, P. G. Mattos, R. Varão. The Orbit Space Approach for Piecewise Smooth Vector Fields. 2026.

In this work we develop a well-defined theory of orbit spaces for piecewise smooth vector fields (PSVFs). This approach is inspired by the techniques already used in the study of endomorphisms, namely inverse limit analysis, and has been used before for PSVFs. We then apply the construction of our theory to understanding transitivity in PSVFs. Our results prove that the known examples of transitive PSVFs in the literature, the bean model and the sphere model, are indeed transitive in the orbit space.

Publications in other areas

P. G. Mattos, F. A. M. Mattos. Hiperconcentração da renda e da riqueza: como medir e por que é importante ampliar a discussão além do aspecto econômico. Revista Simetria, 2023.

O artigo propõe uma discussão interdisciplinar e multidimensional da desigualdade, destacando a necessidade de unir análise qualitativa da crescente desigualdade econômica que vem ocorrendo nos países desenvolvidos com uma análise dos desdobramentos dessa desigualdade econômica sobre outras formas de manifestação de desigualdades, como a desigualdade social, ambiental, urbana entre outras. Na última seção é incorporada uma discussão sobre Urbanismo e as formas de disputa em torno dos espaços das cidades e do orçamento público.

Projects

Non-Deterministic Chaos in Piecewise Smooth Dynamical Systems

A piecewise smooth vector field is a vector field that varies smoothly outside a discontinuity hypersurface $\Sigma$ , but changes discontinuously when crossing it. According to Filippov’s convention, solutions may “slide” along $\Sigma$ , which generates non-uniqueness of solutions — the system is said to be non-deterministic. This non-uniqueness prevents the use of a continuous flow to describe the dynamics, and therefore classical tools from ergodic theory and symbolic dynamics cannot be applied directly.

The central approach of this research line is the orbit space: to the non-deterministic system $$(M, F)$$ one associates a space $\tilde{M}$ whose points are the maximal orbits of the system, endowed with a metric and a continuous flow $\tilde{\Phi}$ . The system $(\tilde{M}, \tilde{\Phi})$ is deterministic, which allows the use of classical tools. The main results obtained show that topological transitivity and non-deterministic chaos in the base space $$M$$ are related to the analogous properties in the orbit space $\tilde{M}$ , and that transitive two-dimensional piecewise smooth systems have strictly positive topological entropy.

Open problems include the complete characterization of the equivalence between transitivity in $$M$$ and in $\tilde{M}$ , the generalization of these results to higher dimensions (especially using the classification of fold-fold T-singularities), and the study of topological mixing and invariant measures in the orbit space.

Entropy in Extended Symbolic Shifts

Bernoulli shifts are prototypical models for chaotic behavior in dynamical systems. This research line studies a generalization: the $$(m_-, m_+)$$ -extended shifts, which encode non-invertible systems using two distinct alphabets. Given a backward alphabet with $$m_-$$ symbols and a forward alphabet with $$m_+$$ symbols, and a surjective transition function $\phi$ , one defines a dynamics $\sigma_\phi$ that generalizes the bilateral Bernoulli shift (recovered when $$m_- = m_+$$ ).

The main results obtained explicitly compute the metric entropy and the folding entropy of $\sigma_\phi$ in terms of the probability distributions over the two alphabets. In particular, the metric entropy does not depend on the transition function $\phi$ , and the folding entropy measures the non-invertibility of the system — being zero exactly in the invertible case. These systems are especially useful for encoding non-invertible baker-type endomorphisms and for representing the non-deterministic dynamics of Filippov systems.

Events

Talks

2026 International Mathematics Days V, UFV, Folding and Metric Entropies for Extended Shifts
2025 Afternoon's Entropy at ICMC, ICMC, USP, Folding and Metric Entropies for Extended Shifts
2024 XIII Workshop on Dynamical Systems (MAT80), IMECC, Unicamp, Entropy and Non-Deterministic Chaos for Piecewise Smooth Vector Fields
2022 I Encontro Paulista da Pós-Graduação em Matemáticas, Entropia em Sistemas Dinâmicos
2016 DivulgaMat, IMECC, Unicamp, Um problema relacionado à rotação do círculo

Posters

2024 International Mathematics Days IV, UFV, Folding Entropy of the Zip Shift Map
2024 DinAmicI in Rio, IMPA, Entropy and Non-Deterministic Chaos for Piecewise Smooth Vector Fields
2023 VII Escola Brasileira de Sistemas Dinâmicos, UFC, The Orbit Space Approach for Piecewise Smooth Vector Fields
2023 XII Workshop on Dynamical Systems, IBILCE, Unesp, The Orbit Space Approach for Piecewise Smooth Vector Fields
2019 V Escola Brasileira de Sistemas Dinâmicos, UFMG, Unicidade Ergódica de Folheações Horocíclicas
2018 São Paulo Dynamical Systems Days, ICMC, USP, Introdução à Teoria Ergódica
2016 XXIV Congresso de Iniciação Científica da Unicamp, Unicamp, Dinâmica Simbólica: uma Introdução via Exemplos Hiperbólicos

Groups

Dynamics Working Group IMECC–Unicamp
- (2026–) https://www.ime.unicamp.br/~regisvarao/grupo-de-trabalho/
- (2024–25) https://daniel.mat.br/pt-br/gdt/
TeQA Dynamical Systems Network
- https://teqa.ime.unicamp.br/

Researchers

Here I list some of my colleagues and collaborators and their personal pages, in the order I met them.