Piecewise smooth vectors fields: closing lemma, topological entropy and Shifts
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Data
2021-02-25
Autores
Antunes, André do Amaral
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
Recentemente, a teoria sobre campos vetoriais suaves por partes (CVSPs) tem passado por importantes desenvolvimentos. Uma linha de investigação desses campos vetoriais é procurar estabelecer resultados análogos aos já clássicos sobre campos vetoriais suaves, como os teoremas de Poincaré-Bendixson e Índice de Poincaré. Nessa linha de trabalho, nós abordamos o clássico problema do Closing Lemma para CVSPs e apresentamos uma resposta positiva para o caso C0. Outra possível linha de investigação é estudar as diferenças entre CVSPs e suaves. A maior parte delas surge do fato de que não há unicidade de trajetória por um ponto para CVSPs. Isto implica resultados como a existência de um CVSP planar caótico. Nesta linha de trabalho, propomos um novo modo de abordar CVSPs, através da construção de um espaço métrico de todas as possíveis trajetórias, usamos isso para definir entropia topológica de um CVSP; provamos a existências de CVSPs com entropia positiva (finita e infinita) e damos condições suficientes para que um CVSP tenha entropia infinita. Além disso, a partir desse espaço métrico, propomos um modo de conjugar a dinâmica de um CVSP com a aplicação shift em espaços de sequências.
Recently, the theory concerning piecewise smooth vector fields (PSVFs for short) has been undergoing important improvements. One line of investigation of these vector fields is to seek to establish results analogous to those already well known for the smooth case, such as Poincaré Bendixson and Poincaré Index Theorems. On this line of work, we tackle the classical problem of Closing Lemma in the setting of PSVFs and provide a positive answer for the case C0. Another possible line of investigation is studying the differences between PSVFs and smooth vector fields. Most of them arises from the fact that there is no uniqueness of trajectory passing through a point for a PSVF. It implies results like the existence of a planar PSVF that is chaotic. On this line, we propose a new way of looking at PSVFs, by the construction of a metric space of all possible trajectories, we use it to define topological entropy of a PSVF; prove the existence of PSVFs of positive (finite and infinite) entropy and give a sufficient condition for a PSVF to have infinite topological entropy. Moreover, from this metric space, we propose a way of conjugating the dynamics of PSVFs and shift spaces.
Recently, the theory concerning piecewise smooth vector fields (PSVFs for short) has been undergoing important improvements. One line of investigation of these vector fields is to seek to establish results analogous to those already well known for the smooth case, such as Poincaré Bendixson and Poincaré Index Theorems. On this line of work, we tackle the classical problem of Closing Lemma in the setting of PSVFs and provide a positive answer for the case C0. Another possible line of investigation is studying the differences between PSVFs and smooth vector fields. Most of them arises from the fact that there is no uniqueness of trajectory passing through a point for a PSVF. It implies results like the existence of a planar PSVF that is chaotic. On this line, we propose a new way of looking at PSVFs, by the construction of a metric space of all possible trajectories, we use it to define topological entropy of a PSVF; prove the existence of PSVFs of positive (finite and infinite) entropy and give a sufficient condition for a PSVF to have infinite topological entropy. Moreover, from this metric space, we propose a way of conjugating the dynamics of PSVFs and shift spaces.
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Palavras-chave
Sistemas dinâmicos, Campos de vetores suaves por partes, Closing Lemma, Entropia topológica, Shifts, Dynamical systems, Piecewise smooth vector fields, Topological entropy