On Robust Expansiveness for Sectional Hyperbolic Attracting Sets

IF 0.6 4区数学 Q3 MATHEMATICS

Moscow Mathematical Journal Pub Date : 2019-10-26 DOI:10.17323/1609-4514-2023-23-1-11-46

V. Araújo, J. Cerqueira

引用次数: 1

Abstract

We prove that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive (under an open technical condition of strong dissipative for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in $3$-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly expansive non-singular vector field is uniformly hyperbolic; and a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a $3$-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions).

查看原文本刊更多论文

关于分段双曲吸引集的鲁棒可扩展性

我们证明了$C^1$向量场的截面双曲吸引集是鲁棒扩张的（在高余维情况下的强耗散的开放技术条件下）。这扩展了$3$-流中奇异双曲吸引子的可扩展性的已知结果，即使在这种低维环境中也是如此。利用星向量场研究的最新进展，我们推导出一些相反的结果：一个鲁棒扩张的非奇异向量场是一致双曲的；并且一个鲁棒传递吸引子是区间双曲的当且仅当，它是鲁棒扩张的。在低维环境中，我们证明了$3$-流的吸引集是奇异双曲的，当且仅当它是鲁棒混沌的（对初始条件鲁棒敏感）。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Moscow Mathematical Journal 数学-数学

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The Moscow Mathematical Journal (MMJ) is an international quarterly published (paper and electronic) by the Independent University of Moscow and the department of mathematics of the Higher School of Economics, and distributed by the American Mathematical Society. MMJ presents highest quality research and research-expository papers in mathematics from all over the world. Its purpose is to bring together different branches of our science and to achieve the broadest possible outlook on mathematics, characteristic of the Moscow mathematical school in general and of the Independent University of Moscow in particular. An important specific trait of the journal is that it especially encourages research-expository papers, which must contain new important results and include detailed introductions, placing the achievements in the context of other studies and explaining the motivation behind the research. The aim is to make the articles — at least the formulation of the main results and their significance — understandable to a wide mathematical audience rather than to a narrow class of specialists.