Abstract
In this paper we study two properties related to the structure of hyperbolic sets. First we construct new examples answering in the negative the following question posed by Katok and Hasselblatt in [[12], p. 272]Question. Let $\Lambda$ be a hyperbolic set, and let $V$ be an open neighborhood of $\Lambda$. Does there exist a locally maximal hyperbolic set $\widetilde{\Lambda}$ such that $\Lambda \subset \widetilde{\Lambda} \subset V $?We show that such examples are present in linear Anosov diffeomorophisms of $\mathbb{T}^3$, and are therefore robust.Also we construct new examples of sets that are not contained in any locally maximal hyperbolic set. The examples known until now were constructed by Crovisier in [7] and by Fisher in [9], and these were either in dimension equal or bigger than 4 or they were not transitive. We give a transitive and robust example in $\mathbb{T}^3$. And show that such examples cannot be build in dimension 2.
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
