A $C^r$-connecting lemma for Lorenz attractors and its application on the space of ergodic measures
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917212183330816 |
|---|---|
| author | Shi, Yi Tian, Xueting Wang, Xiaodong |
| author_facet | Shi, Yi Tian, Xueting Wang, Xiaodong |
| contents | For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\geq 2$) vector field, a heteroclinic orbit associated to the singularity and a critical element can be created through arbitrarily small $C^r$-perturbations. As an application, we show that for $C^r$-dense geometric Lorenz attractors, the Dirac measure of the singularity is isolated inside the space of ergodic measures and thus the ergodic measure space is not connected; while for $C^r$-generic geometric Lorenz attractors, the space of ergodic measures is path connected with dense periodic measures. In particular, the generic part proves a conjecture proposed by C. Bonatti in $C^r$-topology for Lorenz attractors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_08193 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A $C^r$-connecting lemma for Lorenz attractors and its application on the space of ergodic measures Shi, Yi Tian, Xueting Wang, Xiaodong Dynamical Systems For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\geq 2$) vector field, a heteroclinic orbit associated to the singularity and a critical element can be created through arbitrarily small $C^r$-perturbations. As an application, we show that for $C^r$-dense geometric Lorenz attractors, the Dirac measure of the singularity is isolated inside the space of ergodic measures and thus the ergodic measure space is not connected; while for $C^r$-generic geometric Lorenz attractors, the space of ergodic measures is path connected with dense periodic measures. In particular, the generic part proves a conjecture proposed by C. Bonatti in $C^r$-topology for Lorenz attractors. |
| title | A $C^r$-connecting lemma for Lorenz attractors and its application on the space of ergodic measures |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2006.08193 |