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A 
$C^r$-CONNECTING LEMMA FOR LORENZ ATTRACTORS AND ITS APPLICATION ON THE SPACE OF ERGODIC MEASURES
Published online by Cambridge University Press: 26 August 2025
Abstract
For every 
, 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 [11, Conjecture 2] in 
$C^r$-topology for Lorenz attractors.
Information
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 25 , Issue 1 , January 2026 , pp. 1 - 44
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press
Footnotes
Y. Shi was partially supported by National Key R&D Program of China (2021YFA1001900) and NSFC (12090015).
X. Tian was partially supported by NSFC (12471182, 12071082).
X. Wang was the corresponding author and was partially supported by National Key R&D Program of China (2021YFA1001900), NSFC (12071285) and Innovation Program of Shanghai Municipal Education Commission (No. 2021-01-07-00-02-E00087)
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