Work

A $C^r$ -CONNECTING LEMMA FOR LORENZ ATTRACTORS AND ITS APPLICATION ON THE SPACE OF ERGODIC MEASURES

Year: 2025

Type: article

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.

Source: Journal of the Institute of Mathematics of Jussieu

DOI: 10.1017/s147474802510114x

Authors: Yi Shi, Xueting Tian, Xiaodong Wang

Institutions: Sichuan University, Fudan University, Shanghai Jiao Tong University

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Shanghai Municipal Education Commission, Science and Technology Commission of Shanghai Municipality