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

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.