A Cʳ -CONNECTING LEMMA FOR LORENZ ATTRACTORS AND ITS APPLICATION ON THE SPACE OF ERGODIC MEASURES

For every , we prove a Cʳ -connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$ -dimensional Cʳ ( r≥ 2 ) vector field, a heteroclinic orbit associated to the singularity and a critical element can be created through arbitrarily small Cʳ -perturbations. As an application, we show that for Cʳ -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 Cʳ -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 Cʳ -topology for Lorenz attractors.