A $C^r$-connecting lemma for Lorenz attractors and its application on the space of ergodic measures

For every $r\in {\mathbb{N}}_{\ge 2}\cup \{\mathrm{\infty }\}$, we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\ge 2$) 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.