The space of ergodic measures for Lorenz attractors

For every $rinmathbb{N}_{geq 2}cup{infty}$, we show that the space of ergodic measures is path connected for $C^r$-generic Lorenz attractors while it is not connected for $C^r$-dense Lorenz attractors. Various properties of the ergodic measure space for Lorenz attractors have been showed. In particular, a $C^r$-connecting lemma ($rgeq2$) for Lorenz attractors also has been proved. In $C^1$-topology, we obtain similar properties for singular hyperbolic attractors in higher dimensions.