Measures of intermediate pressures for geometric Lorenz attractors

Pressure measures the complexity of a dynamical system concerning a continuous observation function. A dynamical system is called to admit the intermediate pressure property if for any observation function, the measure theoretical pressures of all ergodic measures form an interval. We prove that the intermediate pressure property holds for $C^r (r\geq 2)$ generic geometric Lorenz attractors while it fails for $C^r (r\geq 2)$ dense geometric Lorenz attractors, which gives a sharp contrast. Similar results hold for $C^1$ singular hyperbolic attractors.