Measures of intermediate entropies for star vector fields

We prove that all star vector fields, including Lorenz attractors and multi-singular hyperbolic vector fields, admit the intermediate entropy property. To be precise, if X is a star vector field with htop(X) > 0, then for any h ∈ [0, htop(X)), there exists an ergodic invariant measure μ of X such that hμ(X)= h. Moreover, we show that the topological entropy is lower semi-continuous for star vector fields.