Examples of almost-holomorphic and totally real laminations in complex surfaces

We show that there exists a Lipschitz almost-complex structure J   on CP2CP2, arbitrarily close to the standard one, and a compact lamination by J  -holomorphic curves satisfying the following properties: it is minimal, it has hyperbolic holonomy and it is transversally Lipschitz. Its transverse Hausdorff dimension can be any number δδ in an interval (0,δmax)(0,δmax) where δmax=1.6309…. We also show that there is a compact lamination by totally real surfaces in C2C2 with the same properties, unless the transverse dimension can be any number 0<δ<10<δ<1. Our laminations are transversally totally disconnected.