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

We show that there exists a Lipschitz almost-complex structure J on C P², 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 δ in an interval ( 0, δ_max ) where δ_max = 1.6309 ... . We also show that there is a compact lamination by totally real surfaces in C² with the same properties, unless the transverse dimension can be any number 0 < δ < 1. Our laminations are transversally totally disconnected.