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

We show that there exists a Lipschitz almost-complex structure J on $C{P}^2$, 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 $\delta$ in an interval $(0,{\delta }_{\max })$ where ${\delta }_{\max }=1.6309\dots$. We also show that there is a compact lamination by totally real surfaces in $C^2$ with the same properties, unless the transverse dimension can be any number $0<\delta <1$. Our laminations are transversally totally disconnected.