Hausdorff and packing dimensions and measures for nonlinear transversally non-conformal thin solenoids

We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\epsilon }$ hyperbolic, (partially) linear solenoids $\mathrm{\Lambda }$ over the circle embedded in ${\mathbb{R}}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear ones. Under an assumption of transversality and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing \emph{thinness}, assuming also there is an invariant $C^{1+\epsilon }$ strong stable foliation, we prove that Hausdorff dimension $\mathrm{H}\mathrm{D}(\mathrm{\Lambda }\cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else $\mathrm{H}\mathrm{D}(\mathrm{\Lambda })=t_0+1$. We prove also that for the packing measure $0<{\mathrm{\Pi }}_{t_0}(\mathrm{\Lambda }\cap W^s)<\mathrm{\infty }$ but for Hausdorff measure ${\mathrm{H}\mathrm{M}}_{t_0}(\mathrm{\Lambda }\cap W^s)=0$ for all $W^s$. Also $0<{\mathrm{\Pi }}_{1+t_0}(\mathrm{\Lambda })<\mathrm{\infty }$ and ${\mathrm{H}\mathrm{M}}_{1+t_0}(\mathrm{\Lambda })=0$. A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every $W^s$ has measure ${\mathrm{H}\mathrm{M}}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$. The latter holds due to a large deviations phenomenon.