Box dimension of stable sub-slices of fractal graphs over Anosov automorphisms

We consider fractal graphs invariant by a skew product F:Tk×R→Tk×R of the form F(x,y)=(Ax,λy+p(x)) where 0<λ<1, p:Tk→R is a Ck+1 function, and A is an Anosov automorphism of Tk admitting k distinct eigenvalues with respective eigenvectors forming a basis of Rk. We extend the techniques introduced by [J.L. Kaplan, J.Mallet-Paret and J.A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergodic Theory Dyn. Syst. 4 (1984), pp. 261–281] to calculate the box dimension of sub-slices of the invariant graph of F. We show that the stable sub-slices have a different fractal structure in comparison to the graph. We exhibit conditions on the skew product that ensure the box dimension of the graph is smaller than the sum of the box dimensions of its stable/unstable sub-slices. These conditions hold for generic functions p∈Ck+1.