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

Abstract

We consider fractal graphs invariant by a skew product $F:{\mathbb{T}}^k\times \mathbb{R}\to {\mathbb{T}}^k\times \mathbb{R}$ of the form $F(x,y)=(Ax,\lambda y+p(x))$ where $0<\lambda <1$, $p:{\mathbb{T}}^k\to \mathbb{R}$ is a $C^{k+1}$ function, and $A$ is an Anosov diffeomorphism of ${\mathbb{T}}^k$ admitting $k$ distinct eigenvalues with respective eigenvectors forming a basis of ${\mathbb{R}}^k$. We note that the stable sub-slices can give information of the fractal structure of the graph that is not captured by the box dimension of the graph. Using the results of Kaplan,Mallet-Paret, and York [6], 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. We prove that these conditions hold for generic functions $p\in C^{k+1}$.

23 pages, 2 figures