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

Abstract

We consider fractal graphs invariant by a skew product $F:\mathbb{T}^k\times \mathbb{R}\rightarrow \mathbb{T}^k\times \mathbb{R}$ of the form $F(x,y)=(Ax, \lambda y+p(x))$ where $0<\lambda<1$, $p\colon\mathbb{T}^k\to\mathbb{R}$ 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 $p\in C^{k+1}$.