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

TL;DR

This paper investigates the fractal geometry of invariant graphs under skew products over Anosov diffeomorphisms, revealing conditions where the graph's box dimension is less than the sum of its stable and unstable slices, with generic functions satisfying these conditions.

Contribution

It establishes conditions under which the box dimension of fractal graphs over Anosov systems is strictly less than the sum of slice dimensions, and shows these conditions are generically met.

Findings

Box dimension of graphs can be smaller than sum of slice dimensions.

Conditions for dimension gap are satisfied for generic functions.

Provides new insights into fractal structure of invariant graphs.

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 $p\in C^{k+1}$.