A Dichotomy for the dimension of solenoidal attractors on high dimensional space

TL;DR

This paper establishes a dichotomy for the solenoidal attractors of certain skew product dynamical systems in high dimensions, showing they are either real analytic graphs or have a Hausdorff dimension determined by system parameters.

Contribution

It proves a dichotomy for solenoidal attractors in high-dimensional skew products, characterizing when they are analytic graphs or have a specific Hausdorff dimension, depending on the rotation parameter.

Findings

Attractors are either real analytic graphs or have a Hausdorff dimension given by a formula.

The analytic graph case occurs only for countably many parameters unless the function is constant.

The Hausdorff dimension depends on the base map and contraction rate.

Abstract

We study dynamical systems generated by skew products: $$T: [0,1)\times\mathbb{C}\to [0,1)\times\mathbb{C} \quad\quad T(x,y)=(bx\mod1,\gamma y+\phi(x))$$ where integer $b\ge2$, $\gamma\in\mathbb{C}$ such that $0<|\gamma|<1$, and $\phi$ is a real analytic $\mathbb{Z}$-periodic function. Let $\Delta\in[0,1) $ such that $\gamma=|\gamma|e^{2\pi i\Delta}$. For the case $\Delta\notin\mathbb{Q}$ we prove the following dichotomy for the solenoidal attractor $K^{\phi}_{b,\,\gamma}$ for $T$: Either $K^{\phi}_{b,\,\gamma}$ is a graph of real analytic function, or the Hausdorff dimension of $K^{\phi}_{b,\,\gamma}$ is equal to $\min\{3,1+\frac{\log b}{\log1/|\gamma|}\}$. Furthermore, given $b$ and $\phi$, the former alternative only happens for countable many $\gamma$ unless $\phi$ is constant.