Dimension of a class of intrinsically transversal solenoidal attractors in high dimensions

TL;DR

This paper investigates the fractal dimensions of high-dimensional solenoidal attractors, establishing conditions under which their Hausdorff and box-counting dimensions coincide and are continuous, extending Bowen's formula to this class.

Contribution

It introduces a new class of high-dimensional solenoidal attractors with proven dimension properties under specific geometric and dynamical conditions.

Findings

Hausdorff and box-counting dimensions coincide for the attractors

Dimension is continuous within the class of attractors studied

Dimension equals the zero of the topological pressure as per Bowen's formula

Abstract

We study the fractal dimension of a class of solenoidal attractors in dimensions greater or equal than 3, proving that if the contraction is sufficiently strong, the expansion is close to conformal and the attractor satisfy a geometrical condition of transversality between its components, then the Hausdorff and box-counting dimension of every stable section of the attractor have the same value, which corresponds to the zero of the topological pressure as in Bowen's formula. We also calculate the dimension of the attractor and prove that it is continuous in this class.