Hausdorff Dimension, Lagrange and Markov Dynamical Spectra for Geometric Lorenz Attractors

TL;DR

This paper proves that geometric Lorenz attractors have Hausdorff dimension greater than 2 and demonstrates that their associated Lagrange and Markov spectra often have non-empty interior, revealing complex dynamical properties.

Contribution

It establishes the Hausdorff dimension of geometric Lorenz attractors and links this to the structure of their dynamical spectra, a novel connection in dynamical systems theory.

Findings

Hausdorff dimension of Lorenz attractors > 2

Lagrange and Markov spectra often have non-empty interior

Spectral properties are linked to geometric dimension

Abstract

In this paper, we show that geometric Lorenz attractors have Hausdorff dimension strictly greater than $2$. We use this result to show that for a "large" set of real functions the Lagrange and Markov Dynamical spectrum associated to these attractors has persistently non-empty interior.