Generalization of the multiplicative and additive compounds of square matrices and contraction in the Hausdorff dimension

TL;DR

This paper introduces generalized matrix compounds for real indices and applies them to extend contraction theory to systems that contract sets with Hausdorff dimension greater than a real number, broadening the scope of dynamical system analysis.

Contribution

The paper defines and studies $oldsymbol{ extit{ extalpha}}$-multiplicative and additive compounds for real $oldsymbol{ extalpha}$, extending classical integer-based compounds and contraction concepts.

Findings

Generalized compounds for real indices are mathematically characterized.

Application to Douady and Oesterlé Theorem demonstrates usefulness.

Extension of contraction theory to $ extit{ extalpha}$-contraction systems.

Abstract

The $k$ multiplicative and $k$ additive compounds of a matrix play an important role in geometry, multi-linear algebra, the asymptotic analysis of nonlinear dynamical systems, and in bounding the Hausdorff dimension of fractal sets. These compounds are defined for integer values of $k$. Here, we introduce generalizations called the $\alpha$ multiplicative and $\alpha$ additive compounds of a square matrix, with $\alpha$ real. We study the properties of these new compounds and demonstrate an application in the context of the Douady and Oesterl\'{e} Theorem. This leads to a generalization of contracting systems to $\alpha$ contracting systems, with $\alpha$ real. Roughly speaking, the dynamics of such systems contracts any set with Hausdorff dimension larger than $\alpha$. For $\alpha=1$ they reduce to standard contracting systems.