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

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.