Equivalent Conditions for Domination of $\mathrm{M}(2,\mathbb{C})$-sequences

TL;DR

This paper characterizes the domination of matrix sequences in M(2,C) through equivalent conditions involving singular values and establishes a version of the Avalanche Principle for these sequences.

Contribution

It extends known hyperbolicity and dominated splitting conditions to M(2,C)-sequences using singular value criteria and introduces an Avalanche Principle adaptation.

Findings

Equivalent singular value conditions for domination in M(2,C)-sequences.

A new version of the Avalanche Principle for these sequences.

Connections between exponential growth, splitting, and singular values.

Abstract

It is well known that a $\mathrm{SL}(2,\mathbb{C})$-sequence is uniformly hyperbolic if and only it satisfies a uniform exponential growth condition. Similarly, for $\mathrm{GL}(2,\mathbb{C})$-sequences whose determinants are uniformly bounded away from zero, it has dominated splitting if and only if it satisfies a uniform exponential gap condition between the two singular values. Inspired by [QTZ], we provide a similar equivalent description in terms of singular values for $\mathrm{M}(2,\mathbb{C})$-sequences that admit dominated splitting. We also prove a version of the Avalanche Principle for such sequences.