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

Chang Sun, Zhenghe Zhang

TL;DR

The paper provides a sharp singular-value–based criterion for dominated splitting of bounded ${ m M}(2,\mathbb{C})$-sequences by proving domination is equivalent to the conjunction of a singular-value gap $(SVG)$ and a fast-invertibility condition $(FI)$. It develops the projective-geometry and SVD toolkit to construct invariant stable/unstable directions, establishes their separation, and extends these results to ${ m M}(2,\mathbb{C})$-cocycles over dynamical bases. It further introduces an Avalanche Principle for singular ${ m M}(2,\mathbb{C})$-sequences, providing a quantitative bound on growth deviations and linking to Lyapunov exponent regularity. Together, these results generalize prior work on ${ m SL}(2,\mathbb{C})$ and ${ m GL}(2,\mathbb{C})$ cases and offer a framework for non-invertible matrix cocycles and their dynamical consequences.

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.

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

TL;DR

The paper provides a sharp singular-value–based criterion for dominated splitting of bounded -sequences by proving domination is equivalent to the conjunction of a singular-value gap and a fast-invertibility condition . It develops the projective-geometry and SVD toolkit to construct invariant stable/unstable directions, establishes their separation, and extends these results to -cocycles over dynamical bases. It further introduces an Avalanche Principle for singular -sequences, providing a quantitative bound on growth deviations and linking to Lyapunov exponent regularity. Together, these results generalize prior work on and cases and offer a framework for non-invertible matrix cocycles and their dynamical consequences.

Abstract

It is well known that a -sequence is uniformly hyperbolic if and only it satisfies a uniform exponential growth condition. Similarly, for -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 -sequences that admit dominated splitting. We also prove a version of the Avalanche Principle for such sequences.

Paper Structure

This paper contains 8 sections, 20 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. ${\mathbb C}{\mathbb P}^1$ and Singular Value Decomposition
  3. Domination Implies $\mathrm{(SVG)}$ and $\mathrm{(FI)}$
  4. $\mathrm{(SVG)}$ and $\mathrm{(FI)}$ Imply Domination
  5. Existence of $E^s$ and $E^u$
  6. Separation of $E^s$ and $E^u$
  7. Domination for ${\mathrm M}(2,{\mathbb C})$-Cocycles
  8. An Avalanche Principle for ${\mathrm M}(2,{\mathbb C})$-sequences

Key Result

Theorem 1

Let $B\in\ell^{\infty}({\mathbb Z},\mathrm{M}(2,{\mathbb C}))$. $B \in {\mathcal{D}}{\mathcal{S}}$ if and only if $B$ satisfies both $\mathrm{(SVG)}$ and $\mathrm{(FI)}$.

Theorems & Definitions (45)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Example 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 35 more