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Geometric conditions for matrix domination in two dimensions

Argyrios Christodoulou

TL;DR

This work addresses the problem of when a finite subset of $\mathrm{SL}(2,\mathbb{R})$ is dominated (uniformly hyperbolic) in two dimensions by exploiting purely geometric data. It develops a cross-ratio framework $C(f,g)$ that encodes the relative configuration of the axes of hyperbolic elements and connects contraction rates to traces through $\cosh\left(\tfrac{1}{2}T_f\right)=\tfrac{1}{2}|\mathrm{tr}(f)|$, situating the analysis in the hyperbolic geometry of the Möbius action on $\mathbb{H}$ and $\mathbb{D}$. The paper provides explicit necessary and sufficient conditions for domination: a Jørgensen-type necessary condition, a pairwise sufficient criterion based on cross-ratios and traces, and a constructive theorem (Theorem MAIN) that yields dominated sets with prescribed eigenvectors by tuning contraction rates via trace-based inequalities. This yields practical, computable criteria for domination in $2$-D, with potential applications to spectral theory, dynamics on projective spaces, and representation theory of $\mathrm{SL}(2,\mathbb{R})$.

Abstract

In this article we prove a necessary and a sufficient condition for a finite subset of the special linear group to be dominated. These conditions are purely geometric in nature, as they only involve the trace and the eigenvectors of the matrices, and can be computed explicitly. Our sufficient condition, in particular, provides a simple algorithm for constructing a dominated set with prescribed eigenvectors. The techniques involved in our proofs take advantage of the interaction between dominated sets and two-dimensional hyperbolic geometry.

Geometric conditions for matrix domination in two dimensions

TL;DR

This work addresses the problem of when a finite subset of is dominated (uniformly hyperbolic) in two dimensions by exploiting purely geometric data. It develops a cross-ratio framework that encodes the relative configuration of the axes of hyperbolic elements and connects contraction rates to traces through , situating the analysis in the hyperbolic geometry of the Möbius action on and . The paper provides explicit necessary and sufficient conditions for domination: a Jørgensen-type necessary condition, a pairwise sufficient criterion based on cross-ratios and traces, and a constructive theorem (Theorem MAIN) that yields dominated sets with prescribed eigenvectors by tuning contraction rates via trace-based inequalities. This yields practical, computable criteria for domination in -D, with potential applications to spectral theory, dynamics on projective spaces, and representation theory of .

Abstract

In this article we prove a necessary and a sufficient condition for a finite subset of the special linear group to be dominated. These conditions are purely geometric in nature, as they only involve the trace and the eigenvectors of the matrices, and can be computed explicitly. Our sufficient condition, in particular, provides a simple algorithm for constructing a dominated set with prescribed eigenvectors. The techniques involved in our proofs take advantage of the interaction between dominated sets and two-dimensional hyperbolic geometry.

Paper Structure

This paper contains 8 sections, 17 theorems, 92 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background material
  3. Domination and the action on projective space
  4. Hyperbolic geometry of the Möbius group
  5. Cross-ratios
  6. A necessary condition for domination
  7. Geometric conditions on pairs
  8. Proof of Theorem \ref{['MAIN']}

Key Result

Theorem 1.2

Let $u_1,u_2,\dots,u_N,s_1,s_2,\dots s_N\in\mathbb{R}\mathbb{P}^1$ be $2N$ pairwise distinct points in the projective real line. Suppose that $A_1,A_2,\dots A_N\in\mathrm{SL}(2,\mathbb{R})$ are hyperbolic matrices with $u(A_i)=u_i$ and $s(A_i)=s_i$, for $i=1,2,\dots,N$, and consider the constants for all pairs $i,j\in\{1,2,\dots,N\}$ with $i\neq j$. If then the set $\{A_1,A_2,\dots, A_N\}$ is do

Figures (7)

  • Figure 1: A hyperbolic transformation depicted by its axis directed towards the attracting fixed point.
  • Figure 2: Formulas for the cross-ratio in Lemma \ref{['crl']}.
  • Figure 3: Cross-ratio and geometric configuration
  • Figure 4: The regions $C$ and $D$
  • Figure 5: The hyperbolic transformations in Lemma \ref{['newlem']}.
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 3.1
  • ...and 19 more