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Dedieu-Shub Measures

Joshua Paik

TL;DR

The paper investigates Dedieu--Shub measures, objects that assign to each group element a $X$-probability measure whose $K$-averaged pushforward equals a fixed stationary measure. It develops a general, rigorous framework for existence, well-definedness, and projection of DS measures, and then constructs explicit DS measures for $G=GL(d,\mathbb{C})$ on the complex flag variety $\mathbb{F}_d$ as well as for $G=GL(2,\mathbb{R})$ on $\mathbb{RP}^1$, distinguishing positive and negative determinant cases with detailed computations (including Blaschke-factor dynamics and Poisson kernels). The work applies DS measures to inequalities relating random and deterministic Lyapunov exponents, recovers known results (e.g., Rivin, Avila–Bochi, Pujals), and provides new explicit density formulas and a double fibration approach that yields the DS weights in high dimensions. A key experimental aim examines the Arnold family to probe the existence of DS measures for diffeomorphism groups, finding numerical obstructions that hint DS measures may fail to exist in that setting. Overall, the paper offers a comprehensive treatment of DS measures, their algebraic/dynamical structure, and their role in connecting random and deterministic growth rates in linear and projective dynamics.

Abstract

This paper introduces Dedieu-Shub measures and surveys their appearance in the literature.

Dedieu-Shub Measures

TL;DR

The paper investigates Dedieu--Shub measures, objects that assign to each group element a -probability measure whose -averaged pushforward equals a fixed stationary measure. It develops a general, rigorous framework for existence, well-definedness, and projection of DS measures, and then constructs explicit DS measures for on the complex flag variety as well as for on , distinguishing positive and negative determinant cases with detailed computations (including Blaschke-factor dynamics and Poisson kernels). The work applies DS measures to inequalities relating random and deterministic Lyapunov exponents, recovers known results (e.g., Rivin, Avila–Bochi, Pujals), and provides new explicit density formulas and a double fibration approach that yields the DS weights in high dimensions. A key experimental aim examines the Arnold family to probe the existence of DS measures for diffeomorphism groups, finding numerical obstructions that hint DS measures may fail to exist in that setting. Overall, the paper offers a comprehensive treatment of DS measures, their algebraic/dynamical structure, and their role in connecting random and deterministic growth rates in linear and projective dynamics.

Abstract

This paper introduces Dedieu-Shub measures and surveys their appearance in the literature.

Paper Structure

This paper contains 27 sections, 28 theorems, 105 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and Results
  3. General Properties of Dedieu--Shub measures and Application to Lyapunov Exponents
  4. General properties
  5. Dedieu--Shub measures are well defined
  6. Fubini--Like property
  7. Projecting Dedieu--Shub measures
  8. An application to Lyapunov exponents
  9. Proof of Theorem \ref{['GL2R']}
  10. Proof strategy
  11. Facts about the projective action
  12. Positive determinant
  13. What is the physical measure of a $B_\theta$?
  14. Finishing the argument
  15. Negative determinant
  16. ...and 12 more sections

Key Result

theorem 1.2

Let $G = GL(d,\mathbb{C})$, $K = \mathrm{SU}(d)$, and $X = \mathbb{F}_d$. Let $\mathrm{Sym}(d)$ be the symmetric group on $\{1,...,d\}$. For a given $A \in \mathrm{GL}(d,\mathbb{C})$, let Let $p:\mathrm{Sym}(d) \times GL(d,\mathbb{C}) \to [0,1]$ be defined as Define $m: \mathrm{GL}(d,\mathbb{C}) \to \mathrm{Prob}(\mathbb{F}_d)$ as Then $m_A$ is a Dedieu--Shub measure.

Figures (2)

  • Figure 1:
  • Figure 2: In black, which we call the leftover, we depict the density of $\mathcal{E}(x) = 1 - \frac{d}{d\mu} \int_{r \in [0,1] \smallsetminus \mathbb{Q}} \tau^+_{r,0.05} dr.$ For there to be a Dedieu Shub measure for the Arnold family, $\sum_{p/q \in [0,1] \cap \mathbb{Q}} \mu(I_{p,q}) \frac{d}{d\mu}\left(\int_{c \in I_{p/q}} \tau^+_{c,\varepsilon} + \tau^i_{c,\varepsilon} \ dc\right)(x) \geq \mathcal{E}(x)$ for every $x \in X$. However, in the grey shaded box, we see this is not possible.

Theorems & Definitions (58)

  • definition 1.1
  • theorem 1.2: dedieu2003mike
  • theorem 1.3: Dedieu--Shub dedieu2003mike
  • theorem 1.4
  • conjecture 1.5: dedieu2003mike,burns2001recent
  • theorem 1.6: armentanorandom
  • lemma 2.1
  • proof
  • lemma 2.2
  • proof
  • ...and 48 more