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Lyapunov exponents for random products of non-negative matrices

Nima Alibabaei

Abstract

We first study i.i.d. products of finitely many invertible $2 \times 2$ matrices with positive entries, and prove that the top Lyapunov exponent admits an explicit, rapidly convergent Neumann-series-type representation involving an infinite matrix. We further show that non-negative invertible $2 \times 2$ matrices are simultaneously conjugate to positive matrices if and only if ``generalized'' heteroclinic connections do not occur among products of length at most $2$. These results yield a series formula for the Hausdorff dimension of the intersection of the middle-$n$th Cantor set with a random translate of itself, for every natural number $n$ except $4$. Furthermore, our method applies to the intersection of thick Cantor sets under random translation. We also determine the almost sure growth rate of i.i.d. three-term recurrences with finitely many positive coefficients.

Lyapunov exponents for random products of non-negative matrices

Abstract

We first study i.i.d. products of finitely many invertible matrices with positive entries, and prove that the top Lyapunov exponent admits an explicit, rapidly convergent Neumann-series-type representation involving an infinite matrix. We further show that non-negative invertible matrices are simultaneously conjugate to positive matrices if and only if ``generalized'' heteroclinic connections do not occur among products of length at most . These results yield a series formula for the Hausdorff dimension of the intersection of the middle-th Cantor set with a random translate of itself, for every natural number except . Furthermore, our method applies to the intersection of thick Cantor sets under random translation. We also determine the almost sure growth rate of i.i.d. three-term recurrences with finitely many positive coefficients.
Paper Structure (21 sections, 29 theorems, 254 equations, 3 tables)

This paper contains 21 sections, 29 theorems, 254 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Application I: Intersection of translated Cantor sets
  5. Application II: Three-term recurrences
  6. Kernel expansion, Definitions, and Basic Lemmas
  7. Heuristic explanation
  8. Definitions and Basic Lemmas
  9. Proof of Theorem \ref{['thm: main theorem 1']}
  10. Proof of Theorem \ref{['thm: main theorem 2']}
  11. Cutting the tail of the series
  12. Integral representation
  13. Integral representation for iterates
  14. Integral operators and telescoping
  15. Proof of Theorem \ref{['thm: main theorem 3']} and Applications
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\{A_i\}_{i\in I}$ be a finite family of invertible positive $2 \times 2$ matrices, and let $\mu$ be the Bernoulli measure on $I^{\mathbb N}$ associated to a probability vector $(w_i)_{i\in I}$. Then the top Lyapunov exponent $\lambda$ admits the explicit representation Here, letting $f_i = [F(A_i)]$, and the linear operator $T=(b_{k,n})_{k,n\in\mathbb N_0}: V \to V$ is defined by

Theorems & Definitions (80)

  • Theorem 1.1: Kernel expansion
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6: Hawkes
  • Theorem 1.7: Kenyon--Peres: translated Cantor sets
  • Definition 1.8
  • Proposition 1.9
  • Example 1.10
  • ...and 70 more