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Linear dynamics of random products of operators

Valentin Gillet

TL;DR

This work investigates the linear dynamics of random operator cocycles specified by $T_n(\omega)=T(\tau^{n-1}\omega)\cdots T(\omega)$ on a separable Fréchet space under an ergodic $\tau$, focusing on two-valued cocycles with $T(\omega)=T_1$ on $A_1$ and $T(\omega)=T_2$ on $A_2$. It blends ergodic theory (Birkhoff sums and CLTs) with functional-analytic models on $H^2(\mathbb{D})$ and $H(\mathbb{C})$, using the Universality Criterion to establish when random products are universal or topologically mixing. The main contributions include explicit universality/mixing criteria for random adjoints of multiplication operators when $m(A_1)=m(A_2)=1/2$, extensions to cases with unequal measures, and noncommuting examples; it also obtains CLTs for Birkhoff sums of centered indicator functions under irrational rotations and the doubling map. These results advance the understanding of random cocycle dynamics in infinite-dimensional settings and hint at broader implications for random differential/difference equations and operator cocycles in analytic function spaces.

Abstract

We study the linear dynamics of the random sequence $(T_n(.))_{n \geq 1}$ of the operators $T_n(ω) = T(τ^{n-1}ω) \dotsm T(τω) T(ω), n \geq 1$. These products depend on an ergodic measure-preserving transformation $τ: \mathbb{T} \to \mathbb{T}$ on the probability space $(\mathbb{T}, m)$ and on a strongly measurable map $T : \mathbb{T} \to \mathcal{B}(X)$, where $X$ is a separable Fréchet space. We will be focusing on the case where $T(ω)$ is equal to an operator $T_1$ on $X$ for every $ω\in A_1$ and equal to an operator $T_2$ on $X$ for every $ω\in A_2$, where $A_1, A_2$ are two disjoint Borel subsets of $[0,1)$ such that $A_1 \cup A_2 = [0,1)$ and $m(A_k) > 0$ for $k = 1,2$. More precisely, we will be focusing on the case where the operators $T_1$ and $T_2$ are adjoints of multiplication operators on the Hardy space $H^2(\mathbb{D})$, as well as the case where $T_1$ and $T_2$ are entire functions of exponential type of the derivation operator on the space of entire functions. Finally, we will study the linear dynamics of a case of a random product $T_n(ω)$ for which the operators $T(τ^i ω), i \geq 0$, do not commute. We will give particular importance to the case where the ergodic transformation is an irrational rotation or the doubling map on $\mathbb{T}$.

Linear dynamics of random products of operators

TL;DR

This work investigates the linear dynamics of random operator cocycles specified by on a separable Fréchet space under an ergodic , focusing on two-valued cocycles with on and on . It blends ergodic theory (Birkhoff sums and CLTs) with functional-analytic models on and , using the Universality Criterion to establish when random products are universal or topologically mixing. The main contributions include explicit universality/mixing criteria for random adjoints of multiplication operators when , extensions to cases with unequal measures, and noncommuting examples; it also obtains CLTs for Birkhoff sums of centered indicator functions under irrational rotations and the doubling map. These results advance the understanding of random cocycle dynamics in infinite-dimensional settings and hint at broader implications for random differential/difference equations and operator cocycles in analytic function spaces.

Abstract

We study the linear dynamics of the random sequence of the operators . These products depend on an ergodic measure-preserving transformation on the probability space and on a strongly measurable map , where is a separable Fréchet space. We will be focusing on the case where is equal to an operator on for every and equal to an operator on for every , where are two disjoint Borel subsets of such that and for . More precisely, we will be focusing on the case where the operators and are adjoints of multiplication operators on the Hardy space , as well as the case where and are entire functions of exponential type of the derivation operator on the space of entire functions. Finally, we will study the linear dynamics of a case of a random product for which the operators , do not commute. We will give particular importance to the case where the ergodic transformation is an irrational rotation or the doubling map on .

Paper Structure

This paper contains 20 sections, 56 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Objectives of the article and motivation
  3. Linear dynamics
  4. Main results
  5. Organization of the paper
  6. Notation and definition
  7. Behavior of Birkhoff sums associated to a centered real function
  8. Boundedness of Birkhoff sums
  9. Central limit theorems for ergodic transformations
  10. CLTs for an irrational rotation $R_\alpha$ with $\alpha$ having bounded partial quotients.
  11. CLTs for an irrational rotation $R_\alpha$ with $\alpha$ having unbounded partial quotients.
  12. CLTs for the doubling map
  13. Random products of adjoint multipliers on the Hardy space $H^2(\mathbb D)$
  14. Case $m(A_1) = m(A_2) = 1/2$
  15. Case where possibly $m(A_1) \ne m(A_2)$
  16. ...and 5 more sections

Key Result

Proposition 1.1

Let $T_n : X \to X$, $n \geq 1$, be commuting continuous maps on a separable complete metric space $X$. Suppose that each $T_n$, $n \geq 1$, has a dense range. The following assertions are equivalent: In this situation, the set of points in $X$ with a dense orbit under $(T_n)_{n \geq 1}$ is a dense $G_{\delta }$-set.

Theorems & Definitions (93)

  • Proposition 1.1
  • Proposition 1.2: GEPM
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 83 more