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A Kakutani-Rokhlin decomposition for conditionally ergodic process in the measure-free setting of vector lattices

Youssef Azouzi, Marwa Masmoudi, Bruce Alastair Watson

TL;DR

This work extends Rokhlin-type decomposition to the measure-free setting of vector lattices by formulating a Kakutani-Rokhlin decomposition for conditionally ergodic dynamics on Riesz spaces using a conditional expectation preserving system $(E,T,S,e)$. It proves both an ε-free Kakutani-Rokhlin lemma and, under aperiodicity, an ε-bounded decomposition, and shows that any aperiodic CEPS can be arbitrarily well approximated by periodic CEPS. The framework integrates Poincaré recurrence and the Kac formula into the Riesz-space context, enabling decomposition and approximation of conditionally ergodic processes without reference to a measure space. These results extend Rokhlin tower concepts to measure-free settings, broadening applicability to stochastic processes in vector lattices and offering new tools for analyzing conditional expectations in $L^1(T)$-like modules. Overall, the paper provides a rigorous bridge between classical ergodic decomposition and the vector-lattice generalization, with potential applications to a wide class of conditional-ergodic processes.

Abstract

Recently the Kac formula for the conditional expectation of the first recurrence time of a conditionally ergodic conditional expectation preserving system was established in the measure free setting of vector lattices (Riesz spaces). We now give a formulation of the Kakutani-Rokhlin decomposition for conditionally ergodic systems in terms of components of weak order units in a vector lattice. In addition, we prove that every aperiodic conditional expectation preserving system can be approximated by a periodic system.

A Kakutani-Rokhlin decomposition for conditionally ergodic process in the measure-free setting of vector lattices

TL;DR

This work extends Rokhlin-type decomposition to the measure-free setting of vector lattices by formulating a Kakutani-Rokhlin decomposition for conditionally ergodic dynamics on Riesz spaces using a conditional expectation preserving system . It proves both an ε-free Kakutani-Rokhlin lemma and, under aperiodicity, an ε-bounded decomposition, and shows that any aperiodic CEPS can be arbitrarily well approximated by periodic CEPS. The framework integrates Poincaré recurrence and the Kac formula into the Riesz-space context, enabling decomposition and approximation of conditionally ergodic processes without reference to a measure space. These results extend Rokhlin tower concepts to measure-free settings, broadening applicability to stochastic processes in vector lattices and offering new tools for analyzing conditional expectations in -like modules. Overall, the paper provides a rigorous bridge between classical ergodic decomposition and the vector-lattice generalization, with potential applications to a wide class of conditional-ergodic processes.

Abstract

Recently the Kac formula for the conditional expectation of the first recurrence time of a conditionally ergodic conditional expectation preserving system was established in the measure free setting of vector lattices (Riesz spaces). We now give a formulation of the Kakutani-Rokhlin decomposition for conditionally ergodic systems in terms of components of weak order units in a vector lattice. In addition, we prove that every aperiodic conditional expectation preserving system can be approximated by a periodic system.
Paper Structure (5 sections, 14 theorems, 97 equations)

This paper contains 5 sections, 14 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kakutani-Rokhlin Lemma - $\epsilon$-free
  4. Aperiodicity and an $\epsilon$-bounded decomposition
  5. Approximation of aperiodic maps

Key Result

Lemma 2.2

If $(E,T,S,e)$ is a conditional expectation preserving system and $T$ is strictly positive then $Sg=g$ for all $g\in R(T)$. In the case of $E$ being an $R(T)$ module this invariance gives that $S(gf)=gSf$ for all $g\in R(T)$ and $f\in E$.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Lemma 2.2
  • Proof 1
  • Definition 2.4: Recurrence
  • Theorem 2.3: Poincaré
  • Corollary 2.4
  • Lemma 2.5
  • ...and 22 more