Weak Closure Theorem for Double Staircase Actions

TL;DR

This paper introduces a double staircase construction in ergodic theory and characterizes the weak closure of its powers, revealing a specific structure involving convex combinations of the transformation and the integral.

Contribution

It presents a novel double staircase construction and explicitly describes the weak closure of its iterates, advancing understanding of ergodic transformations.

Findings

Weak closure consists of convex combinations of T^n and the integral.

Explicit description of the weak closure for the double staircase transformation.

Provides new insights into the structure of ergodic transformations.

Abstract

We introduce a double staircase construction $T$ and show that the weak closure of $\{T^n\}$ is $\{\int, 2^{-m}T^n+(1-2^{-m})\int \ : m\in\N,\ n\in \Z\}.$