Young Towers for Product Systems

TL;DR

This paper demonstrates that the direct product of systems with Young towers also admits a Young tower with decay rates bounded by the slowest component, enabling analysis of complex systems' statistical properties.

Contribution

It establishes that the direct product of systems with Young towers inherits a Young tower structure with decay rates constrained by the slowest component.

Findings

The decay rate of the product system's Young tower is bounded by the slowest decay rate of the component systems.

The result applies to various complex systems, including Lorenz-like, multimodal, and Hénon maps.

Statistical properties of these systems can be derived using the Young tower framework.

Abstract

We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $ C^2 $ interval maps with critical points and singularities, H\'enon maps and partially hyperbolic systems.