Historical behavior of skew products and arcsine laws

TL;DR

This paper investigates how skew product systems exhibit historical behavior, showing that under certain conditions, Birkhoff averages fail to converge for most points, linking this to the arcsine law and random walk dynamics.

Contribution

It establishes a general framework connecting historical behavior in skew products to the arcsine law, extending previous examples and providing new insights into orbit distributions.

Findings

Almost every orbit exhibits non-convergent Birkhoff averages under certain ergodic conditions.

The arcsine law describes the distribution of orbit behaviors in these systems.

Known skew product examples are confirmed to meet the conditions for historical behavior.

Abstract

We study the occurrence of historical behavior for almost every point in the setting of skew products with one-dimensional fiber dynamics. Under suitable ergodic conditions, we establish that a weak form of the arcsine law leads to the non-convergence of Birkhoff averages along almost every orbit. As an application, we show that this phenomenon occurs for one-step skew product maps over a Bernoulli shift, where the stochastic process induced by the iterates of the fiber maps is conjugate to a random walk. Furthermore, we revisit known examples of skew products that exhibit historical behavior almost everywhere, verifying that they fulfill the required ergodic and probabilistic conditions. Consequently, our results provide a unified and generalized framework that connects such behaviors to the arcsine distribution of the orbits.