Random-like properties of chaotic forcing

TL;DR

This paper demonstrates that skew systems with expanding bases exhibit approximate exponential decay of correlations, even with mild assumptions on the fiber maps, and the approximation improves as the base expansion increases.

Contribution

It establishes approximate decay of correlations for skew systems with minimal assumptions, extending understanding beyond hyperbolic dynamics.

Findings

Decay of correlations is approximately exponential with error bounds.

Error diminishes as the base expansion tends to infinity.

Results apply to broader systems via acceleration or conjugation.

Abstract

We prove that skew systems with a sufficiently expanding base have approximate exponential decay of correlations, meaning that the exponential rate is observed modulo an error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. However, the error in the approximation goes to zero when the expansion of the base tends to infinity. The result can be applied beyond the original setup when combined with acceleration or conjugation arguments, as our examples show.