Conditional mixing in deterministic chaos

TL;DR

This paper introduces the concept of conditional mixing in chaotic systems, showing that certain conditional measures rapidly converge to the full measure, impacting long-term prediction and linear response theory.

Contribution

It provides rigorous and numerical evidence for conditional mixing in specific maps, advancing understanding of probabilistic predictability in chaotic dynamics.

Findings

Conditional measures converge exponentially to SRB measures.

Conditional mixing holds in generalized baker's maps.

Numerical evidence of conditional mixing in non-Markovian maps.

Abstract

While on the one hand, chaotic dynamical systems can be predicted for all time given exact knowledge of an initial state, they are also in many cases rapidly mixing, meaning that smooth probabilistic information (quantified by measures) on the system's state has negligible value for predicting the long-term future. However, an understanding of the long-term predictive value of intermediate kinds of probabilistic information is necessary in various physical problems, and largely remains lacking. Of particular interest in data assimilation and linear response theory are the conditional measures of the SRB measure on zero sets of general smooth functions of the phase space. In this paper we give rigorous and numerical evidence that such measures generically converge back under the dynamics to the full SRB measures, exponentially quickly. We call this property conditional mixing. We will prove that conditional mixing holds in a class of generalised baker's maps, and demonstrate it numerically in some non-Markovian piecewise hyperbolic maps. Conditional mixing provides a natural limit on the effectiveness of long-term forecasting of chaotic systems via partial observations, and appears key to proving the existence of linear response outside the setting of smooth uniform hyperbolicity.