Conditional mixing in deterministic chaos

DOI10.1017/ETDS.2023.55arXiv2206.09291OpenAlexW4385984667MaRDI QIDQ6402483

Publication date: 18 June 2022

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.

Full work available at URL: https://doi.org/10.1017/etds.2023.55

Mathematics Subject Classification ID

Smooth ergodic theory, invariant measures for smooth dynamical systems (37C40) Thermodynamic formalism, variational principles, equilibrium states for dynamical systems (37D35) Generic properties, structural stability of dynamical systems (37C20)