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Conditional mixing in deterministic chaos

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  18 August 2023

CAROLINE L. WORMELL Open the ORCID record for CAROLINE L. WORMELL [Opens in a new window]
CAROLINE L. WORMELL*
Affiliation:
Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, CNRS, Paris, France (e-mail: caroline.wormell@anu.edu.au)
*
e-mail: ca.wormell@gmail.com
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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 Sinai–Ruelle–Bowen (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. While conditional mixing typically cannot be proven from standard transfer operator theory, we will prove that conditional mixing holds in a class of generalized 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.

Keywords

mixingfilteringCantor measureFourier decay

MSC classification

Primary: 37C40: Smooth ergodic theory, invariant measures 37C20: Generic properties, structural stability 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 31
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Copyright
© Australian National University, 2023. Published by Cambridge University Press

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