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Conditional mixing in deterministic chaos

Published online by Cambridge University Press:  18 August 2023

Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, CNRS, Paris, France (e-mail:


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.

Original Article
© Australian National University, 2023. Published by Cambridge University Press

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Baladi, V.. Optimality of Ruelle’s bound for the domain of meromorphy of generalized zeta functions. Port. Math. 49(1) (1992), 6983.Google Scholar
Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.CrossRefGoogle Scholar
Baladi, V.. Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps. Springer, Cham, Switzerland, 2018.CrossRefGoogle Scholar
Baladi, V. and Gouëzel, S.. Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 14531481.CrossRefGoogle Scholar
Barański, K., Gutman, Y. and Śpiewak, A.. A probabilistic Takens theorem. Nonlinearity 33(9) (2020), 4940.CrossRefGoogle Scholar
Bonatti, C., Crovisier, S., Díaz, L. J. and Wilkinson, A.. What is…a blender? Notices Amer. Math. Soc. 63(10) (2016), 11751178.CrossRefGoogle Scholar
Bourgain, J. and Dyatlov, S.. Fourier dimension and spectral gaps for hyperbolic surfaces. Geom. Funct. Anal. 27(4) (2017), 744771.CrossRefGoogle Scholar
Butterley, O., Canestrari, G. and Jain, S.. Discontinuities cause essential spectrum. Comm. Math. Phys. 398(2) (2023), 627653.CrossRefGoogle Scholar
Chernov, N. I.. Limit theorems and Markov approximations for chaotic dynamical systems. Probab. Theory Related Fields 101(3) (1995), 321362.CrossRefGoogle Scholar
Demers, M. and Liverani, C.. Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc. 360(9) (2008), 47774814.CrossRefGoogle Scholar
Dolgopyat, D.. On decay of correlations in Anosov flows. Ann. of Math. (2) 147(2) (1998), 357390.CrossRefGoogle Scholar
Doucet, A., De Freitas, N. and Gordon, N. J.. Sequential Monte Carlo Methods in Practice. Springer, New York, 2001.CrossRefGoogle Scholar
Gottwald, G. A., Wormell, C. L. and Wouters, J.. On spurious detection of linear response and misuse of the fluctuation–dissipation theorem in finite time series. Phys. D 331 (2016), 89101.CrossRefGoogle Scholar
Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26(1) (2006), 189217.CrossRefGoogle Scholar
Hochman, M. and Shmerkin, P.. Equidistribution from fractal measures. Invent. Math. 202(1) (2015), 427479.CrossRefGoogle Scholar
Misiurewicz, M.. Strange attractors for the Lozi mappings. Ann. New York Acad. Sci. 357(1) (1980), 348358.CrossRefGoogle Scholar
Misiurewicz, M. and Štimac, S.. Symbolic dynamics for Lozi maps. Nonlinearity 29(10) (2016), 3031.CrossRefGoogle Scholar
Mosquera, C. A. and Olivo, A.. Fourier decay of self-similar measures on the complex plane. Preprint, 2022, arXiv:2207.11570.CrossRefGoogle Scholar
Mosquera, C. A. and Shmerkin, P.. Self-similar measures: asymptotic bounds for the dimension and Fourier decay of smooth images. Ann. Acad. Sci. Fenn. Math. 43 (2018), 823834.CrossRefGoogle Scholar
Oljača, L., Kuna, T. and Bröcker, J.. Exponential stability and asymptotic properties of the optimal filter for signals with deterministic hyperbolic dynamics. Preprint, 2021, arXiv:2103.01190.Google Scholar
Ruelle, D.. Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds—A contribution to the Gallavotti–Cohen chaotic hypothesis. Nonlinearity 31(12) (2018), 5683.CrossRefGoogle Scholar
Sahlsten, T. and Stevens, C.. Fourier transform and expanding maps on Cantor sets. Preprint, 2022, arXiv:2009.01703.Google Scholar
Schmeling, J. and Troubetzkoy, S.. Dimension and invertibility of hyperbolic endomorphisms with singularities. Ergod. Th. & Dynam. Sys. 18(5) (1998), 12571282.CrossRefGoogle Scholar
Wormell, C. L.. On convergence of linear response formulae in some piecewise hyperbolic maps. Preprint, 2022, arXiv:2206.09292.Google Scholar
Young, L.-S.. Bowen–Ruelle measures for certain piecewise hyperbolic maps. The Theory of Chaotic Attractors. Eds. Hunt, B. R., Li, T.-Y., Kennedy, J. A. and Nusse, H. E.. Springer, New York, 1985, pp. 265272.CrossRefGoogle Scholar