What Are SRB Measures, and Which Dynamical Systems Have Them?

By Lai-Sang Young. 2002

602 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/851bc17f5114e62ee9c4282b624b0c555de40c12

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Citations in the corpus

What Are the New Implications of Chaos for Unpredictability? (2009, background): “As we will see in the next section, typically for systems proven to be chaotic physical measures can be proven to exist (Lyubich [2002]; Young [2002] ).”
A review of linear response theory for general differentiable dynamical systems (2009, unknown): “See Young [51] for further discussion.”
Fluctuation Relations for Diffusion Processes (2007, background): “In this theory, the stationary states correspond to invariant measures of the SRB type [81] and the entropy production is described by phase-space contraction [73].”
Mathematical and physical ideas for climate science (2013, methodology): “ts with theory. It is advantageous to assume that the forced and dissipative systems under our consideration possess a unique Sinai-Ruelle-Bowen (SRB) measure (Eckmann and Ruelle, 1985; Ruelle, 1989; Young, 2002), which guarantees that, regardless of the initial state of the system, the steady state”
Spectral Properties of Dynamical Systems, Model Reduction and Decompositions (2005, background): “This clearly includes systems that preserve a smooth invariant measure such as Hamiltonian systems, but also includes systems that possess a physical measure [31].”
The physics of climate variability and climate change (2019, background): “e ensemble average almost surely with respect to Lebesgue measure; moreover physical implies natural. A particular class of invariant measures of interest are Sinai-Ruelle-Bowen (SRB) measures (e.g., Young, 2002), and an important result is that, for a system with no null Lyapunov exponents, an ergo”
Linear response, or else (2014, background): “, while in general they do not admit any absolutely continuous invariant measure. Physical measures are sometimes called SRB 1 measures after Sinai, Ruelle, and Bowen, who studied them in the sixties [65]. Instead of a single discrete-time dynamical system f, let us now consider a one-parameter fami”
Statistical inference for dynamical systems: A review (2012, background): “Often the measure μ has some additional properties (it has no zero Lyapunov exponents and absolutely continuous conditional measures with respect to λ on unstable manifolds), and in this case μ may be called an SRB (Sinai-RuelleBowen) measure [127].”
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator (2016, background): “system eq. (4) with a basin of attraction B. The existence of ergodic average in eq. (19) can be extended to trajectories starting in Bby assuming that the invariant measure on Ais a physical measure [9,49]. To formalize this notion, let denote the standard Lebesgue measure on B. We assume that ther”
SRB measures for partially hyperbolic systems whose central direction is weakly expanding (2014, unknown): “…exponents µ -almost everywhere and admitting a system of conditional measures such that the conditional measures on unstable manifolds are absolutely continuous with respect to the Lebesgue measures Leb γ on these manifolds induced by the restriction of the Riemannian structure [14, 37, 39, 50].”
Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems (2007, methodology): “26a) with the qGFDT approximation, using the fact that an SRB measure is smooth along the uniformly expanding directions [6,22].”