Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions

By M. Field, Ian Melbourne, Andrew Török. Ergodic Theory and Dynamical Systems, 2003

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Large deviations for nonuniformly hyperbolic systems (2008, background): “There is a Young tower with base Y = [12 , 1] and it follows from Hu [16] that the optimal return time decay rate is given by Leb(y ∈ Y : R(y) > n) ≈ n−(β+1) where β = 1 α − 1.”
A new test for chaos in deterministic systems (2002, background): “Moreover, the p and q components each behave like a Brownian motion on the line if the chaotic attractor is uniformly hyperbolic (Field et al. 2002). A nondegeneracy result of Nicol et al. (2001) ensures that the variance of the Brownian motion is nonzero for almost all choices of observable φ. Rece”
Almost sure invariance principle for sequential and non-stationary dynamical systems (2014, methodology): “ The basic strategy is now the standard technique of coding ﬁrst by a two sided shift and then reducing to a non-invertible one-sided shift. There is a good description in Field, Melbourne and Toro¨k [13]. We use a Markov partition to code (T,X,µ) by a 2-sided shift (σ,Ω,ν) in a standard way [8, 29]”
ALMOST SURE INVARIANCE PRINCIPLE FOR DYNAMICAL SYSTEMS BY SPECTRAL METHODS (2009, unknown): “However, our method does not apply to the whole class of rapid-mixing hyperbolic ﬂows, contrary to the martingale arguments of Melbourne and T¨or¨ok [17].”
On the validity of the 0–1 test for chaos (2009, unknown): “Write v : X → R as an m-dimensional Fourier series v(x) = ∑ ∈Zm v ei ·x where v− = v̄ .”