Almost sure invariance principle for dynamical systems with stretched exponential mixing rates

A vector-valued almost sure invariance principle for hyperbolic dynamical systems

“…The scalar ASIP has been shown to hold for large classes of dynamical systems [14,17,18,21,24,25,30,31,35]. Chernov & Dolgopyat [10, Problem 1] asked for a proof of the ASIP for R d -valued observables, and it is this problem that is solved in this paper.…”

“…These systems include Axiom A diffeomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates.In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian motion. 7 15 +ǫ ).Remark 1.8 (a) A number of authors [9,30,35] have independently established scalar ASIPs for one-dimensional projections of q(t). In hindsight, the scalar ASIP for the Lorentz…”

Sinai Billiards under Small External Forces II

“…We should note that our bound on the error term in (2.13) is not optimal; better bounds, e.g., O(N −1/4+ε ), can be obtained by using methods of recent works [14,26]. A more general (vector) version of the ASIP can be derived based on the results of the manuscript [25] (not yet published).…”

Rates of Convergence in the Strong Invariance Principle for Non-adapted Sequences Application to Ergodic Automorphisms of the Torus

“…To conclude, let us mention some previous works in the context of dynamical systems: several results have been established with the rate o(n 1/2−ε ) for some ε > 0 (see [10,3,4,22,17]). Results giving a rate in o(n 1/4+ε ) for every ε > 0 can be found in [19,5,18,7].…”

“…Up to our knowledge, the present work gives the first strong approximations results for such partially hyperbolic transformations T for unbounded (and then non continuous) functions f . To conclude, let us mention some previous works in the context of dynamical systems: several results have been established with the rate o(n 1/2−ε ) for some ε > 0 (see [10,3,4,22,17]). Results giving a rate in o(n 1/4+ε ) for every ε > 0 can be found in [19,5,18,7].…”