Almost Sure Invariance Principle For Nonuniformly Hyperbolic Systems

We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon. Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.Comment: 21 pages, To appear in Communications in Mathematical Physic

Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas

In this paper, we show how the Gordin martingale approximation method fits into the anisotropic Banach space framework. In particular, for the time-one map of a finite horizon planar periodic Lorentz gas, we prove that Holder observables satisfy statistical limit laws such as the central limit theorem and associated invariance principles.Comment: Final version, to appear in Nonlinearity. Corrected some minor typos from previous versio

Annealed and quenched limit theorems for random expanding dynamical systems

In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure invariance principle. We also discuss the quenched central limit theorem, dynamical Borel-Cantelli lemmas, Erd\"os-R\'enyi laws and concentration inequalities.Comment: Appeared online in Probability Theory and Related Field

Recurrence statistics for the space of Interval Exchange maps and the Teichm\"uller flow on the space of translation surfaces

In this note we show that the transfer operator of a Rauzy-Veech-Zorich renormalization map acting on a space of quasi-H\"older functions is quasicompact and derive certain statistical recurrence properties for this map and its associated Teichm\"uller flow. We establish Borel-Cantelli lemmas, Extreme Value statistics and return time statistics for the map and flow. Previous results have established quasicompactness in H\"older or analytic function spaces, for example the work of M. Pollicott and T. Morita. The quasi-H\"older function space is particularly useful for investigating return time statistics. In particular we establish the shrinking target property for nested balls in the setting of Teichm\"uller flow. Our point of view, approach and terminology derives from the work of M. Pollicott augmented by that of M. Viana