Large deviations in non-uniformly hyperbolic dynamical systems

Abstract

We prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

Differentiability of thermodynamical quantities in non-uniformly expanding dynamics

“…Let us mention that local rate functions have also been used in [RY08] and let us refer the reader to Section 5.3 for more details.…”

Sharp large deviations for some hyperbolic systems

“…Moreover, J ( p) = 0 if and only if p = dm . Many results on large deviations for hyperbolic (discrete and continuous) dynamical systems have been established in both the uniformly hyperbolic case (see [G, Kif, L, OP, Y] and the references given there) and the non-uniformly hyperbolic case [AP,MN,RY,PSY,PoS3].…”

An Expansion Estimate for Dispersing Planar Billiards with Corner Points

“…In fact, in [4] a Young tower (introduced in the seminal work of Young [16]) with an exponential tail of the return times is constructed. For such "Young systems", many further statistical properties have been proved, including large deviations ( [13], [11]), local limit laws ( [14]), almost sure invariance principles ( [10]), and Berry-Esséen type theorems ( [12]). It is worthwhile to mention that the same strong statistical properties can also be obtained by means of the more geometrical coupling approach (introduced in [17] and further developed in [6,5]); the reader can find a detailed exposition of the application of this technique to billiard systems in [7,Section 7].…”