Weak specification properties and large deviations for non-additive potentials

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0143385713000667How to cite this article: PAULO VARANDAS and YUN ZHAO (2015). Weak specication properties and large deviations for non-additive potentials. Ergodic Theory and Dynamical Systems, 35, pp 968-993Abstract. We obtain large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. In particular, a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some applications to the study of the convergence of Lyapunov exponents are given. IntroductionThe purpose of the theory of large deviations is to study the rates of convergence of sequences of random variables to some limit distribution. Some applications of these ideas into the realm of dynamical systems have been particularly useful to estimate the velocity at which time averages of typical points of ergodic invariant measures converge to the space average as guaranteed by Birkhoff's ergodic theorem. More precisely, given a continuous transformation f on a compact metric space M and a reference measure ν, one interesting question is to obtain sharp estimates for the ν-measure of the deviation sets {x ∈ M : (1/n) n−1 j=0 g( f j (x)) > c} for all continuous functions g : M → R and real numbers c. We refer the reader to [1, 11, 16-18, 24-27, 31, 33, 35, 42, 43, 45] and the references therein for an account of recent large deviations results.Since many relevant quantities in dynamical systems arise from non-additive sequences, e.g. the largest Lyapunov exponent for higher-dimensional dynamical systems, Kingman's ergodic theorem becomes in many situations crucial in the study of deviation sets {x ∈ M : ϕ n (x) > cn} with respect to some not necessarily additive sequence = {ϕ n } n of continuous functions. Inspired by the pioneering work of Young [43] our purpose in this direction is to provide sharp large deviations estimates for a wide class of nonadditive sequences of continuous potentials. Our approach uses ideas from the non-additive

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

“…In [36], Zhao and the second author obtained large deviations results for weak Gibbs measures and sub-additive observables with the weak Bowen condition. We say the sequence Φ = {φ n } satisfies the tempered distortion condition if…”

“…From [36,Theorem B] we know that if F * (Ψ, µ Φ ) / ∈ J δ then L J δ ,µΦ > 0 and, consequently, the topological pressure of both sets X J and X J is strictly smaller…”

“…Since the potentials ϕ n = log M ιn(x) are constant in n-cylinders the family of potentials Φ clearly satisfies the bounded distortion condition. It follows as a consequence of the large deviations bound in [36] and Theorem B that taking Ψ = Φ with q = 1 and c > 0, the set X c = {x ∈ Σ : lim sup n→∞ 1 n log M ιn(x) − M * (µ Φ ) > c} of points whose exponential growth of M ιn(x) is c-far away from the maximal Lyapunov exponent M * (µ Φ ) for infinitely many values of n has topological pressure strictly smaller than P top (f, Φ). Moreover, with respect to the maximal entropy measure µ 0 Corollary B yields that the topological pressure function c → h X c (f ) is strictly decreasing and concave for small positive c.…”

“…Free energy function and Legendre transforms. We are interested in the regularity of the rate function in the large deviations principles obtained in [36].…”

“…Then it follows from [2, Theorem 9] that there exists a unique equilibrium state µ i for (f, Φ i ) which is a weak Gibbs measure with respect to the family of potentials Φ i , for i = 1, 2. Moreover, from [36,Example 4.6], for any c > 0 the tail of the convergence to the largest or smallest Lyapunov exponent (corresponding respectively to j = 1 or j = 2)…”