Given a non-conformal repeller Λ of a C 1+γ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok's approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure. 0

No 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

AbstractWithout any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

In this paper, we define the topological pressure for sub-additive potential via separated sets in random dynamical systems and give a proof of the relativized variational principle for the topological pressure.