Pressures for asymptotically sub-additive potentials under a mistake function

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.

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“…So, Theorem A extends the results in [15,22] to the subadditive case. Also, the last equality was proved in [11]. [22].…”

Section: Remark 22mentioning

confidence: 80%

Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

Cao

Yun

2012

Ergod. Th. Dynam. Sys.

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“…A generalization of formulas (2.3) and (2.4) to measure theoretic pressure was given in [13] and [14]. We refer the reader to [4,12,21,29] for the proof of this variational principle and further details on topological pressure of non-additive potentials.…”

Section: 2mentioning

confidence: 98%

Weak specification properties and large deviations for non-additive potentials

Varandas

Yun

2013

Ergod. Th. Dynam. Sys.

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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

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“…In 2009, Zhao and Cao [15] gave a definition of measure-theoretic pressure of sub-additive potentials for ergodic measures, and generalized the above results in [13] and [6]. Moreover, we refer to [2,1] for more pressure versions of Katok's entropy formula. In 2009, Zhu [17] established Katok's entropy formula in the case of random dynamical systems.…”

Section: Introductionmentioning

confidence: 93%

Katok's Entorpy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

Huang,

Chen,

Wang

2018

Preprint

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Pressures for asymptotically sub-additive potentials under a mistake function

Cited by 28 publications

References 23 publications

Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

Weak specification properties and large deviations for non-additive potentials

Katok's Entorpy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

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