On the topological pressure of random bundle transformations in sub-additive case

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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“…[2], [8].) It satisfies a variational principle (see [8] for a proof and [32] for its random version).…”

Section: Resultsmentioning

confidence: 99%

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

Cao

Yun

2012

Ergod. Th. Dynam. Sys.

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“…The last quantity is called the sub-additive topological pressure of Φ with respect to T . We just mention that the above definition is reasonable, see [22] for details. (ω, x)) for some function ϕ ∈ L 1 E (Ω, C(M)), then we call π T (Φ) the random additive topological pressure, see [4,14] for details, denote it simply by π T (ϕ).…”

Section: Random Notationsmentioning

confidence: 99%

“…We first check the new defined potential F • ψ = {f n • ψ} n≥1 is indeed subadditive. Precisely, we have [5]) and lim n→∞ 1 n f n • ψdµ = lim n→∞ 1 n f n dµ ψ , by the variational principle of random sub-additive topological pressure in [22] we have π ϕ 1 (F • ψ) ≤ π ϕ 2 (F ). By symmetry we get the reverse inequality and hence the desired result.…”

Section: Random Notationsmentioning

confidence: 99%

The Hausdorff dimension of average conformal repellers under random perturbation

Yun¹,

Cao²,

Ban

2009

Nonlinearity

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“…For this purpose, in Sections 4 and 5, we will use relative pressure theory. Relative pressure for continuous functions and the relative variational principle (see [18,30]) were extended to subadditive potentials under a certain condition (see [37,35]). Here we state some basic results from [37,35] that we need.…”

Section: Introductionmentioning

confidence: 99%

On factors of Gibbs measures for almost additive potentials

Yayama¹

2014

Ergod. Th. Dynam. Sys.

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Let $(X, \sigma_X), (Y, \sigma_Y)$ be one-sided subshifts with the specification property and $\pi:X\rightarrow Y$ a factor map. Let $\mu$ be a unique invariant Gibbs measure for a sequence of continuous functions $\F=\{\log f_n\}_{n=1}^{\infty}$ on $X$, which is an almost additive potential with bounded variation. We show that $\pi\mu$ is also a unique invariant Gibbs measure for a sequence of continuous functions $\G=\{\log g_n\}_{n=1}^{\infty}$ on $Y$. When $(X, \sigma_X)$ is a full shift, we characterize $\G$ and $\mu$ by using relative pressure. This almost additive potential $\G$ is a generalization of a continuous function found by Pollicott and Kempton in their work on the images of Gibbs measures for continuous functions under factor maps. We also consider the following question: Given a unique invariant Gibbs measure $\nu$ for a sequence of continuous functions $\F_2$ on $Y$, can we find an invariant Gibbs measure $\mu$ for a sequence of continuous functions $\F_1$ on $X$ such that $\pi\mu=\nu$? We show that such a measure exists under a certain condition. If $(X, \sigma_X)$ is a full shift and $\nu$ is a unique invariant Gibbs measure for a function in the Bowen class, then we can find a preimage $\mu$ of $\nu$ which is a unique invariant Gibbs measure for a function in the Bowen class.Comment: 33 pages, To appear in Ergodic Theory and Dynamical System

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On the topological pressure of random bundle transformations in sub-additive case

Abstract: 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.

Cited by 12 publications

References 17 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

The Hausdorff dimension of average conformal repellers under random perturbation

On factors of Gibbs measures for almost additive potentials

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