The thermodynamic formalism for sub-additive potentials

Cao, Yongluo; Ding, Feng; Huang, Wen

doi:10.3934/dcds.2008.20.639

Cited by 145 publications

(151 citation statements)

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“…By the variational principle for sub-additive topological pressure (see [12]), we have that P (A, s) = sup h µ (T ) + lim n→∞ 1 n log ϕ s (A(y, n)) dµ .…”

Section: Application: a Comment On Feng And Shmerkin's Resultsmentioning

confidence: 99%

“…is called the sub-additive topological pressure of Φ. One can show (see [12]) that it satisfies the following variational principle:…”

Section: Markov Partitions and Gibbs Measuresmentioning

confidence: 99%

“…In [54], Zhang introduced a new version of Bowen's equation which involves the limit of a sequence of topological pressures for singular valued potentials, and proved that the unique solution of this equation is an upper bound of the Hausdorff dimension of repellers. Finally, in [2], using thermodynamic formalism for sub-additive potentials developed in [12], the authors showed that the zero of the topological pressure of sub-additive singular valued potentials Φ f (t) = {−ϕ t (·, f n )} n≥1 (see precise definition in Section 3.1) gives an upper bound of the Hausdorff dimension of repellers, and furthermore, that the upper bounds obtained in the previous works [2,19,54] are all equal. We refer the reader to [14] and [7] for a detailed description of the recent progress in dimension theory of dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

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Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

2019

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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

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“…By the variational principle for sub-additive topological pressure (see [12]), we have that P (A, s) = sup h µ (T ) + lim n→∞ 1 n log ϕ s (A(y, n)) dµ .…”

Section: Application: a Comment On Feng And Shmerkin's Resultsmentioning

confidence: 99%

“…is called the sub-additive topological pressure of Φ. One can show (see [12]) that it satisfies the following variational principle:…”

Section: Markov Partitions and Gibbs Measuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

2019

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“…The singular value potential Φ A is an example of a subadditive potential Ψ = {log ψ n } n∈N which can be thought of as a generalization of the Birkhoff sum S n ψ for a potential ψ ∈ C(Σ, R). The usual thermodynamical notions of the pressure and the equilibrium states of a potential ψ extend to subadditive potentials [CFH08].…”

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confidence: 99%

Quasi-multiplicativity of Typical Cocycles

Park

2020

Commun. Math. Phys.

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We show that typical (in the sense of [BV04] and [AV07]) Hölder and fiber-bunched GL d (R)-valued cocycles over a subshift of finite type are uniformly quasimultiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles.

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“…The existence of the above limit follows from a sub-additive argument. The authors in [8] proved the following variational principle.…”

Section: Furthermore a Sequence Of Continuous Functionsmentioning

confidence: 99%

Dimensions of $ C^1- $average conformal hyperbolic sets

Wang¹,

Wang²,

Cao³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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This paper introduces the concept of average conformal hyperbolic sets, which admit only one positive and one negative Lyapunov exponents for any ergodic measure. For an average conformal hyperbolic set of a C 1 diffeomorphism, utilizing the techniques in sub-additive thermodynamics formalism and some geometric arguments with unstable/stable manifolds, a formula of the Hausdorff dimension and lower (upper) box dimension is given in this paper, which are exactly the sum of the dimensions of the restriction of the hyperbolic set to a stable and unstable manifolds. Furthermore, the dimensions of an average conformal hyperbolic set varies continuously with respect to the

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The thermodynamic formalism for sub-additive potentials

Cited by 145 publications

References 30 publications

Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

Quasi-multiplicativity of Typical Cocycles

Dimensions of $ C^1- $average conformal hyperbolic sets

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