Dimension of non-conformal repellers: a survey

Chen, Jianyu; Pesin, Yakov

doi:10.1088/0951-7715/23/4/r01

Cited by 38 publications

(34 citation statements)

References 37 publications

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“…In fact, there are many settings in which s(Λ) is equal to the Hausdorff dimension. See Section 5 and a survey [CP10] for more details on the number s(Λ). From its definition, it follows that s(Λ) varies upper semi-continuously under small perturbations of the repeller Λ.…”

mentioning

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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confidence: 99%

Quasi-multiplicativity of Typical Cocycles

Park

2020

Commun. Math. Phys.

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“…Recently, in [9] the authors introduced the super-additive topological pressure, and showed that the zero of super-additive topological pressure gives a lower bound of the Hausdorff dimension of repellers. We refer the reader to [10] and [5] for a detailed description of the recent progress in dimension theory of dynamical systems.…”

Section: Introductionmentioning

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 question of which dynamical systems have ergodic invariant measures of full Hausdorff dimension has generated substantial interest over the past few decades, see e.g. [4,9,17,18,21,24,25,27,30,31,32,33,42,45,52,55], as well as the survey articles [7,13,22,51] and the books [5,6]. Most of the results are positive, proving the existence and uniqueness of a measure of full dimension under appropriate hypotheses on the dynamical system.…”

Section: Introductionmentioning

confidence: 99%

“…Progress beyond these cases, and in particular in the case where the system is expanding but may have different rates of expansion in different directions, has been much slower and of more limited scope, see e.g. [7,13,22,51]. Such systems, called "expanding repellers", form another large and much-studied class of examples.…”

Section: Introductionmentioning

confidence: 99%

The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result

Das

Simmons

2017

Invent. math.

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We construct a self-affine sponge in R 3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space.

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Dimension of non-conformal repellers: a survey

Cited by 38 publications

References 37 publications

Quasi-multiplicativity of Typical Cocycles

Quasi-multiplicativity of Typical Cocycles

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

The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result

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