The Hausdorff dimension of average conformal repellers under random perturbation

Yun, Zhao; Cao, Yongluo; Ban, Jung-Chao

doi:10.1088/0951-7715/22/10/005

Cited by 9 publications

(7 citation statements)

References 20 publications

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“…We just remark that the dimension of average conformal repeller can be computed by the zero of the sub-additive topological pressure, see [16] for details. And authors have proved that the Hausdorff dimension of average conformal repeller is stable under random perturbation, see [17] for details.…”

Section: Remarkmentioning

confidence: 99%

The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials

Yun

Zhang

Cao

2011

Nonlinear Analysis: Theory, Methods & Applications

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Section: Remarkmentioning

confidence: 99%

The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials

Yun

Zhang

Cao

2011

Nonlinear Analysis: Theory, Methods & Applications

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“…The formula in the previous theorem was first established by Barreira and Wolf for conformal repellers of a C 1+α map (see [2]). Here, we relax the smoothness to C 1 and extend their result for average conformal repellers which are indeed non-conformal (see [25] for an example).…”

Section: 2mentioning

confidence: 81%

Measure theoretic pressure and dimension formula for non-ergodic measures

Fang¹,

Cao²,

Yun³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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This paper first studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carathéodory-Pesin structure described in [13], and show that this quantity is equal to the essential supremum of the free energy of the measures in an ergodic decomposition. To the best of our knowledge, this formula is new even for entropy. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the free energy if the measure is ergodic. Particularly, if the dynamical system satisfies the uniform separation condition and the ergodic measures are entropy dense, this quantity is still equal to the the free energy even if the measure is non-ergodic. As an application of the main result, we find that the Hausdorff dimension of an invariant measure supported on an average conformal repeller is given by the zero of the measure theoretic pressure of this measure. Furthermore, if a hyperbolic diffeomorphism is average conformal and volume-preserving, the Hausdorff dimension of any invariant measure on the hyperbolic set is equal to the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions. 0 N →∞ R(Z, ϕ, α, N, ǫ), r(Z, ϕ, α, ǫ) = lim sup N →∞ R(Z, ϕ, α, N, ǫ) and define the jump-up points of r(Z, ϕ, α, ǫ) and r(Z, ϕ, α, ǫ) as CP Z (f, ϕ, ǫ) = inf{α : r(Z, ϕ, α, ǫ) = 0} = sup{α : r(Z, ϕ, α, ǫ) = +∞}, CP Z (f, ϕ, ǫ) = inf{α : r(Z, ϕ, α, ǫ) = 0} = sup{α : r(Z, ϕ, α, ǫ) = +∞} respectively. Definition 2.2. We call the quantities CP Z (f, ϕ) = lim ǫ→0 CP Z (f, ϕ, ǫ) and CP Z (f, ϕ) = lim ǫ→0 CP Z (f, ϕ, ǫ) the lower and upper topological pressures of (f, ϕ) on the set Z.

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“…Ban 等 [9] 引入了一类特殊的非共形排斥子-平均共形排斥子, 这推广了文献 [1,27] 中拟共形排斥子和渐近共形排斥子的概念. 有例子表明存在非共形的平均共形排斥子, 参见文献 [45]. 特别地, Ban 等 [9] 得到了关于平均共形排斥子维数的 Bowen 方程公式.…”

Section: 平均共形排斥子unclassified

“…Bogenschütz 和 Ochs [47] 证明了共形排斥子的 Hausdorff 维数在随机扰动下是稳定的. 进一步, Zhao 等 [45] 证明了平均共形排斥子的 Hausdorff 维数在随机扰动下也是稳定的. 最近, Feng 和 Shmerkin [48] 证明了自仿情形下次可加拓扑压的连续性, 进而得到了自仿集维数的连续性.…”

Section: 非共形排斥子上拓扑压的连续性unclassified

Topological pressure and dimension estimation

YongLuo¹,

Yun²

2016

Sci. Sin.-Math.

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In this paper, we give a survey of recent results in the dimension theory of dynamical systems, with emphasis on the dimension of repellers and hyperbolic sets and the dimension of invariant measures. In the case of conformal dynamics, the theory is completely well understood. However, it still lacks today a satisfactory general approach for the non-conformal case, although a number of interesting and nontrivial developments have been obtained. Indeed, we only well understand some particular non-conformal repellers, e.g., generalized Sierpiński carpets and average conformal repellers.

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The Hausdorff dimension of average conformal repellers under random perturbation

Abstract: We prove that the Hausdorff dimension of an average conformal repeller is stable under random perturbations. Our perturbation model uses the notion of a bundle random dynamical system.

Cited by 9 publications

References 20 publications

The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials

The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials

Measure theoretic pressure and dimension formula for non-ergodic measures

Topological pressure and dimension estimation

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