Hausdorff dimension of limit sets

Dufloux, Laurent

doi:10.1007/s10711-017-0240-2

Cited by 8 publications

(16 citation statements)

References 21 publications

(19 reference statements)

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“…We will denote also these subspaces of BH d H chains for notational ease. Arguments analogue to the ones presented in [21,Section 7.2] imply that well positioned Schottky groups are hyperconvex representations: Proposition 8.6. Let ρ : Γ Ñ PO K p1, dq be a well positioned Schottky subgroup.…”

Section: Examples Of Locally Conformal Representationsmentioning

confidence: 92%

“…Furthermore we know that for every element g P PO K p1, dq, we have p 2 pgq " d, and hence every point in BΓ is locally conformal for β. Theorem 5.14 then applies and gives the second statement. [20,21]. He says that a Schottky subgroup Γ ă PUp1, dq generated by a symmetric set W is well positioned if, for every w P W there is an open subsets Bpwq Ă BH d C such that ‚ the closures Bpwq are pairwise disjoint; ‚ wpBH d C zBpw ´1qq Ă Bpwq; ‚ no chain passes through three of these open subsets Bpwq.…”

Section: Examples Of Locally Conformal Representationsmentioning

confidence: 99%

“…Let us also note that only very recently Bárány-Hochman-Rapaport [2] provided a complete answer for Iterated-Function-Systems on the plane. On the discrete groups side, Dufloux [21] has studied a class of Schottky subgroups of isometries of the complex hyperbolic n-space, that he calls well positioned, and proves the analogue of Sullivan's result for the Hausdorff dimension of the limit set with respect to any Riemannian metric.…”

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confidence: 95%

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Conformality for a robust class of non-conformal attractors

Pozzetti¹,

Sambarino²,

Wienhard³

2019

Preprint

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In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent. In the appendix, in collaboration with M. Bridgeman, we extend a classical result on the Hessian of the Hausdorff dimension on purely imaginary directions. Contents 42 Appendix A. Continuity of entropy for non-Archimedean Anosov representations 46 Appendix B. Hessian of Hausdorff dimension on pure imaginary directions 49 References 54 (Cited on pages 5 and 35.

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Section: Examples Of Locally Conformal Representationsmentioning

confidence: 92%

Section: Examples Of Locally Conformal Representationsmentioning

confidence: 99%

mentioning

confidence: 95%

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Conformality for a robust class of non-conformal attractors

Pozzetti¹,

Sambarino²,

Wienhard³

2019

Preprint

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“…Remark 4•9. Other families of fractals that enjoy equality in equation ( 4•3) can be found in [2] ("horizontal fractals") and [5] (limit sets of Schottky groups in "good position" at the boundary of the complex hyperbolic plane).…”

Section: Consider the Continuous Piecewise Linear Functionmentioning

confidence: 99%

Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere

Dufloux¹,

Суомала²

2021

Math. Proc. Camb. Phil. Soc.

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We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group \[\mathbb{H} = \mathbb{C} \times \mathbb{R}\] , endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection \[\pi (E)\] (projection along the center of \[\mathbb{H}\] ) almost surely equals \[\min \{ 2,{\dim _\operatorname{H} }(E)\} \] and that \[\pi (E)\] has non-empty interior if \[{\dim _{\text{H}}}(E) > 2\] . As a corollary, this allows us to determine the Hausdorff dimension of E with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension \[{\dim _{\text{H}}}(E)\] . We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere \[{{\text{S}}^3}\] endowed with the visual metric d obtained by identifying \[{{\text{S}}^3}\] with the boundary of the complex hyperbolic plane. In \[{{\text{S}}^3}\] , we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in \[{{\text{S}}^3}\] satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.

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“…Let us now describe briefly the construction of this mapping σ. For more details, we refer the reader to [2] or [3].…”

Section: General Factsmentioning

confidence: 99%

Projections of Patterson-Sullivan measures and the dichotomy of Mohammadi-Oh

Dufloux

2017

Isr. J. Math.

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Let Γ be some discrete subgroup of SO o (n + 1, R) with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure measure with respect to closed connected subspaces of the N component in some Iwasawa decomposition SO o (n + 1, R) = KAN . We also study the dimension of projected Patterson-Sullivan measures along some fixed direction.Definition. Let µ be some (Borel) probability measure on R n (n ≥ 2). Assume that µ has exact dimension δ. We say that µ is regular if for any m-plane V in R n (1 ≤ m ≤ n − 1) the orthogonal projection of µ onto V has dimension inf{δ, m} almost everywhere.Obviously, if this is the case, then the orthogonal projection of µ onto V is in fact exact dimensional.Theorem B (Theorem 5.2). Let Γ be a discrete non-elementary subgroup of G = SO o (1, n + 1). Assume that Γ is Zariski-dense and has finite BMS measure. Let µ be the Patterson-Sullivan measure (of exponent δΓ) associated with Γ. For µ-almost every ξ ∈ ∂H n+1 , the push-forward of µ through the inverse stereographic mapping ∂H n+1 \ {ξ} → R n is a regular measure.The proof of Theorem B relies on results of Hillel Furstenberg, Pablo Shmerkin and Michael Hochman.

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Hausdorff dimension of limit sets

Cited by 8 publications

References 21 publications

Conformality for a robust class of non-conformal attractors

Conformality for a robust class of non-conformal attractors

Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere

Projections of Patterson-Sullivan measures and the dichotomy of Mohammadi-Oh

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