We exhibit a class of Schottky subgroups of PU(1, n) (n ≥ 2) which we call well-positioned and show that the Hausdorff dimension of the limit set ΛΓ associated with such a subgroup Γ, with respect to the spherical metric on the boundary of complex hyperbolic n-space, is equal to the growth exponent δΓ.For general Γ we establish (under rather mild hypotheses) a lower bound involving the dimension of the Patterson-Sullivan measure along boundaries of complex geodesics.Our main tool is a version of the celebrated Ledrappier-Young theorem.

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.

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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We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group H = C × R, endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection π(E) (projection along the center of H) almost surely equals min{2, dim H (E)} and that π(E) has non-empty interior if dim 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 H (E).We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere S 3 endowed with the visual metric d obtained by identifying S 3 with the boundary of the complex hyperbolic plane. In 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 S 3 satisfy a strong version of the Marstrand's projection theorem, without any exceptional directions.