arXiv subjects

Laurent Dufloux

Publications and source records attributed to Laurent Dufloux.

6 recordsLinked to original sources

arXiv2019

Radial projections along chains

We state strong Marstrand properties for two related families of fractals in Heisenberg groups $\mathcal{H}^d$: limit sets of Schottky groups in good position, and attractors of self-similar IFS enjoying the open set condition in the quotient $\mathcal{H}^d/Z$. For such a fractal $X$, we show that the dimension of $π_x X$ does not depend on $x \in \mathcal{H}^d$, where $π_x$ denotes the radial projection along chains passing through $x$. This follows from a local entropy averages argument due to Hochman and Shmerkin.

math.MG↗

arXiv2018

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

We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group, endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection $π(E)$ (projection along the center of the Heisenberg group) 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.

math.PR↗

arXiv2018

Linear foliations of complex spheres I. Chains

We provide coordinate-free versions of the classical projection Theorem of Marstrand-Kaufman-Mattila. This allows us to generalize this Theorem to the complex setting; in restriction to complex spheres, we obtain further projection Theorems along so-called complex chains.

math.MG↗

arXiv2017

Hausdorff dimension of limit sets

We exhibit a class of Schottky subgroups of $\mathbf{PU}(1,n)$ ($n \geq 2$) which we call well-positioned and show that the Hausdorff dimension of the limit set $Λ_Γ$ 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 $δ_Γ$. 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.

math.DS↗

arXiv2017

The case of equality in the dichotomy of Mohammadi-Oh

If $n \geq 3$ and $Γ$ is a convex-cocompact Zariski-dense discrete subgroup of $\mathbf{SO}^o(1,n+1)$ such that $δ_Γ=n-m$ where $m$ is an integer, $1 \leq m \leq n-1$, we show that for any $m$-dimensional subgroup $U$ in the horospheric group $N$, the Burger-Roblin measure associated to $Γ$ on the quotient of the frame bundle is $U$-recurrent.

math.DS↗

arXiv2016

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

Let $Γ$ be some discrete subgroup of $\mathbf{SO}^o(n+1,\mathbf{R})$ with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the $N$ component in some Iwasawa decomposition $\mathbf{SO}^o(n+1,\mathbf{R})=KAN$. We also study the dimension of projected Patterson-Sullivan measures along some fixed small circle.

math.DS↗