arXiv subjects

Andrés Sambarino

Publications and source records attributed to Andrés Sambarino.

13 recordsLinked to original sources

arXiv2026

On local minimality of Hausdorff dimension for round limit sets

We address a question of Canary--Taylor and Canary--Minsky--Taylor for sufficiently symmetric compact hyperbolic $3$-orbifolds with totally geodesic boundary. We give sufficient conditions on the symmetries of such an orbifold to show that, among the complete convex cocompact structures on the interior, the structure with Fuchsian ends is a critical point (respectively, a strict local minimum) for the critical exponent. We also provide an infinite family of non-rigid examples fulfilling these conditions.

math.GT↗

arXiv2025

Asymptotic properties of infinitesimal characters and applications

Inspired by Benoist, we study objects linked to integrable tangent vectors on the character variety of a semi-group $Γ$ with values in a semi-simple real-algebraic group $\mathsf G$. We prove the \emph{cone of Jordan variations} has non-empty interior and, when $\mathsf G$ is split, establish non-empty interior of the set of \emph{length-normalized variations}. We apply these techniques to pressure forms on Anosov representations and higher-rank Teichmüller spaces. We identify an explicit functional $φ\in\mathfrak a^*$ whose pressure form is compatible with Goldman's symplectic form at Fuchsian points in the Hitchin component. Finally, we show the degeneration of the Hausdorff dimension of \emph{higher}-quasi-circles is governed by a Diophantine equation.

math.GR↗

arXiv2023

Metric properties of boundary maps, Hilbert entropy and non-differentiability

We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal F(\mathbb R^3)$. Generalizations for regularity properties of boundary maps between locally conformal representations are also discussed. An ingredient for the proofs is the concept of hyperplane conicality that we introduce for a $θ$-Anosov representation into a reductive real-algebraic Lie group $G$. In contrast with directional conicality, hyperplane-conical points always have full mass for the corresponding Patterson-Sullivan measure.

math.GR↗

arXiv2023

Infinitesimal Zariski closures of positive representations

We classify the (semi-simple parts of the) Lie algebra of the Zariski closure of a discrete subgroup of a split simple real-algebraic Lie group, whose limit sets are minimal and such that the limit set in the space of full flags contains a positive triple of flags (as in Lusztig). We then apply our result to obtain a new proof of Guichard's classification of Zariski closures of Hitchin representations into $PSL_d(\mathbb{R}).$

math.RT↗

arXiv2022

A report on an ergodic dichotomy

We establish (some directions) of a Ledrappier correspondence between Hölder cocycles, Patterson-Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson-Sullivan measures for $θ$-Anosov representations over a local field and show that these are parametrized by the $θ$-critical hypersurface of the representation. We use these Patterson-Sullivan measures to establish a dichotomy concerning directions in the interior of the $θ$-limit cone of the representation in question: if $u$ is such a half-line, then the subset of $u$-conical limit points has either total-mass if $|θ|\leq2$ or zero-mass if $|θ|\geq4.$ The case $|θ|=3$ remains unsettled.

math.DS↗

arXiv2021

Anosov representations with Lipschitz limit set

We study Anosov representations whose limit set has intermediate regularity, namely is a Lipschitz submanifold of a flag manifold. We introduce an explicit linear functional, the unstable Jacobian, whose orbit growth rate is integral on this class of representations. We prove that many interesting higher rank representations, including $Θ$-positive representations, belong to this class, and establish several applications to rigidity results on the orbit growth rate in the symmetric space.

math.DG↗

arXiv2020

Hessian of Hausdorff dimension on purely imaginary directions

We extend classical results of Bridgeman-Taylor and McMullen on the Hessian of the Hausdorff dimension on quasi-Fuchsian space to the class of (1,1,2)-hyperconvex representations, a class introduced in arXiv:1902.01303 which includes small complex deformations of Hitchin representations and of $Θ$-positive representations. We also prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion $Γ\to PO(n,1) \to PU(n,1)$ is positive definite when $Γ$ is co-compact in $PO(n,1)$ (unless $n=2$ and the deformation is tangent to $\mathfrak{X}(Γ,PO(2,1))).$

math.DG↗

arXiv2020

Conformality for a robust class of non-conformal attractors

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.

math.DG↗

arXiv2020

Anosov representations and dominated splittings

We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Guéritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti by different methods. We also give characterizations in terms of multicones and cone-types inspired in the work of Avila-Bochi-Yoccoz and Bochi-Gourmelon. Finally we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti.

math.GR↗

arXiv2017

An introduction to pressure metrics on higher Teichmüller spaces

We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman-Canary-Labourie-Sambarino "\emph{The pressure metric for Anosov representations}" and discusses questions and open problems which arise.

math.DG↗

arXiv2017

Eigenvalues and Entropy of a Hitchin representation

We show that the critical exponent of a representation in the Hitchin component of $PSL(d,\mathbb{R})$ is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $ρ\backslash X,$ where $X$ is the symmetric space of $PSL(d,\mathbb{R}).$ The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of $ρ$ is smooth, then $ρ$ is Fuchsian.

math.GR↗

arXiv2014

On entropy, regularity and rigidity for convex representations of hyperbolic manifolds

Given a convex representation $ρ:Γ\to\textrm{PGL}(d,\mathbb{R})$ of a convex co-compact group $Γ$ of $\mathbb{H}^k$ we find upper bounds for the quantity $αh_ρ,$ where $h_ρ$ is the entropy of $ρ$ and $α$ is the Hölder exponent of the equivariant map $\partialΓ\to\mathbb{P}(\mathbb{R}^d).$ We also give rigidity statements when the upper bound is attained. We then study Hitchin representations and prove that if $ρ:π_1Σ\to\textrm{PSL}(d,\mathbb{R})$ is in the Hitchin component then $αh_ρ\leq 2/(d-1)$ (where $α$ is the Hölder exponent of the map $ζ:\partial\mathbb{H}^2\to\mathscr{F}$) with equality if and only if $ρ$ is Fuchsian.

math.GR↗

arXiv2012

Quantitative properties of convex representations

Let $Γ$ be a discrete subgroup of $\textrm{PGL}(d,\R)$ and fix some euclidean norm $\|\ \|$ on $\R^d.$ Let $N_Γ(t)$ be the number of elements in $Γ$ whose operator norm is $\leq t.$ In this article we prove an asymptotic for the growth of $N_Γ(t)$ when $t\to\infty$ for a class of $Γ$'s which contains, in particular, Hitchin representations of surface groups and groups dividing a convex set of $¶(\R^d).$ We also prove analogue counting theorems for the growth of the spectral radii. More precise information is given for Hitchin representations.

math.GR↗