Hausdorff dimension of limit sets for projective Anosov representations

Glorieux, Olivier; Monclair, Daniel; Tholozan, Nicolas

doi:10.48550/arxiv.1902.01844

Cited by 5 publications

(9 citation statements)

References 18 publications

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“…In this section, we show that critical exponents and entropies agree for cusped P θ -Anosov representations of geometrically finite Fuchsian groups. This generalizes results of Glorieux-Montclair-Tholozan [27,Thm. 3.1] and Pozzetti-Sambarino-Weinhard [43,Prop.…”

Section: Critical Exponents and Entropiessupporting

confidence: 90%

“…If k ∈ θ and Γ is P θ -transverse, we define Λ k,c (Γ) = π k (Λ θ,c (Γ)), where π k : F θ → Gr k (R d ) is the projection map. Our result generalizes work of Glorieux-Montclair-Tholozan [27] and Pozzetti-Sambarino-Wienhard [43] in the Anosov setting.…”

supporting

confidence: 86%

“…As a consequence we obtain the following generalization of results of Burger [16], Glorieux-Montclair-Tholozan [27] and Kim-Minsky-Oh [34] for pairs of convex cocompact representations.…”

supporting

confidence: 68%

“…We obtain upper bounds on the Hausdorff dimensions of conical limit sets of P k -transverse groups, generalizing results of Glorieux-Montclair-Tholozan [27] and Pozzetti-Sambarino-Wienhard [43] from the Anosov setting.…”

supporting

confidence: 66%

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Entropy rigidity for cusped Hitchin representations

Canary¹,

Zhang²,

Zimmer³

2022

Preprint

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We establish an entropy rigidity theorem for Hitchin representations of all geometrically finite Fuchsian groups which generalizes a theorem of Potrie and Sambarino for Hitchin representations of closed surface groups. In the process, we introduce the class of (1, 1, 2)hypertransverse groups and show for such a group that the Hausdorff dimension of its conical limit set agrees with its (first) simple root entropy, providing a common generalization of results of Bishop and Jones, for Kleinian groups, and Pozzetti, Sambarino and Wienhard, for Anosov groups. We also introduce the theory of transverse representations of projectively visible groups as a tool for studying discrete subgroups of linear groups which are not necessarily Anosov or relatively Anosov.

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Section: Critical Exponents and Entropiessupporting

confidence: 90%

supporting

confidence: 86%

“…As a consequence we obtain the following generalization of results of Burger [16], Glorieux-Montclair-Tholozan [27] and Kim-Minsky-Oh [34] for pairs of convex cocompact representations.…”

supporting

confidence: 68%

supporting

confidence: 66%

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Entropy rigidity for cusped Hitchin representations

Canary¹,

Zhang²,

Zimmer³

2022

Preprint

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“…While working on this article, we came to know about two recent developments by Pozzetti-Sambarino-Wienhard ( [PSW19]) and Glorieux-Monclair-Tholozan ( [GMT19]) which are related to our work. In these articles, the authors proved that the Hausdorff dimension of the limit set of a projective Anosov subgroup Γ in the real projective space is bounded above by a certain critical exponent, called the "simple root critical exponent" in the second article.…”

mentioning

confidence: 99%

Patterson-Sullivan theory for Anosov subgroups

Dey,

Kapovich

2019

Preprint

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We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.Let G be a noncompact real semisimple Lie group, X = G/K be the associated symmetric space and Γ be a τ mod -Anosov subgroup of G. We will be assuming several conditions on G and X; they are labeled as "assumption" in Section 1. We consider two types of G-invariant (pseudo-)metrics on X, namely, one is the Riemannian metric of the symmetric space X and the other one is Finslerian. The critical exponents of Γ with respect to these two metrics, denoted by δ R and δ F , respectively, are defined in the usual fashion, i.e., as the exponents of convergence of associated Poincaré series (see Section 2). Using the classical construction of Patterson, we define a Γ-invariant conformal density on the flag limit set of Γ (see Section 3).Throughout this paper, the Finsler metric is given more emphasis than its Riemannian counterpart. For example, the construction of the above mentioned Patterson-Sullivan density is carried out in terms of the Finsler metric. The main reason for this choice is that Finsler metrics reflect the asymptotic geometry of Γ better than the Riemannian metric.We should note that many of the results in this paper are often proven for more general classes of discrete subgroups of G with the hope that the results may be useful, for instance, in the study of relatively Anosov subgroups.1 Regarding Anosov subgroups, the main results of this paper are summarized below.Let σ mod be a maximal simplex in the Tits building of X, ι : σ mod → σ mod be the opposition involution, τ mod be an ι-invariant face of σ mod , P be the maximal parabolic subgroup of G that stabilizes τ mod , and Flag(τ mod ) = G/P be the partial flag manifold associated to the face τ mod (see Subsection 1.2). Main theorem.Let Γ be a nonelementary τ mod -Anosov subgroup of G and δ F be the Finsler critical exponent for the action of Γ on the symmetric space X = G/K. Then the Patterson-Sullivan density µ (constructed with respect to the Finsler metric on X) on the flag limit set Λ τ mod (Γ) ⊂ Flag(τ mod ) is the unique (up to a constant factor) Γ-invariant conformal density. Moreover, (i) The density µ is non-atomic and its dimension equals to δ F .(ii) The support of µ is Λ τ mod (Γ) and the action Γ Λ τ mod (Γ) is ergodic with respect to µ.(iii) The critical exponent δ F (as well as the Riemannian critical exponent δ R ) is positive and finite.(iv) The Poincaré series of Γ diverges at the critical exponent δ F . In other words, Γ has (Finsler) divergence type.(v) The δ F -dimensional Hausdorff measure on Λ τ mod (Γ) with respect to a Gromov (pre-)metric2 is a member of a Γ-invariant confo...

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A report on an ergodic dichotomy

Sambarino

2023

Ergod. Th. Dynam. Sys.

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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 $\vartheta $ -Anosov representations over a local field and show that these are parameterized by the $\vartheta $ -critical hypersurface of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the $\vartheta $ -limit cone of the representation in question: if ${\mathsf {u}}$ is such a half-line, then the subset of ${\mathsf {u}}$ -conical limit points has either total mass if $|\vartheta |\leq 2$ or zero mass if $|\vartheta |\geq 4.$ The case $|\vartheta |=3$ remains unsettled.

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Hausdorff dimension of limit sets for projective Anosov representations

Cited by 5 publications

References 18 publications

Entropy rigidity for cusped Hitchin representations

Entropy rigidity for cusped Hitchin representations

Patterson-Sullivan theory for Anosov subgroups

A report on an ergodic dichotomy

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