Conformality for a robust class of non-conformal attractors

Pozzetti, Beatrice; Sambarino, Andrés; Wienhard, Anna

doi:10.48550/arxiv.1902.01303

Cited by 7 publications

(18 citation statements)

References 34 publications

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“…If n ≥ 3 and ρ : π 1 Σ → PGL(n, R) is (1, 1, 2)-hyperconvex in the sense of Pozzetti-Sambarino-Wienhard [10], then we establish in this paper the following Basmajian's identity:…”

Section: Introductionmentioning

confidence: 78%

Identities for hyperconvex Anosov representations

He¹

2019

Preprint

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In this paper, we establish Basmajian's identity for (1, 1, 2)hyperconvex Anosov representations from a free group into PGL(n, R). We then study our series identities on holomorphic families of Cantor non-conformal repellers associated to complex (1, 1, 2)-hyperconvex Anosov representations. We show that the series is absolutely summable if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit nontrivial monodromy.

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“…If n ≥ 3 and ρ : π 1 Σ → PGL(n, R) is (1, 1, 2)-hyperconvex in the sense of Pozzetti-Sambarino-Wienhard [10], then we establish in this paper the following Basmajian's identity:…”

Section: Introductionmentioning

confidence: 78%

Identities for hyperconvex Anosov representations

He¹

2019

Preprint

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“…triangle inequality. On the other hand, ℓ α k is, in many ways, a better generalization of the hyperbolic length function, at least for representation ρ : Γ → SL(E) satisfying property H k : for example it is proven in [PSW19b] that the associated entropy is constant and equal to one, and in [PSW19b, Appendix A] that the pressure metric associated to the first root has, on the Hitchin component, more similarities to the Weyl-Petersson metric than the usual pressure metric. Theorem 1.1 can be reformulated in terms of these geometric quantities:…”

Section: Thus Theorem 11 Yieldsmentioning

confidence: 99%

“…It is also possible to deduce Theorem 1.3 following the lines of Labourie's proof for Hitchin representations [Lab07, Section 4.4] using that, whenever a representation ρ has property H k , the image of its associated boundary map is a C 1 -circle in Gr k (E) [PSW19b,Proposition 8.11]. The argument we provide here is, however, more direct and closer to the circle of ideas important in the rest of the paper.…”

Section: Introductionmentioning

confidence: 99%

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A collar lemma for partially hyperconvex surface group representations

Beyrer¹,

Pozzetti²

2020

Preprint

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We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into SL(n, R) that satisfy partial hyperconvexity properties inspired from Labourie's work. This is the case for several open sets of Anosov representations not contained in higher rank Teichmüller spaces, as well as for Θ-positive representations into SO(p, q) if p ≥ 4. We moreover show that 'positivity properties' known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations. Contents1. Introduction 2 2. Preliminaries 7 2.1. Anosov representations 8 3. Cross ratios 12 3.1. Projective cross ratios 12 3.2. Grassmannian cross ratio 13 3.3. Relations of the cross ratios 15 4. Partial hyperconvexity 15 4.1. Property H k 15 4.2. Property C k 18 4.3. Reducible representations and Fuchsian locii 22 4.4. Θ-positive representations 23 5. Positively ratioed representations 26 6. Proof of the collar lemma 28 7. A counterexample to the strong collar lemma 36 References 39

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“…• The boundary map ξ l has C 1 image for l < k (Proposition 6.18 and [PSW,Proposition 8.11]). • The l-collar lemma holds for l < k − 1 (Definition 2.19, Corollary 6.20 and [BP17, Theorem 1.3]) • The representation ρ is l-positively ratioed for l < k (Definition 2.13, Corollary 6.19).…”

Section: Introductionmentioning

confidence: 99%

Degenerations of k-positive surface group representations

Beyrer,

Pozzetti

2021

Preprint

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We introduce k-positive representations, a large class of {1, . . . , k}-Anosov surface group representations into PGL(E) that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non-discrete representations, but any limit is at least (k − 3)-positive and irreducible limits are (k − 1)-positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations. JONAS BEYRER AND BEATRICE POZZETTI 6.2. k-positive representations 27 6.3. Transversality properties of k-positive representations 28 6.4. k-tridirect representations 30 7. Degenerations of k-positive representations 31 7.1. Anosov limits of k-positive representations 31 7.2. Limits of k-positive representations 32 7.3. Coherence of the boundary map in the limit 33 8. Examples and open questions 36 8.1. All transversality could get lost in the limit 37 8.2. The set of positively ratioed representations is not open 37 8.3. Limits of k-positive representations 38 References 39

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Conformality for a robust class of non-conformal attractors

Cited by 7 publications

References 34 publications

Identities for hyperconvex Anosov representations

Identities for hyperconvex Anosov representations

A collar lemma for partially hyperconvex surface group representations

Degenerations of k-positive surface group representations

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