Deux caractérisations de la mesure d'équilibre d'un endomorphisme de P k (C)

Briend, Jean-Yves; Duval, Jean-Luc

doi:10.1007/s10240-001-8190-4

Cited by 79 publications

(90 citation statements)

References 11 publications

(15 reference statements)

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“…Let us see how we deduce (6), thus completing the proof of Theorem 2.1. Since the sets of P n (x) are pairwise disjoint, we have:…”

Section: Proposition 57mentioning

confidence: 99%

“…Fornaess and Sibony [14] proved that μ is mixing and that log Jac f ∈ L 1 (μ). Briend and Duval [5] established that the exponents of μ are bounded below by log √ d and that μ is the unique measure of maximal entropy (h μ = log d k ) [6]. Concerning the Hausdorff dimension of μ, Mañé's formula asserts that dim H μ = log d/λ when k = 1.…”

Section: Application To the Equilibrium Measure μ Of Fmentioning

confidence: 99%

See 1 more Smart Citation

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Dupont

2010

Math. Ann.

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Let f be an endomorphism of CP k and ν be an f -invariant measure with positive Lyapunov exponents (λ 1 , . . . , λ k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f , the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim H μ ≥ log d λ 1 + log d λ 2 , which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f n in CP k . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in CP k and a normalization theorem for the ν-generic inverse branches of f n .

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“…Let us see how we deduce (6), thus completing the proof of Theorem 2.1. Since the sets of P n (x) are pairwise disjoint, we have:…”

Section: Proposition 57mentioning

confidence: 99%

Section: Application To the Equilibrium Measure μ Of Fmentioning

confidence: 99%

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Dupont

2010

Math. Ann.

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“…This exceptional set is still not well understood, its study seems to require new ideas. Very recently, Briend-Duval [2] showed that using a dynamical argument, as in our work [6,8,10], one can obtain a short proof of Corollary 1.4, see also [6,Prop. 2.4] for a more general setting.…”

Section: Introductionmentioning

confidence: 99%

Equidistribution speed for endomorphisms of projective spaces

Dinh

Sibony

2009

Math. Ann.

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“…Fornaess-Sibony proved that µ is mixing [22] and Briend-Duval that µ is the unique measure of maximal entropy [7].…”

mentioning

confidence: 99%

Bernoulli coding map and almost sure invariance principle for endomorphisms of $${\mathbb P^k}$$

Dupont

2008

Probab. Theory Relat. Fields

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Let f be an holomorphic endomorphism of P k and µ be its measure of maximal entropy. We prove an almost sure invariance principle for the systems (P k , f, µ). Our class U of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map ω : ( , s, ν) → (P k , f, µ). We obtain the invariance principle for an observable ψ on (P k , f, µ) by applying Philipp-Stout's theorem for χ = ψ • ω on ( , s, ν). The invariance principle implies the central limit theorem as well as several statistical properties for the class U. As an application, we give a direct proof of the absolute continuity of the measure µ when it satisfies Pesin's formula. This approach relies on the central limit theorem for the unbounded observable log Jac f ∈ U.

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Deux caractérisations de la mesure d'équilibre d'un endomorphisme de P k (C)

Cited by 79 publications

References 11 publications

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Equidistribution speed for endomorphisms of projective spaces

Bernoulli coding map and almost sure invariance principle for endomorphisms of $${\mathbb P^k}$$

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