Collet, Eckmann and the bifurcation measure

Astorg, Matthieu; Gauthier, Thomas; Mihalache, N; Vigny, Gabriel

doi:10.1007/s00222-019-00874-5

Cited by 17 publications

(23 citation statements)

References 24 publications

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“…Theorem 1 of [19] and its revision, Theorem 1 of the present note, include or imply many of the previous results in this direction, e.g. in [29], [16], [23], [30], [5], and have found new applications in [12], [1], [3].…”

supporting

confidence: 61%

Fixed points of the Ruelle–Thurston operator and the Cauchy transform

Levin

2021

Fund. Math.

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supporting

confidence: 61%

Fixed points of the Ruelle–Thurston operator and the Cauchy transform

Levin

2021

Fund. Math.

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“…Gauthier [59] extended Shishikura's theorem to show that Supp(µ bif ) has maximal Hausdorff dimension at each of its points. Let us also note that by using advanced non-uniform hyperbolicity techniques, it was shown by Astorg, Gauthier, Mihalache and Vigny [6] that in the space M d of rational maps of degree d, Supp(µ bif ) has positive volume.…”

Section: Sketch Of Proofmentioning

confidence: 99%

Geometric methods in holomorphic dynamics

Dujardin

2021

Preprint

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In this note we review a selection of contemporary research themes in holomorphic dynamics. The main topics that will be discussed are: geometric (laminar and woven) currents and their applications, bifurcation theory in one and several variables, and the problem of wandering Fatou components.that one-dimensional rational maps are generically structurally stable, using surprisingly elementary arguments. For the quadratic family z 2 + c, c ∈ C, the bifurcation locus is the celebrated Mandelbrot set, whose intricate structure was thoroughly studied since then, using a variety of combinatorial and geometric methods. This research area was profoundly renewed in the 2000's by the systematic investigation of higher dimensional phenomena, and in particular with the introduction of bifurcation currents by DeMarco [33]. The bifurcation theory of holomorphic dynamical systems is nowadays a very active research domain, and a meeting point between the communities of one and several variable dynamicists. We relate this continuing story in Section 2.Finally, one recent breakthrough is the construction of wandering Fatou components in higher dimensional polynomial dynamics, which at the same time solves an old problem and raises many questions. We review these recent developments in Section 3.Let us conclude this introduction with a little notice: some important theorems will be mentioned only in passing, while other are isolated within numbered environments: this is meant to keep the reading flow, not to reflect a hierarchy of importance. Likewise, the list of references is already quite long, but not exhaustive, and we apologize in advance for any serious omission.

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“…To conclude the proof of Theorem 2.3, we rely on the following purely dynamical result, which is an immediate adaptation of [AGMV,Lemma 3.5].…”

Section: Properly Prerepelling Marked Points Bifurcatementioning

confidence: 99%

Dynamical pairs with an absolutely continuous bifurcation measure

Gauthier

2018

Preprint

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In this article, we study algebraic dynamical pairs (f, a) parametrized by an irreducible quasi-projective curve Λ having an absolutely continuous bifurcation measure. We prove that, if f is non-isotrivial and (f, a) is unstable, this is equivalent to the fact that f is a family of Lattès maps. To do so, we prove the density of transversely prerepelling parameters in the bifucation locus of (f, a) and a similarity property, at any transversely prerepelling parameter λ0, between the measure µ f,a and the maximal entropy measure of f λ 0 . We also establish an equivalent result for dynamical pairs of P k , under an additional assumption.

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Collet, Eckmann and the bifurcation measure

Cited by 17 publications

References 24 publications

Fixed points of the Ruelle–Thurston operator and the Cauchy transform

Fixed points of the Ruelle–Thurston operator and the Cauchy transform

Geometric methods in holomorphic dynamics

Dynamical pairs with an absolutely continuous bifurcation measure

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