Lattès maps and the interior of the bifurcation locus

Biebler, Sébastien

doi:10.3934/jmd.2019014

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…From (12), it follows that ρ satisfies d −1 L ρ = ρ, i.e., ρ is an eigenvector of the operator L , of eigenvalue d . Standard arguments (see for instance [9, End of the Proof of Theorem 4.1 and Proposition 4.11]) imply the uniqueness of such ρ satisfying the above property, and the uniqueness of the conformal measure m. In particular, for any continuous function g : J H → R, we have…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…More strikingly, in [21,36], Dujardin and Taflin construct open sets in the bifurcation locus in the family H d (P k ) of all endomorphisms of P k , k ≥ 1, of a given degree d ≥ 2 (see also [12] for further examples). Their strategy also works when considering the subfamily of polynomial skew products (and actually these open sets are built close to this family).…”

Section: Introductionmentioning

confidence: 99%

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Higher bifurcations for polynomial skew products

Astorg¹,

Bianchi²

2022

JMD

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<p style='text-indent:20px;'>We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula> critical points bifurcate <i>independently</i>, with <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula> up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior. As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.</p>

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Higher bifurcations for polynomial skew products

Astorg¹,

Bianchi²

2022

JMD

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“…After this result, a natural question is that of the abundance of robust bifurcations in Hol d (P k ). Taflin [83] showed that robust bifurcations are abundant near product polynomial maps of C 2 , and Biebler [17] showed that Lattès maps of sufficiently large degree are accumulated by robust bifurcations. Blenders are involved directly or indirectly in both cases, and seem to appear quite naturally when a repelling periodic point bifurcates to a saddle.…”

Section: Robust Bifurcationsmentioning

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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“…A C r -blender for a C r -endomorphism F of a surface S is a hyperbolic basic set K s.t. an union of its local unstable manifolds has C r -robustly a non-empty interior: there exists a non-empty open set U ⊂ S included in an union of local unstable manifolds of the continuation K of K for any map F C r -close to F. This property turned out to have many other powerful applications: for example C 1density of stable ergodicity [ACW], robust homoclinic tangencies [BD2,Bie1] and thus Newhouse phenomenon, the existence of generic families displaying robustly infinitely many sinks [Be1], robust bifurcations in complex dynamics [Du,Taf,Bie2], fast growth of the number of periodic points [Be2,AST] ... Thus the following question is of fundamental interest: when do blenders appear ?…”

Section: Blenders and Almost Blendersmentioning

confidence: 99%

Almost blenders and parablenders

Biebler¹

2020

Preprint

Self Cite

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A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds contains robustly an open set. Introduced by Bonatti and Díaz in the 90s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper, we introduce analogous notions in a measurable point of view. We define an almost blender as a hyperbolic basic set for which a prevalent perturbation has a local unstable set having positive Lebesgue measure. Almost parablenders are defined similarly in terms of jets. We study families of endomorphisms of R 2 leaving invariant the continuation of a hyperbolic basic set. When some inequality involving the entropy and the maximal contraction along stable manifolds is satisfied, we obtain an almost blender or parablender. This answers partially a conjecture of Berger. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak and Urbański, we study families of fiberwise unipotent skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.

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Lattès maps and the interior of the bifurcation locus

Cited by 6 publications

References 9 publications

Higher bifurcations for polynomial skew products

Higher bifurcations for polynomial skew products

Geometric methods in holomorphic dynamics

Almost blenders and parablenders

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