Hedgehogs in higher dimensions and their applications

Lyubich, Mikhail; Radu, Remus; Tanase, Raluca

doi:10.24033/ast.1115

Cited by 3 publications

(4 citation statements)

References 12 publications

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“…If µ = +∞, then we say F is in asymptotic ultraresonant form. Theorem 3.10.5 ([LRT20]). Let F ∈ End( 2 , 0) be elliptic-attracting with elliptic multiplier λ ∈ S 1 and a hedgehog K in B ⊆ 2 .…”

Section: Parabolic-elliptic Fixed Pointsmentioning

confidence: 94%

“…More precise information in dimension 2 was recently established by Firsova, Lyubich, Radu, and Tanase, and in [FLRT20] and [LRT20], who use the centre manifold reduction of Theorem 3.7.14 to recover Perez-Marco's hedgehogs: Theorem 3.10. 3 ([FLRT20]).…”

Section: Elliptic-attracting Fixed Pointsmentioning

confidence: 97%

“…Recently, Lyubich, Radu, and Tanase gave an alternative proof of Theorem 3.7.9 in [LRT20] via centre manifold reduction, that extends to the case of mixed hyperbolic multipliers. An important ingredient is the quasiconformal conjugation of F to a holomorphic map on the (non-analytic) centre manifold: Theorem 3.7.14 ([LRT20]). Let F ∈ Aut( d , 0) with one neutral multiplier λ and hyperbolic multipliers otherwise.…”

Section: Mixed Hyperbolic Multipliersmentioning

confidence: 99%

“…This facilitates the examination of the dynamics on the centre manifold with classical one-dimensional techniques to obtain parabolic manifolds with local Fatou coordinates. Though, a priori, these are only 1 , the fact that the petals lie in the holomorphic part of W c (see [Aba01a]), ensures holomorphicity: Theorem 3.7.15 ([LRT20]). Let F ∈ Aut( d , 0) be parabolic-attracting with multiplier 1 of algebraic multiplicity 1, one attracting multiplier λ, the remaining multipliers repelling, and finite semi-parabolic order k +1.…”

Section: If All Hyperbolic Multipliers Of F Are Attracting the Conjug...mentioning

confidence: 99%

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Small divisors in discrete local holomorphic dynamics

Reppekus

2023

Boll Unione Mat Ital

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We give an introduction to the study of local dynamics of iterated holomorphic mappings near a fixed point via local conjugations in one and several complex variables. Starting with the systematic construction of formal conjugations to Poincaré–Dulac normal form in general and formal linearisations in particular, we discuss conditions for convergence of the normalising series in terms of the linear part. The convergence is closely related to the size of denominators that show up in the normalising series and depend only on the linear part of the mapping. Hence, we speak of a small-divisor problem. The central result on the linearisation problem is the Brjuno condition, that ensures holomorphic linearisability. The first part of these notes is a survey of the local dynamics in one variable from the viewpoint of local normalisations. In this case, the picture is fairly complete. In the second part we introduce the additional obstacles to both formal and holomorphic normalisations that emerge from interactions of multiple eigenvalues, such as resonances, that preclude the Brjuno condition in particular. For these cases, we proceed with several generalisations of the Brjuno condition, that allow us to find convergent conjugations to at least partial normalisations. The last part reviews a recent application of such a partial normalisation to illuminate the local dynamics near certain one-resonant fixed points.

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Section: Parabolic-elliptic Fixed Pointsmentioning

confidence: 94%

Section: Elliptic-attracting Fixed Pointsmentioning

confidence: 97%

Section: Mixed Hyperbolic Multipliersmentioning

confidence: 99%

Section: If All Hyperbolic Multipliers Of F Are Attracting the Conjug...mentioning

confidence: 99%

See 2 more Smart Citations