Polynomial diffeomorphisms of C2: currents, equilibrium measure and hyperbolicity

Bedford, Eric; Smillie, John

doi:10.1007/bf01239509

Cited by 177 publications

(280 citation statements)

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“…We show that the set Fix(F 2 ) is discrete, and (if δ = −1) that F admits infinitely many saddle points of period 1 or 2, which implies that the Julia set is not empty. We also show that there is no irreducible invariant algebraic curve (the same was proved by Bedford-Smillie for polynomial Hénon maps in [4]). The dynamical behavior can be restricted even further by considering transcendental Hénon maps whose map f has a given order of growth.…”

Section: Introductionmentioning

confidence: 80%

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Dynamics of transcendental Hénon maps

Arosio

Benini

Fornæss³

et al. 2018

Math. Ann.

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The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental Communicated by Ngaiming Mok.

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Section: Introductionmentioning

confidence: 80%

“…It follows from a result of Bedford-Smillie [4] that a polynomial Hénon map does not have any invariant algebraic curve. Indeed, given any algebraic curve, the normalized currents of integration on the push-forwards of this curve converge to the (1, 1) current μ − , whose support does not lie on an algebraic curve.…”

Section: Invariant Algebraic Curvesmentioning

confidence: 97%

Dynamics of transcendental Hénon maps

Arosio

Benini

Fornæss³

et al. 2018

Math. Ann.

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“…There has been considerable progress with respect to the classification of periodic Fatou components, see for example [2][3][4]7,13,18]. However, the existence question for wandering Fatou components has been until recently almost untouched.…”

Section: Introductionmentioning

confidence: 99%

“…However, the existence question for wandering Fatou components has been until recently almost untouched. Only for a few specific cases, the non-existence of wandering Fatou components is known, for example for polynomial automorphism that act hyperbolically on their Julia sets, see [2], and for polynomial skew-products near a superattracting invariant fiber.…”

Section: Introductionmentioning

confidence: 99%

Polynomial skew-products with wandering Fatou-disks

Peters

Vivas

2016

Math. Z.

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Until recently, little was known about the existence of wandering Fatou components for rational maps in more than one complex variables. In 2014, examples of wandering Fatou components were constructed in Astorg et al. [1] for polynomial skew-products with an invariant parabolic fiber. In 2004 Lilov already proved the non-existence of wandering Fatou components for polynomial skew-products in the basin of an invariant super-attracting fiber. In the current work we investigate in how far his methods carry over to the geometrically attracting case. Lilov proved a stronger statement, namely that the orbit of any horizontal disk in the super-attracting basin must eventually intersect a fattened Fatou components. Here we give explicit constructions to show that this stronger result is false in the geometrically attracting case. However, we also prove that the constructed disks do not give rise to wandering Fatou components. Our construction therefore leaves open the existence of wandering Fatou components in the geometrically attracting case, showing however that the situation is significantly more complicated than in the super-attracting case.

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“…We will use some standard facts from the dynamics of polynomial automorphisms of C 2 . General references are [FM,BS1,BLS1,Si1]. As usual K + (resp.…”

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confidence: 99%