Polynomial Diffeomorphisms of C 2 . II: Stable Manifolds and Recurrence

Bedford, Eric; Smillie, John

doi:10.2307/2939284

Cited by 73 publications

(119 citation statements)

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“…If no iterate of F is the identity on , then is a biholomorphic to a circular domain A, and by the same proof as in [5,Proposition 6] there exists a biholomorphism from to A × C which conjugates the map F to (z, w) → e iθ z, δ e iθ w .…”

Section: Proposition 435 If Some Iterate F J | Is the Identity Thenmentioning

confidence: 99%

“…4 the invariant recurrent components of the Fatou set of a transcendental Hénon map, that is, components which admits an orbit accumulating to an interior point. Invariant recurrent components have been described for polynomial Hénon maps in [5]; our classification holds not only for transcendental Hénon maps but also for the larger class of holomorphic automorphisms with constant Jacobian. Moreover, using the fact that f is a transcendental holomorphic function, we obtain in Sect.…”

Section: Introductionmentioning

confidence: 99%

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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: Proposition 435 If Some Iterate F J | Is the Identity Thenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamics of transcendental Hénon maps

Arosio

Benini

Fornæss³

et al. 2018

Math. Ann.

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“…Bedford and Smillie ( [6]) have shown that for f a polynomial diffeomorphism of C 2 , with d(f ) > 1, f is hyperbolic on its Julia set, J, iff f is hyperbolic on its chain recurrent set, R, iff f is hyperbolic on its nonwandering set, Ω. Thus we say f is hyperbolic if any of these conditions holds.…”

Section: Drawing Meaningful Picturesmentioning

confidence: 99%

A Numerical Method for Constructing the Hyperbolic Structure of Complex Henon Mappings

Hruska

2006

Found Comput Math

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Abstract. For complex parameters a, c, we consider the Hénon mapping Ha,c : C 2 → C 2 , given by (x, y) → (x 2 + c − ay, x), and its Julia set, J. In this paper, we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J, which is preserved by DH, and a continuous norm in which DH (and DH −1 ) uniformly expands the cones (and their complements). We show a consequence of a successful construction is a proof that H is hyperbolic on J. We give several new examples of hyperbolic maps, produced with our computer program, Hypatia, which implements our methods.

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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%

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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Polynomial Diffeomorphisms of C 2 . II: Stable Manifolds and Recurrence

Cited by 73 publications

References 0 publications

Dynamics of transcendental Hénon maps

Dynamics of transcendental Hénon maps

A Numerical Method for Constructing the Hyperbolic Structure of Complex Henon Mappings

Polynomial skew-products with wandering Fatou-disks

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