Smooth combs inside hedgehogs

Biswas, Kingshook

doi:10.3934/dcds.2005.12.853

Cited by 9 publications

(7 citation statements)

References 3 publications

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“…As a notable resurgence of smoothess, it was proved in [Bis05] that some nonlinearizable holomorphic maps have hedgehogs that contain a Cantor set of smooth hairs. 14 1.5.…”

Section: 4mentioning

confidence: 99%

Smooth Siegel disks everywhere

Ávila¹,

Buff²,

Chéritat³

2020

Astérisque

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“…As a notable resurgence of smoothess, it was proved in [Bis05] that some nonlinearizable holomorphic maps have hedgehogs that contain a Cantor set of smooth hairs. 14 1.5.…”

Section: 4mentioning

confidence: 99%

Smooth Siegel disks everywhere

Ávila¹,

Buff²,

Chéritat³

2020

Astérisque

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“…This k is necessarily equal to the winding number in C m of the curve φ n (x + iε) that runs along the lower edges of the "rectangles" defining (e ı ), which is equal to the winding number of J ε in C m , that is 1, so we get condition (3). If condition (4) were not satisfied for all m , n big enough, then taking subsequences would contradict the fact that J ε is stably crooked in C m .…”

Section: Pseudo Circlesmentioning

confidence: 99%

“…In the same spirit, in [PM97] Pérez Marco was able, using tube-log Riemann surfaces, to construct examples of injective holomorphic maps defined in a subset U of C that have a Siegel disk compactly contained in U whose boundary is a C ∞ Jordan curve, which came as a surprise. Again the method is versatile and Kingshook Biswas used Pérez-Marco's construction to produce a set of interesting examples: [Bis05,Bis08]. Here we add an ingredient to this construction and get: Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

Relatively compact Siegel disks with non-locally connected boundaries

Chéritat

2010

Math. Ann.

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“…When f is non-linearizable, these are called hedgehogs. The topology and dynamics of hedgehogs have been studied by Pérez-Marco [PM94,PM96], who also developed techniques using 'tube-log Riemann surfaces' [BPM15a,BPM15b,BPM13] for the construction of interesting examples [PM93, PM95, PM00] of indifferent germs and hedgehogs, which were also used by the author [Bis05,Bis08,Bis16] and Chéritat [Ch11] to construct further examples.…”

Section: Introductionmentioning

confidence: 99%

Loewner evolution of hedgehogs and 2-conformal measures of circle maps

Biswas¹

2020

Ergod. Th. Dynam. Sys.

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Let f be a germ of a holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $${\mathbb C}$$ (i.e. $$f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$$ ). Pérez-Marco [Fixed points and circle maps. Acta Math.179(2) (1997), 243–294] showed the existence of a unique continuous monotone one-parameter family of non-trivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families $$(g_t)$$ of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps $$(g_t)$$ is the orbit of a locally defined semigroup $$(\Phi _t)$$ on the space of analytic circle maps, which we show has a well-defined infinitesimal generator X. The explicit form of X is obtained by using the Loewner equation associated to the family of hulls $$(K_t)$$ . We show that the Loewner measures $$(\mu _t)$$ driving the equation are 2-conformal measures on the circle for the circle maps $$(g_t)$$ .

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Smooth combs inside hedgehogs

Cited by 9 publications

References 3 publications

Smooth Siegel disks everywhere

Smooth Siegel disks everywhere

Relatively compact Siegel disks with non-locally connected boundaries

Loewner evolution of hedgehogs and 2-conformal measures of circle maps

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