Hedgehogs of Hausdorff dimension one

Biswas, Kingshook

doi:10.1017/s0143385707000879

Cited by 7 publications

(8 citation statements)

References 6 publications

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“…For an arbitrary germ of a holomorphic map with an irrationally indifferent fixed point, it is likely that hedgehogs come in variety of topologies and geometries. A general strategy to build germs of holomorphic maps with nontrivial hedgehogs is developed by Perez-Marco and Biswas in [PM97b] and [Bis08], see also [Che11]. In particular, examples of hedgehogs of dimension one and positive area have been presented in [Bis08] and [Bis16].…”

Section: Introductionmentioning

confidence: 99%

“…A general strategy to build germs of holomorphic maps with nontrivial hedgehogs is developed by Perez-Marco and Biswas in [PM97b] and [Bis08], see also [Che11]. In particular, examples of hedgehogs of dimension one and positive area have been presented in [Bis08] and [Bis16]. However, those examples have a very different nature, and are not likely to occur for a rational map of the Riemann sphere or an entire holomorphic map of the complex plane.…”

Section: Introductionmentioning

confidence: 99%

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Dimension paradox of irrationally indifferent attractors

Cheraghi¹,

Dezotti²,

Yang³

2020

Preprint

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In this paper we study the geometry of the attractors of holomorphic maps with an irrationally indifferent fixed point. We prove that for an open set of such holomorphic systems, the local attractor at the fixed point has Hausdorff dimension two, provided the asymptotic rotation at the fixed point is of sufficiently high type and does not belong to Herman numbers. As an immediate corollary, the Hausdorff dimension of the Julia set of any such rational map with a Cremer fixed point is equal to two. Moreover, we show that for a class of asymptotic rotation numbers, the attractor satisfies Karpińska's dimension paradox. That is, the the set of end points of the attractor has dimension two, but without those end points, the dimension drops to one.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dimension paradox of irrationally indifferent attractors

Cheraghi¹,

Dezotti²,

Yang³

2020

Preprint

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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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“…When f is non-linearizable these are called hedgehogs. The topology and dynamics of hedgehogs has been studied by Perez-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], [Bis15]) and Cheritat ([Che11]) to construct further examples.…”

Section: Introductionmentioning

confidence: 99%

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

Biswas

2016

Preprint

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Let f be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin inPerez-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 nontrivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families (gt) of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps (gt) is the orbit of a locally-defined semigroup (Φt) 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 (Kt). We show that the Loewner measures (µt) driving the equation are 2-conformal measures on the circle for the circle maps (z → gt(z)).

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Hedgehogs of Hausdorff dimension one

Abstract: We present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting non-linearisable maps, a common hedgehog of Hausdorff dimension 1, the minimum possible.

Cited by 7 publications

References 6 publications

Dimension paradox of irrationally indifferent attractors

Dimension paradox of irrationally indifferent attractors

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

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

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