Rigorous bounds on the Hausdorff dimension of Siegel disc boundaries

Andrew Burbanks; Andrew Osbaldestin; A. Stirnemann

doi:10.1007/s002200050507

Rigorous bounds on the Hausdorff dimension of Siegel disc boundaries

Research output: Contribution to journal › Article › peer-review

Abstract

We calculate rigorous bounds on the Hausdorff dimension of Siegel disc boundaries for maps that are attracted to the critical fixed point of the renormalization operator. This is done by expressing (a piece of) the universal invariant curve of the fixed-point maps as the limit set of an iterated function system. In particular, we prove (by computer-assisted means) that the Hausdorff dimension of these boundary curves is less than 1.08523 for maps that are close enough to the fixed point and attracted to it under renormalization.

Original language	English
Pages (from-to)	417-439
Number of pages	23
Journal	Communications in Mathematical Physics
Volume	199
Issue number	2
DOIs	https://doi.org/10.1007/s002200050507
Publication status	Published - 1998

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10.1007/s002200050507

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Invited speaker at MIRaW – Mathematical Interdisciplinary Research at Warwick event on "Computation in Mathematics and Mathematics in Computation"
Burbanks, A. (Invited speaker)
11 Mar 2024
Activity: Talk or presentation types › Invited talk
British Applied Mathematics Colloquium (BAMC) 2023
Burbanks, A. (Presented paper)
3 Apr 2023 → 5 Apr 2023
Activity: Participating in or organising an event types › Participation in conference

Cite this

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abstract = "We calculate rigorous bounds on the Hausdorff dimension of Siegel disc boundaries for maps that are attracted to the critical fixed point of the renormalization operator. This is done by expressing (a piece of) the universal invariant curve of the fixed-point maps as the limit set of an iterated function system. In particular, we prove (by computer-assisted means) that the Hausdorff dimension of these boundary curves is less than 1.08523 for maps that are close enough to the fixed point and attracted to it under renormalization.",

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AB - We calculate rigorous bounds on the Hausdorff dimension of Siegel disc boundaries for maps that are attracted to the critical fixed point of the renormalization operator. This is done by expressing (a piece of) the universal invariant curve of the fixed-point maps as the limit set of an iterated function system. In particular, we prove (by computer-assisted means) that the Hausdorff dimension of these boundary curves is less than 1.08523 for maps that are close enough to the fixed point and attracted to it under renormalization.

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Rigorous bounds on the Hausdorff dimension of Siegel disc boundaries

Abstract

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Activities

Invited speaker at MIRaW – Mathematical Interdisciplinary Research at Warwick event on "Computation in Mathematics and Mathematics in Computation"

British Applied Mathematics Colloquium (BAMC) 2023

Cite this