Boundaries of Siegel Disks: Numerical Studies of their Dynamics and Regularity
| dc.contributor.utaustinauthor | de la Llave, Rafael | en_US |
| dc.creator | de la Llave, Rafael | en_US |
| dc.creator | Petrov, Nikola P. | en_US |
| dc.date.accessioned | 2016-10-28T19:51:25Z | |
| dc.date.available | 2016-10-28T19:51:25Z | |
| dc.date.issued | 2008-09 | en_US |
| dc.description.abstract | Siegel disks are domains around fixed points of holomorphic maps in which the maps are locally linearizable (i.e., become a rotation under an appropriate change of coordinates which is analytic in a neighborhood of the origin). The dynamical behavior of the iterates of the map on the boundary of the Siegel disk exhibits strong scaling properties which have been intensively studied in the physical and mathematical literature. In the cases we study, the boundary of the Siegel disk is a Jordan curve containing a critical point of the map (we consider critical maps of different orders), and there exists a natural parametrization which transforms the dynamics on the boundary into a rotation. We compute numerically this parameterization and use methods of harmonic analysis to compute the global Holder regularity of the parametrization for different maps and rotation numbers. We obtain that the regularity of the boundaries and the scaling exponents are universal numbers in the sense of renormalization theory (i.e., they do not depend on the map when the map ranges in an open set), and only depend on the order of the critical point of the map in the boundary of the Siegel disk and the tail of the continued function expansion of the rotation number. We also discuss some possible relations between the regularity of the parametrization of the boundaries and the corresponding scaling exponents. (C) 2008 American Institute of Physics. | en_US |
| dc.description.department | Mathematics | en_US |
| dc.description.sponsorship | NSF | en_US |
| dc.identifier | doi:10.15781/T2GX44X5D | |
| dc.identifier.citation | de la Llave, Rafael, and Nikola P. Petrov. "Boundaries of Siegel disks: numerical studies of their dynamics and regularity." Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 18, No. 3 (Sep., 2008): 033135. | en_US |
| dc.identifier.doi | 10.1063/1.2985856 | en_US |
| dc.identifier.issn | 1054-1500 | en_US |
| dc.identifier.uri | http://hdl.handle.net/2152/43242 | |
| dc.language.iso | English | en_US |
| dc.relation.ispartof | en_US | |
| dc.relation.ispartofserial | Chaos | en_US |
| dc.rights | Administrative deposit of works to Texas ScholarWorks: This works author(s) is or was a University faculty member, student or staff member; this article is already available through open access or the publisher allows a PDF version of the article to be freely posted online. The library makes the deposit as a matter of fair use (for scholarly, educational, and research purposes), and to preserve the work and further secure public access to the works of the University. | en_US |
| dc.rights.restriction | Open | en_US |
| dc.subject | critical-points | en_US |
| dc.subject | cylinder renormalization | en_US |
| dc.subject | hausdorff dimension | en_US |
| dc.subject | quasi-periodicity | en_US |
| dc.subject | theorem | en_US |
| dc.subject | maps | en_US |
| dc.subject | curves | en_US |
| dc.subject | proof | en_US |
| dc.subject | sets | en_US |
| dc.subject | mathematics, applied | en_US |
| dc.subject | physics, mathematical | en_US |
| dc.title | Boundaries of Siegel Disks: Numerical Studies of their Dynamics and Regularity | en_US |
| dc.type | Article | en_US |