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Probabilistic approximation of partly filled-in composite Julia sets
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Albaha University, Faculty of Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2017 (English)In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 119, no 3, p. 203-220Article in journal (Refereed) Published
Abstract [en]

We study properties of the metric space of pluriregular sets and of contractions on that space induced by finite families of proper polynomial mappings of several complex variables. In particular, we show that closed balls in the space of pluriregular sets do not have to be compact and we give a simple proof of applicability of the so-called chaos game in the case of composite Julia sets. Part of the construction of those sets also leads to a computationally viable approximation by simpler sets based on Monte-Carlo simulation.

Place, publisher, year, edition, pages
2017. Vol. 119, no 3, p. 203-220
Keywords [en]
composite Julia sets, pluricomplex Green functions, iterated function system, the chaos game, complex dynamics, Monte-Carlo simulation
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-339527DOI: 10.4064/ap4100-8-2017ISI: 000417986400002OAI: oai:DiVA.org:uu-339527DiVA, id: diva2:1176368
Available from: 2018-01-22 Created: 2018-01-22 Last updated: 2018-07-06Bibliographically approved
In thesis
1. Approximation of pluricomplex Green functions: A probabilistic approach
Open this publication in new window or tab >>Approximation of pluricomplex Green functions: A probabilistic approach
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This PhD thesis focuses on probabilistic methods of approximation of pluricomplex Green functions and is based on four papers.

The thesis begins with a general introduction to the use of pluricomplex Green functions in multidimensional complex analysis and a review of their main properties. This is followed by short description of the main results obtained in the enclosed papers.

In Paper I, we study properties of the metric space of pluriregular sets, that is zero sets of continuous pluricomplex Green functions. The best understood non-trivial examples of such sets are composite Julia sets, obtained by iteration of finite families of polynomial mappings in several complex variables. We prove that the so-called chaos game is applicable in the case of such sets. We also visualize some composite Julia sets using escape time functions and Monte Carlo simulation.

In Paper II, we extend results in Paper I to the case of infinite compact families of proper polynomials mappings. With composition as the semigroup operation, we generate families of infinite iterated function systems with compact attractors. We show that such attractors can be approximated probabilistically in a manner of the classic chaos game.

In Paper III, we study numerical approximation and visualisation of pluricomplex Green functions based on the Monte-Carlo integration. Unlike alternative methods that rely on locating a sequence of carefully chosen finite sets of points satisfying some optimal conditions for approximation purposes, our approach is simpler and more direct by relying on generation of pseudorandom points. We examine numerically the errors of approximation for some simple geometric shapes for which the pluricomplex Green functions are known. If the pluricomplex Green functions are not known, the errors in Monte Carlo integration can be expressed with the aid of statistics in terms of confidence intervals.

Finally, in Paper IV, we study how perturbations of an orthonomalization procedure influence the resulting approximate Bergman functions. To this end we consider the concept of near orthonormality of a finite set of vectors in an inner product space, understood as closeness of the Gram matrix of those vectors to the identity matrix. We provide estimates for the errors resulting from using nearly orthogonal bases instead of orthogonal ones. The motivation for this work comes from Paper III: when Gram matrices are calculated via Monte Carlo integration, the outcomes of standard orthogonalisation algorithms are nearly orthonormal bases.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2018. p. 47
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 109
Keywords
pluricomplex Green function, pluriregular sets, Bernstein-Markov property, Bergman function, nearly orthonormal polynomials, orthogonal polynomials, Monte Carlo simulation, composite Julia sets, Julia sets, iterated function systems, the chaos game.
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-355810 (URN)978-91-506-2714-5 (ISBN)
Public defence
2018-09-21, Polhemsalen, 10134, Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:00 (English)
Opponent
Supervisors
Available from: 2018-08-31 Created: 2018-07-06 Last updated: 2018-08-31

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Alghamdi, AzzaKlimek, Maciej

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