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A piecewise contractive dynamical system and Phragmèn's election method
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2019 (English)In: Bulletin de la Société Mathématique de France, ISSN 0037-9484, E-ISSN 2102-622X, Vol. 147, no 3, p. 395-441Article in journal (Refereed) Published
Abstract [en]

We prove some basic results for a dynamical system given by a piece-wise linear and contractive map on the unit interval that takes two possible values at a point of discontinuity. We prove that there exists a universal limit cycle in the non-exceptional cases, and that the exceptional parameter set is very tiny in terms of gauge functions. The exceptional two-dimensional parameter is shown to have Hausdorff-dimension one. We also study the invariant sets and the limit sets; these are sometimes different and there are several cases to consider. In addition, we prove the existence of a unique invariant measure. We apply some of our results for the dynamical system, involving a study of rational and irrational rotation numbers, to a combinatorial problem involving an election method suggested by Phragmen, and we show that the proportion of elected seats for each party converges to a limit, which is a rational number except for a very small exceptional set of parameters.

Place, publisher, year, edition, pages
FRENCH MATHEMATICAL SOC , 2019. Vol. 147, no 3, p. 395-441
Keywords [en]
piecewise contractive dynamical system, Phragmens election method
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-396947DOI: 10.24033/bsmf.2787ISI: 000492745100003OAI: oai:DiVA.org:uu-396947DiVA, id: diva2:1371873
Available from: 2019-11-21 Created: 2019-11-21 Last updated: 2019-11-21Bibliographically approved

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Janson, SvanteÖberg, Anders

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