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Caspar Wessel. On the Analytical Representation of Direction:An Attempt Applied Chiefly to Solving Plane and Spherical Polygons
No full textThis book contains the first complete English translation of Caspar Wessel's paper from 1797 "On the Analytical Representation of Direction". The paper has become famous because it gives the first geometric interpretation of the complex numbers. As the complete translation will show, it is also remarkable for the analytical representation of directions (vectors) in space and the elegant analytic solution of plane and spherical polygons. Wessel's essay is prefaced by two historical papers on the person Caspar Wessel and his work. They shed much new light on this first important Norwegian mathematician and sets his work on "directions" in its proper historical context. Among other things it is shown that the idea of using complex numbers to represent directions in a plane occurred to Wessel as early as 1787 in connection with his work as a surveyor. Moreover the papers address the question why Wessel's essay remained without influence
Siegel discs via trans-quasiconformal surgery:Siegel discs via trans-quasiconformal surgery
No full textSynthetisk fremstilling af holomorfe afbildninger med en Siegel disk ud fra Blaschkemodeller og kvasikonform kirurgi er en i dag klassisk teknik, som i sin oprindelige form først blev udviklet af Ghys. Grundlæggende består teknikken i at udvide en konjugation, af en (typisk) analytisk cirkel-homeomorfi til en stiv rotation, til en kvasikonform homeomorfi af enhedsdisken på sig selv. Ved kirurgi og "pull-back" kan man derefter fremstille en ny afbildning og en invariant næsten kompleks struktur for denne nye afbildning. En klassisk sætning af Morrey-Bojarsky-Ahlfors-Bers sikre at en sådan næsten kompleksstruktur kan integreres, det vil sige definerer en kompleksstruktur. I denne komplekse struktur er den nye afbildning holomorf og har en Siegel disk.Der findes imidlertid masser af tilfælde hvor dette ikke lader sig gøre, men hvor man istedet kan finde en trans-kvasikonform udvidelse til enheds-disken af konjugationen. Hvilket kan vises at være nok til at fuldføre den syntetiske konstruktion af en holomorf afbildning med en Siegel disk. Om en given cirkelhomeomorfi falder i den ene, den anden eller en helt tredie kategori afgøres af cirkelhomeorfiens rotationstal. I dette bidrag redegøres der for, at for en mængde af rotationstal med fuldt mål på cirklen har konjugationen en trans-kvasi-konform udvidelse til disken. Dette er et skridt i beviset for at for næsten alle (i Lebeguemåls forstand) rotations tal, er randen af Siegel-disken for et kvadratisk polynomium med en Siegel disk med et sådant rotationstal en Jordan kurve hvis bane går gennem det kritiske punkt og polynomiets Julia mængde har Lebeguemål 0. <br/
Proceedings of the seventh EWM meeting, European Women in Mathematics
No full textThe proceedings consist of a part concerning EWM and a mathematical part of mainly four series of papers. The series are within the following themes: Holomorphic Dynamics, Algebraic Geometry, Mathematical Physics and Moduli Spaces
Holomorphic Dynamical Systems in the Complex Plane
No full textThe paper reviews some basic properties of Julia sets of polynomials and the Mandelbrot set. In particular we emphasize the concept of normal families, the importance of repelling periodic points and the asymptotic similarity between Juliasets of certain quadratic polynomials and the Mandelbrot set
Topological properties of dynamically defined sets in holomorphic dynamics of one variable
No full textTableaus and their use in Holomorphic Dynamics
No full textMINICOURSE Tableaus and their use in Holomorphic Dynamics. TITLE of class I: Puzzles and para-puzzles, and the divergence property. ABSTRACT: The geometrical part: Puzzles in the most fundamental cases, i.e. associated with a polynomial, which either belongs to the Yoccoz-class of quadratic polynomials or to the bounded/unbounded-class of cubic polynomials (with one bounded and one un-bounded critical orbit). Para-puzzles in the quadratic case. The analytical part: The divergence property and its usefulness. - An infinite set of open disjoint annuli embedded in a bounded open annuli, all of the same homotopy type, is said to have the divergence property, if the infinite series of moduli of these annuli is divergent. TITLE of class II: Tableaus. ABSTRACT: The combinatorial part: Tableaus associated with a polynomial belonging to the Yoccoz-class or the bounded/unbounded class. The tableau rules, classification of critical tableaus, the Fibonacci critical tableau. TITLE of class III: Points are points. ABSTRACT: Combining the geometrical, analytical and combinatorial parts to conclude either local connectivity of the Julia set of a polynomial in the Yoccoz-class or total disconnectivity of the Julia set of a polynomial in the bounded/unbounded class (i.e. the Julia set is a Cantor set). In both cases one proves that (certain) connected components are reduced to point components. Therefore, Adrien Douady liked to say that one proves that "points are points"
Polynomial Vector Fields in One Complex Variable
Full text linkIn recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias
Iterations by odd functions with two extrema
No full textAbstractLet I = [−1, 1] and fI → I be continuous, piecewise monotone and odd with two extrema. A periodic orbit is called symmetric if −x is in the orbit when x is in the orbit. A periodic orbit which is not symmetric is called asymmetric. The first result of this paper proves an ordering of the periods for the symmetric orbits. There are two possibilities depending on how f behaves in a neighbourhood of 0. The second result of this paper proves that for a one-parameter family of odd functions with negative Schwarzian derivative there are three different types of nondegenerate bifurcations: saddle node, period-doubling pitchfork and period-preserving pitchfork. The last type of bifurcation occurs exactly when a symmetric orbit bifurcates to two asymmetric orbits
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