Faculty of Arts and Science - Department of Mathematics and Statistics
André-Aisenstadt Office 4225
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Classification analytique des points fixes paraboliques de germes antiholomorphes et de leurs déploiements Theses and supervised dissertations / 2020-12
We are interested in the dynamics in a neighbourhood of a fixed point of an antiholomorphic function of one variable. First, we want to describe and understand the space of orbits in a neighbourhood of a multiple fixed point, called a parabolic point, and to explore the geometric properties preserved by changes of coordinate. In particular, we solve the problem of analytical classification of parabolic fixed points. To solve this problem, we define a complete modulus of classification that allows to determine whether two germs of antiholomorphic diffeomorphisms are analytically conjugate in a neighbourhood of their parabolic fixed point. We also consider the applications of the modulus to different problems: i) extraction of an n-th antiholomorphic root, ii) existence of an invariant real analytical curve under the dynamics of a parabolic antiholomorphic germ, and iii) centraliser of a parabolic antiholomorphic germ. In the second part, we study generic unfoldings of a double fixed point, i.e. a parabolic point of codimension 1. The questions are similar in nature, namely to understand the space of orbits and the geometric properties of unfoldings. In order to classify generic unfoldings, the modulus of classification of the parabolic point is unfolded, thus providing the necessary and sufficient conditions to determine when two generic unfoldings are equivalent.