Abstract
We return to the subject of local, identity-tangent diffeomorphisms ƒ of ℂ and their analytic invariants A ω (ƒ), under the complementary viewpoints of effective computation and explicit expansions. The latter rely on two basic ingredients: the so-called multizetas (transcendental numbers) and multitangents (transcendental functions), with resurgence monomials and their monics providing the link between the two. We also stress the difference between the collectors (preinvariant but of one piece) and the connectors (invariant but mutually unrelated).
Much attention has been paid to streamlining the nomenclature and notations. On the analysis side, resurgence theory rules the show. On the algebraic or combinatorial side, mould theory brings order and structure into the profusion of objects. Along the way, the paper introduces quite a few novel notions: new alien operators, new forms of resurgence, new symmetry types for moulds. It also broaches the subject of ‘phantom dynamics’ (dealing with formal diffeos that nonetheless possess invariants A ω (ƒ)) and culminates in the comparison of arithmetical and dynamical monics, a distinction that reflects the dual nature of the A ω (ƒ) as Stokes constants and holomorphic invariants.
Access provided by CONRICYT-eBooks. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
X. Buff, J. Ecalle, A. Epstein, Limits of degenerate parabolic quadratic rational maps, Geom. Funct. Anal. 23 (2013), 42–95.
J. Ecalle, “Théorie des invariants holomorphes”, Thèse d’Etat, Orsay-Paris-Sud, 1974.
J. Ecalle, Algèbres de fonctions résurgentes. Pub. Math. Orsay (1981).
J. Ecalle, Les fonctions résurgentes appliquées à l’itération. Pub. Math. Orsay (1981).
J. Ecalle, L’équation du pont et la classification analytique des objets locaux. Pub. Math. Orsay (1985).
J. Ecalle, Six lectures on transseries, analysable functions and the constructive proof of Dulac’s conjecture, In: “Bifurcation and Periodic Orbits of Vector Fields”, D. Schlomiuk ed., Kluwer, 1993, 75–184.
J. Ecalle, Cohesive functions and weak accelerations, Journal d’Analyse Mathématique 60 (1993), 71–97.
P. Fatou, Sur les solutions uniformes de certaines équations fonctionnelles, C. R. Acad. Sci. Paris, 143 (1906), 546–548.
M. Shishikura, On the bifurcation of parabolic fixed points of rational maps, Lecture Notes for the 19th Brazilian Mathematical Colloquium, 1993, IMPA.
Editor information
Rights and permissions
Copyright information
© 2017 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Bouillot, O., Ecalle, J. (2017). Invariants of identity-tangent diffeomorphisms expanded as series of multitangents and multizetas. In: Fauvet, F., Manchon, D., Marmi, S., Sauzin, D. (eds) Resurgence, Physics and Numbers. CRM Series, vol 20. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-613-1_4
Download citation
DOI: https://doi.org/10.1007/978-88-7642-613-1_4
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-612-4
Online ISBN: 978-88-7642-613-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)