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Limit at resonances of linearizations of some complex analytic dynamical systems

Published online by Cambridge University Press:  01 August 2000

ALBERTO BERRETTI ,
STEFANO MARMI  and
DAVID SAUZIN
ALBERTO BERRETTI
Affiliation:
Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy and INFN, Sez. Tor Vergata (e-mail: berretti@roma2.infn.it)
STEFANO MARMI
Affiliation:
Dipartimento di Matematica ‘U. Dini’, Università di Firenze, Viale Morgagni 57a, 50134 Firenze, Italy and INFN, Sez. Firenze (e-mail: marmi@udini.math.unifi.it)
DAVID SAUZIN
Affiliation:
Astronomie et systèmes dynamiques, CNRS – Bureau des longitudes, 77, avenue Denfert-Rochereau, 75014 Paris, France (e-mail: sauzin@bdl.fr)
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Abstract

We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of $({\Bbb C},0)$ and of the semi-standard map.

We prove that for each resonance there exists a suitable blow-up of the Taylor series of the linearization under which it converges uniformly to an analytic function as the multiplier, or rotation number, tends non-tangentially to the resonance. This limit function is explicitly computed and related to questions of formal classification, both for the case of germs and for the case of the semi-standard map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 20 , Issue 4 , August 2000 , pp. 963 - 990
Copyright
2000 Cambridge University Press

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