Communications in Mathematical Physics

, Volume 223, Issue 3, pp 451–464 | Cite as

Total Convergence or General Divergence¶in Small Divisors

  • R. Pérez-Marco


We study generic holomorphic families of dynamical systems presenting problems of small divisors with fixed arithmetic. The characteristic features are delicate problems of convergence of formal power series due to Small Divisors. We prove the following dichotomy: We have convergence for all parameter values, or divergence everywhere except for an exceptional pluri-polar set of parameters. We illustrate this general principle in different problems of Small Divisors. As an application we obtain new richer families of non-linearizable examples in the Siegel problem when the Bruno condition is violated, generalizing and extending to higher dimension previous results of Yoccoz and the author.


Dynamical System Characteristic Feature Power Series General Divergence High Dimension 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • R. Pérez-Marco
    • 1
  1. 1.UCLA, Dept. of Mathematics, 405, Hilgard Ave., Los Angeles, CA 90095-1555, USA.¶E-mail: ricardo@math.ucla.eduUS

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