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Renormalization small divisors continued fractions and geodesic flows

seminar: Linearization of germs: regular dependence on the multiplier

speaker: Carlo Carminati (Dipartimento di Matematica, Università di Pisa)

abstract: We prove that the linearization of germs of holomorphic (or formal) maps of the type $f\lambda(z)=\lambda(z+O(z2))$ have a $C1$--holomorphic dependence on the multiplier $\lambda$.

The linearization is analytic it the modulus of $\lambda$ is different from 1 while the unit circle appears as a natural boundary (because of resonances, i.e. roots of unity); nevertheless solutions are still defined at points on the unit circle which lie "far enough from resonances'', i.e. when the multiplier satisfies a suitable arithmetical condition. We construct an increasing sequence of compacts which avoid resonances and prove that the linearization belongs to the associated space of $C1$--holomorphic functions. These are a special case of Borel's uniform monogenic functions, and their space is arcwise-quasianalytic. Among the consequences of these results, we can prove that the linearizations are defined and admit asymptotic expansions: in fact the asymptotic expansion is of Gevrey type at diophantine points. The regular dependence on the multiplier holds also in the formal ultradifferentiable case.

Wed 18 Apr, 14:30 - 15:30, Sala Conferenze Centro De Giorgi
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