2008/01/18 by Carlo Carminati, Carminati, Carlo, Stefano Marmi +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.0801.2844
openalex publication_date 2008/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the linearization of a germ of holomorphic map of the type Fλ(z)=λ(z+O(z2)) has a C1--holomorphic dependence on the multiplier λ. C1--holomorphic functions are C1--Whitney smooth functions, defined on compact subsets and which belong to the kernel of the ∂ operator. The linearization is analytic for |λ|\not= 1 and the unit circle S1 appears as a natural boundary (because of resonances, i.e. roots of unity). However the linearization is still defined at most points of S1, namely those points 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 spaces of \cal C1--holomorphic functions. This is a special case of Borel's theory of uniform monogenic functions, and the corresponding function space is arcwise-quasianalytic. Among the consequences of these results, we can prove that the linearization admits an asymptotic expansion w.r.t. the multiplier at all points of the unit circle verifying the Brjuno condition: in fact the asymptotic expansion is of Gevrey type at diophantine points.