We reconsider the Schröder–Siegel problem of conjugating an analytic map in ℂ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension n>1 . Assuming a condition which is equivalent to Bruno’s one on the eigenvalues λ1,…,λn of the linear part, we show that the convergence radius ρ of the conjugating transformation satisfies lnρ(λ)≥−CΓ(λ)+C′ with Γ(λ) characterizing the eigenvalues λ , a constant C′ not depending on λ and C=1 . This improves the previous results for n>1 , where the known proofs give C=2 . We also recall that C=1 is known to be the optimal value for n=1 .
Giorgilli, A., Locatelli, U., Sansottera, M. (2015). Improved convergence estimates for the Schröder-Siegel problem. ANNALI DI MATEMATICA PURA ED APPLICATA(194), 995-1013 [10.1007/s10231-014-0408-4].
Improved convergence estimates for the Schröder-Siegel problem
LOCATELLI, UGO;
2015-08-01
Abstract
We reconsider the Schröder–Siegel problem of conjugating an analytic map in ℂ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension n>1 . Assuming a condition which is equivalent to Bruno’s one on the eigenvalues λ1,…,λn of the linear part, we show that the convergence radius ρ of the conjugating transformation satisfies lnρ(λ)≥−CΓ(λ)+C′ with Γ(λ) characterizing the eigenvalues λ , a constant C′ not depending on λ and C=1 . This improves the previous results for n>1 , where the known proofs give C=2 . We also recall that C=1 is known to be the optimal value for n=1 .File | Dimensione | Formato | |
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