We reconsider the Schröder-Siegel problem of conjugating an analytic map in {Mathematical expression} in the neighborhood of a fixed point to its linear part, extending it to the case of dimension {Mathematical expression}. Assuming a condition which is equivalent to Bruno's one on the eigenvalues {Mathematical expression} of the linear part, we show that the convergence radius {Mathematical expression} of the conjugating transformation satisfies {Mathematical expression} with {Mathematical expression} characterizing the eigenvalues {Mathematical expression}, a constant {Mathematical expression} not depending on {Mathematical expression} and {Mathematical expression}. This improves the previous results for {Mathematical expression}, where the known proofs give {Mathematical expression}. We also recall that {Mathematical expression} is known to be the optimal value for {Mathematical expression}.

Improved convergence estimates for the Schröder-Siegel problem / A. Giorgilli, U. Locatelli, M. Sansottera. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 194:4(2015), pp. 1-19. [Epub ahead of print] [10.1007/s10231-014-0408-4]

Improved convergence estimates for the Schröder-Siegel problem

A. Giorgilli
Primo
;
M. Sansottera
2015

Abstract

We reconsider the Schröder-Siegel problem of conjugating an analytic map in {Mathematical expression} in the neighborhood of a fixed point to its linear part, extending it to the case of dimension {Mathematical expression}. Assuming a condition which is equivalent to Bruno's one on the eigenvalues {Mathematical expression} of the linear part, we show that the convergence radius {Mathematical expression} of the conjugating transformation satisfies {Mathematical expression} with {Mathematical expression} characterizing the eigenvalues {Mathematical expression}, a constant {Mathematical expression} not depending on {Mathematical expression} and {Mathematical expression}. This improves the previous results for {Mathematical expression}, where the known proofs give {Mathematical expression}. We also recall that {Mathematical expression} is known to be the optimal value for {Mathematical expression}.
Diophantine conditions; Linearization; Normal forms; Small divisors; Applied Mathematics
Settore MAT/07 - Fisica Matematica
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/250185
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