We study families of holomorphic vector fields, holomorphically depending on parameters,in a neighborhood of an isolated singular point. When the singular point is in the Poincarédomain for every vector field of the family we prove, through a modification of classicalSternberg’s linearization argument, cf. Nelson (1969) [7] too, analytic dependence onparameters of the linearizing maps and geometric bounds on the linearization domain:each vector field of the family is linearizable inside the smallest Euclidean sphere which isnot transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko(2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6],a version of Brjuno’s Theorem in the case of linearization of families of vector fields neara singular point of Siegel type, and apply it to study some 1-parameter families of vectorfields in two dimensions.

Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood ofa singular point / Villarini, Massimo. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - STAMPA. - 373:2(2011), pp. 521-534. [10.1016/j.jmaa.2010.07.041]

Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood ofa singular point

VILLARINI, Massimo
2011

Abstract

We study families of holomorphic vector fields, holomorphically depending on parameters,in a neighborhood of an isolated singular point. When the singular point is in the Poincarédomain for every vector field of the family we prove, through a modification of classicalSternberg’s linearization argument, cf. Nelson (1969) [7] too, analytic dependence onparameters of the linearizing maps and geometric bounds on the linearization domain:each vector field of the family is linearizable inside the smallest Euclidean sphere which isnot transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko(2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6],a version of Brjuno’s Theorem in the case of linearization of families of vector fields neara singular point of Siegel type, and apply it to study some 1-parameter families of vectorfields in two dimensions.
2011
373
2
521
534
Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood ofa singular point / Villarini, Massimo. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - STAMPA. - 373:2(2011), pp. 521-534. [10.1016/j.jmaa.2010.07.041]
Villarini, Massimo
File in questo prodotto:
File Dimensione Formato  
JMAA2011.pdf

Accesso riservato

Tipologia: Versione pubblicata dall'editore
Dimensione 403.97 kB
Formato Adobe PDF
403.97 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

Licenza Creative Commons
I metadati presenti in IRIS UNIMORE sono rilasciati con licenza Creative Commons CC0 1.0 Universal, mentre i file delle pubblicazioni sono rilasciati con licenza Attribuzione 4.0 Internazionale (CC BY 4.0), salvo diversa indicazione.
In caso di violazione di copyright, contattare Supporto Iris

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/715637
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact