Topological rigidity of generic unfoldings of tangent to the identity diffeomorphisms
Annales de l'Institut Fourier, to appear, 54 p.

We prove that a homeomorphism conjugating two generic 1-parameter unfoldings, of local 1-variable tangent to the identity biholomorphisms with a double fixed point at the origin, is real analytic outside the origin by restriction to the unperturbed parameter. Moreover if one of the unfoldings has a restriction to the unperturbed parameter that is not analytically trivial, meaning that is not the time 1 flow of a holomorphic vector field, then the restriction of the conjugating map to the unperturbed parameter is holomorphic or anti-holomorphic. We provide examples that show that the non-analytically trivial hypothesis is necessary. Moreover we characterize the possible behavior of conjugacies for the unperturbed parameter in the analytically trivial case.

We describe the structure of the limits of orbits when we approach the unperturbed parameter. The proof of the rigidity results is based on the study of the action of a topological conjugacy on the limits of orbits.

On considère des germes de biholomorphisme φ 0 et η 0 tangents à l’identité et avec un point fixe double. On montre qu’un homéomorphisme qui conjugue deux déploiements génériques à un paramètre de φ 0 et η 0 est analytique réel si l’on se restreint au paramètre initial (sauf peut-être à l’origine). De plus si φ 0 ou η 0 n’est pas analytiquement trivial, i.e. n’est pas contenu dans un group à un paramètre, la conjugaison induite sur le paramètre initial est holomorphe ou anti-holomorphe. L’hypothèse de non-trivialité est nécessaire. On détermine aussi la nature des conjugaisons sur le paramètre initial se φ 0 ou η 0 ne sont pas analytiquement triviaux.

On décrit la structure des limites d’orbites quand on approche le paramètre initial. Les resultats de rigidité sont conséquences de l’étude de l’action d’une conjugaison topologique sur les limites d’orbites.

Received : 2014-06-12
Revised : 2017-05-22
Accepted : 2018-04-26
Classification:  37F45,  37F75,  37G10,  34E10
Keywords: resonant diffeomorphism, bifurcation theory, topological classification, normal form
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     author = {Rib\'on, Javier},
     title = {Topological rigidity of generic unfoldings of tangent to the identity diffeomorphisms},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Ribón, Javier. Topological rigidity of generic unfoldings of tangent to the identity diffeomorphisms. Annales de l'Institut Fourier, to appear, 54 p.

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