Contracting rigid germs in higher dimensions

Ruggiero, Matteo

Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1913-1950.

En suivant Favre, on dit qu’un germe holomorphe $f:({ℂ}^d,0)\to ({ℂ}^d,0)$ est rigide si l’union de l’ensemble critique de tous ses itérés est à croisement normaux. Nous donnons une classification partielle des germes rigides contractants en toute dimension à conjugaison holomorphe près. On trouve des nouveaux phénomènes de résonance, entre la différentielle de $f$ et son action linéaire sur le groupe fondamental du complémentaire de l’ensemble critique.

Following Favre, we define a holomorphic germ $f:({ℂ}^d,0)\to ({ℂ}^d,0)$ to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of $f$ and its linear action on the fundamental group of the complement of the critical set.

Classification : 37F25
Keywords: holomorphic fixed point germs, contracting rigid germs, normal forms, renormalization, resonances, critical set.
Mot clés : germes holomorphes, point fixe, germes rigides contractants, formes normales, renormalisation, résonances, ensemble critique.

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