TY - JOUR

T1 - Attracting domains of maps tangent to the identity whose only characteristic direction is non-degenerate

AU - Lapan, Sara W.

N1 - Funding Information:
The author would like to thank Mattias Jonsson for his guidance in choosing and studying this problem. The author would also like to thank the referee who provided useful comments. The author was supported in part by the NSF grants DMS-0901073 and DMS-1001740, and by the NSF RTG grant DMS-0602191.

PY - 2013/9

Y1 - 2013/9

N2 - Let f be a holomorphic germ on ℂ2 that fixes the origin and is tangent to the identity. Assume that f has a non-degenerate characteristic direction [v]. Hakim gave conditions that guarantee the existence of attracting domains along [v], however, when f has only one characteristic direction, these conditions are not satisfied. We prove that when [v] is unique, the existence results still hold. In particular, there is a domain Ω whose points converge to the origin along [v] and, on Ω, f is conjugate to a translation. Furthermore, if f is a global automorphism, the corresponding domain of attraction is a Fatou-Bieberbach domain.

AB - Let f be a holomorphic germ on ℂ2 that fixes the origin and is tangent to the identity. Assume that f has a non-degenerate characteristic direction [v]. Hakim gave conditions that guarantee the existence of attracting domains along [v], however, when f has only one characteristic direction, these conditions are not satisfied. We prove that when [v] is unique, the existence results still hold. In particular, there is a domain Ω whose points converge to the origin along [v] and, on Ω, f is conjugate to a translation. Furthermore, if f is a global automorphism, the corresponding domain of attraction is a Fatou-Bieberbach domain.

KW - Fatou-Bieberbach domains

KW - Holomorphic dynamics

KW - domains of attraction

KW - tangent to the identity

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U2 - 10.1142/S0129167X13500833

DO - 10.1142/S0129167X13500833

M3 - Article

AN - SCOPUS:84887451860

VL - 24

JO - International Journal of Mathematics

JF - International Journal of Mathematics

SN - 0129-167X

IS - 10

M1 - 1350083

ER -