Abstract
In this note we extend to arbitrary dimensions a couple of results due respectively to Mattei-Moussu and to Camara-Scardua in dimension 2. We also provide examples of singular foliations having a Siegel-type singularity and answering a question independently raised by Abate and by Genzmer.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abate. The residual index and the dynamics of holomorphic maps tangent to the identity. Duke Math. J., 107(1) (2001), 173–207.
M. Abate. Discrete holomorphic local dynamical systems. Holomorphic dynamical systems, 155, Lecture Notes in Math., 1998, Springer, Berlin (2010).
M. Arizzi and J. Raissy. On Ecalle-Hakim’s theorem in holomorphic dynamics, to appear in proceedings of “Frontiers in Complex Dynamics”, (2011).
S. Bochner. Compact groups of differentiable transformations. Ann. of Math., 46 (1945), 372–381.
F. Brochero-Martínez. Groups of germs of analytic diffeomorphisms in (C2, 0). J. Dynam. Control Systems, 9(1) (2003), 1–32.
C. Camacho, N. Kuiper and J. Palis. The topology of holomorphic flows with singularities. Publ. Math. I. H. E. S., 48 (1978), 5–38.
L. Camara and B. Scardua. On the integrability of holomorphic vector fields. Discrete and Continuous Dynamical Systems, 25(2) (2009), 481–493.
P. Elizarov and Y. Il’yashenko. Remarks in the orbital analytic classification of germs of vector fields. Math. USSR Sb., 123(1) (1983), 111–126.
Y. Genzmer. private communication.
M. Hakim. Analytic transformations of (Cp, 0) tangent to the identity. Duke Math. J., 92(2) (1998), 403–428.
F. Klein, Lectures on the Icosahedron, Dover Phoenix Editions, NY (2003).
N. Ladis. Topological invariants of complex linear flows (in Russian). Differencial’nye Uraunenija, 12(12) (1976), 2159–2169.
J.-F. Mattei and R. Moussu. Holonomie et intégrales premières. Ann. Sc. E. N. S. Série IV, 13(4) (1980), 469–523.
J. Milnor. Dynamics in One Complex Variable, 3rd ed., Annals of Mathematics Studies, 160, Princeton University Press (2006).
V. Naishul. Topological invariants of analytic and area-preserving mappings and their applicationto analytic differential equations in C2 and CP2. Trans. Moscow Math. Soc., 42 (1983), 239–250.
S. Pinheiro and H. Reis. Topological aspects of completely integrable foliations. Journal London Math. Soc. (2), 89(2) (2014), 415–433.
J. C. Rebelo and H. Reis. Local Theory of Holomorphic Foliations and Vector Fields. Lecture Notes available at: http://arxiv. org/abs/1101.4309.
H. Reis. Equivalence and semi-completude of foliations. Nonlinear Anal., 64(8) (2006), 1654–1665.
B. Wehrfritz. Infinite linear groups. Erg. der Mathematik (1976).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Rebelo, J.C., Reis, H. A note on integrability and finite orbits for subgroups of Diff (Cn, 0). Bull Braz Math Soc, New Series 46, 469–490 (2015). https://doi.org/10.1007/s00574-015-0101-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-015-0101-2