Abstract
V. I. Arnold proved in 1993 that the intersection multiplicity between two germs of analytic subvarieties at a fixed point of a holomorphic invertible self-map remains bounded when one of the germs is dragged by iterations of the self-map. The proof is based on the Skolem-Mahler-Lech theorem on zeros in recurrent sequences.
We give a different proof, based on the Noetherianity of certain algebras, which allows one to generalize Arnold’s theorem for local actions of arbitrary finitely generated commutative groups, with both discrete and infinitesimal generators. Simple examples show that for non-commutative groups the analogous assertion fails.
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References
V. I. Arnol’d, Bounds for Milnor numbers of intersections in holomorphic dynamical systems, in Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 379–390. MR1215971 (94i:32039)
V. I. Arnol’d, Dynamics of complexity of intersections, Boletim da Sociedade Brasileira de Matemática (N.S.) 21 (1990), 1–10. DOI 10.1007/BF01236277. MR1139553 (93c:58031).
V. I. Arnol’d, Dynamics of intersections, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 77–84. MR1039340 (91f:58010)
V. I. Arnol’d, Local problems of analysis, Vestnik Moskovskogo Universiteta. Serija I. Matematika, Mehanika 25 (1970), 52–56 (Russian). MR0274875 (43 #633)
V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Vol. I, Monographs in Mathematics, Vol. 82, Birkhäuser Boston Inc., Boston, MA, 1985. The classifcation of critical points, caustics and wave fronts. MR777682 (86f:58018)
G. Binyamini and D. Novikov, Intersection multiplicities of Noetherian functions, Advances in Mathematics 231 (2012), 3079–3093. available at http://arxiv.org/abs/1108.1700.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Mathematical Surveys and Monographs, Vol. 104, American Mathematical Society, Providence, RI, 2003. MR1990179 (2004c:11015).
W. Gignac, On the growth of local intersection multiplicities in holomorphic dynamics: a conjecture of Arnold, arXiv:1212.5272 [math.DS] (2012), available at http://arxiv.org/abs/1212.5272.
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. Graduate Texts in Mathematics, Vol. 9. MR0323842 (48 #2197)
Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, Vol. 86, American Mathematical Society, Providence, RI, 2008. MR2363178 (2009b:34001)
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, Vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by A. Katok and Leonardo Mendoza. MR1326374 (96c:58055)
M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189–191. MR0350782 (50 #3274)
O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York, 1975. Reprint of the 1960 edition; MR0389876 (52 #10706)
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Seigal, A.L., Yakovenko, S. Local dynamics of intersections: V. I. Arnold’s theorem revisited. Isr. J. Math. 201, 813–833 (2014). https://doi.org/10.1007/s11856-014-1065-4
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DOI: https://doi.org/10.1007/s11856-014-1065-4