Fractional iteration of series and transseries
HTML articles powered by AMS MathViewer
- by G. A. Edgar PDF
- Trans. Amer. Math. Soc. 365 (2013), 5805-5832 Request permission
Abstract:
We investigate compositional iteration of fractional order for transseries. For any large positive transseries $T$ of exponentiality $0$, there is a family $T^{[s]}$ indexed by real numbers $s$ corresponding to iteration of order $s$. It is based on Abel’s Equation. We also investigate the question of whether there is a family $T^{[s]}$ all sharing a single support set. A subset of the transseries of exponentiality $0$ is divided into three classes (“shallow”, “moderate” and “deep”) with different properties related to fractional iteration.References
- Matthias Aschenbrenner and Lou van den Dries, Asymptotic differential algebra, Analyzable functions and applications, Contemp. Math., vol. 373, Amer. Math. Soc., Providence, RI, 2005, pp. 49–85. MR 2130825, DOI 10.1090/conm/373/06914
- Irvine Noel Baker, Zusammensetzungen ganzer Funktionen, Math. Z. 69 (1958), 121–163 (German). MR 97532, DOI 10.1007/BF01187396
- I. N. Baker, Permutable power series and regular iteration, J. Austral. Math. Soc. 2 (1961/1962), 265–294. MR 0140666, DOI 10.1017/S1446788700026884
- A. Cayley, On some numerical expansions. Quarterly Journal of Pure and Applied Mathematics 3 (1860) 366–369. Also in: Collected Works vol. IV, pp. 470–472
- P. M. Cohn, Universal algebra, Harper & Row, Publishers, New York-London, 1965. MR 0175948
- O. Costin, Topological construction of transseries and introduction to generalized Borel summability, Analyzable functions and applications, Contemp. Math., vol. 373, Amer. Math. Soc., Providence, RI, 2005, pp. 137–175. MR 2130829, DOI 10.1090/conm/373/06918
- O. Costin, Global reconstruction of analytic functions from local expansions and a new general method of converting sums into integrals. preprint, 2007. http://arxiv.org/abs/math/0612121
- O. Costin, M. D. Kruskal, and A. Macintyre (eds.), Analyzable functions and applications, Contemporary Mathematics, vol. 373, American Mathematical Society, Providence, RI, 2005. Papers from the International Workshop held in Edinburgh, June 17–21, 2002. MR 2130822, DOI 10.1090/conm/373
- Lou van den Dries, Angus Macintyre, and David Marker, Logarithmic-exponential series, Proceedings of the International Conference “Analyse & Logique” (Mons, 1997), 2001, pp. 61–113. MR 1848569, DOI 10.1016/S0168-0072(01)00035-5
- Jean Écalle, Nature du groupe des ordres d’itération complexes d’une transformation holomorphe au voisinage d’un point fixe de multiplicateur $1$, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A261–A263 (French). MR 340569
- G. A. Edgar, Transseries for beginners, Real Anal. Exchange 35 (2010), no. 2, 253–309. MR 2683600, DOI 10.14321/realanalexch.35.2.0253
- G. Edgar Transseries: composition, recursion, and convergence. forthcoming http://arxiv.org/abs/0909.1259v1 orhttp://www.math.ohio-state.edu/∼ edgar/preprints/trans_compo/
- G. Edgar, Transseries: ratios, grids, and witnesses. forthcominghttp://arxiv.org/abs/0909.2430v1 orhttp://www.math.ohio-state.edu/∼ edgar/preprints/trans_wit/
- G. Edgar, Tetration in transseries. forthcoming
- Paul Erdős and Eri Jabotinsky, On analytic iteration, J. Analyse Math. 8 (1960/61), 361–376. MR 125943, DOI 10.1007/BF02786856
- Joris van der Hoeven, Operators on generalized power series, Illinois J. Math. 45 (2001), no. 4, 1161–1190. MR 1894891
- J. van der Hoeven, Transseries and real differential algebra, Lecture Notes in Mathematics, vol. 1888, Springer-Verlag, Berlin, 2006. MR 2262194, DOI 10.1007/3-540-35590-1
- Joris van der Hoeven, Transserial Hardy fields, Astérisque 323 (2009), 453–487 (English, with English and French summaries). MR 2647983
- A. Korkine, Sur un problème d’interpolation. Bulletin des Sciences Mathématiques et Astronomiques (2) 6 (1882) 228–242
- Marek Kuczma, Functional equations in a single variable, Monografie Matematyczne, Tom 46, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. MR 0228862
- Marek Kuczma, Bogdan Choczewski, and Roman Ger, Iterative functional equations, Encyclopedia of Mathematics and its Applications, vol. 32, Cambridge University Press, Cambridge, 1990. MR 1067720, DOI 10.1017/CBO9781139086639
- Salma Kuhlmann, Ordered exponential fields, Fields Institute Monographs, vol. 12, American Mathematical Society, Providence, RI, 2000. MR 1760173, DOI 10.1090/fim/012
- L. S. O. Liverpool, Fractional iteration near a fix point of multiplier $1$, J. London Math. Soc. (2) 9 (1974/75), 599–609. MR 364611, DOI 10.1112/jlms/s2-9.4.599
- Maxwell Rosenlicht, Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc. 299 (1987), no. 1, 261–272. MR 869411, DOI 10.1090/S0002-9947-1987-0869411-2
- Lee A. Rubel, Some research problems about algebraic differential equations. II, Illinois J. Math. 36 (1992), no. 4, 659–680. MR 1215800
- H. F. Wilf, Generatingfunctionology. Academic Press, Boston, 1990.http://www.math.upenn.edu/˜wilf/DownldGF.html
Additional Information
- G. A. Edgar
- Affiliation: Department of Mathematics, The Ohio State University, 231 West Eighteenth Avenue, Columbus, Ohio 43210
- Email: edgar@math.ohio-state.edu
- Received by editor(s): February 25, 2010
- Received by editor(s) in revised form: December 29, 2011
- Published electronically: July 10, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5805-5832
- MSC (2010): Primary 03C64; Secondary 41A60, 39B12, 30B10
- DOI: https://doi.org/10.1090/S0002-9947-2013-05784-4
- MathSciNet review: 3091266