Abstract
In this paper we investigate normalization of rational functions, reducing in the sense of conjugation to monomials or more general power functions. We give conditions for the normalization by computing minimal irreducible decomposition of algebraic varieties. We use those conditions to compute the general n-th order iterates and iterative roots for those rational functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baker, I.N.: The iteration of polynomials and transcendental entire functions. J. Aust. Math. Soc. Ser. A 30(4), 483–495 (1980/81)
Baron, K., Jarczyk, W.: Recent results on functional equations in a single variable, perspectives and open problems. Aequ. Math. 61(1–2), 1–48 (2001)
Beardo, A.F.: Iteration of Rational Functions: Complex Analytic Dynamical Systems. Springer, New York (1991)
Becker, T., Weispfenning, V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, New York (1993)
Belitskii, G., Lubish, Y.: The Abel equation and total solvability of linear functional equations. Stud. Math. 127, 81–89 (1998)
Belitskii, G., Lubish, Y.: The real-analytic solutions of the Abel functional equations. Stud. Math. 134(2), 135–141 (1999)
Bhattacharyya, P., Arumaraj, Y.E.: On the iteration of polynomials of degree 4 with real coeffcients. Ann. Acad. Sci. Fenn. Math. 6(2), 197–203 (1981)
Böttcher, L.E.: The principal laws of convergence of iterates and their application to analysis. Izv. Kazan. Fiz. Mat. Obshch. 14, 155–234 (1904). (in Russian)
Branner, B., Hubbard, J.H.: The iteration of cubic polynomials I. Acta Math. (Djursholm) 160: 143–206 (1988/1992)
Branner, B., Hubbard, J.H.: The iteration of cubic polynomials II. Acta Math. (Djursholm) 169: 229–325 (1988/1992)
Buchberger, B.: Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensional polynomideal. PhD thesis, Universität Innsbruck, Austria (1965)
Chen, X., Shi, Y., Zhang, W.: Planar quadratic degree-preserving maps and their iteration. Results Math. 55, 39–63 (2009)
Chow, S.N., Li, C., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge (2007)
Cowen, C.C.: Analytic solutions of Böttcher’s functional equation in the unit disk. Aequ. Math. 24, 187–194 (1982)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Springer, New York (2007)
Douady, A., Hubbard, J.H.: Etude Dynamique des Polynomes Complexes I, II. Publ. Math. Orsay, Paris, (1984–1985)
Eljoseph, N.: On the iteration of linear fractional transformations. Am. Math. Mon. 75(4), 362–366 (1968)
Fatou, P.: Sur lés équations fonctionnelles. Bulletin de la Société Mathématique de France 47, 161–271 (1919)
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Boston (1994)
Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomials. J. Symb. Comput. 6, 146–167 (1988)
Goetgheluck, P.: Computing binomial coefficients. Am. Math. Mon. 94, 360–365 (1987)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, New York (1994)
Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra, 2nd edn. Springer, Berlin (2008)
Koenigs, G.: Recherches sur les intégrals certains équations fonctionnelles. Ann. Sci. Ecole Norm. Sup. 1(3), 3–41 (1884)
Kuczma, M.: Functional Equations in a Single Variable. PWN, Warszawa (1968)
Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)
Laitochová, J.: Group iteration for Abels functional equation. Nonlinear Anal. Hybrid Syst. 1, 95–102 (2007)
Li, L., Yang, D., Zhang, W.: A note on iterative roots of PM functions. J. Math. Anal. Appl. 341(2), 1482–1486 (2008)
Liu, L., Jarczyk, W., Li, L., Zhang, W.: Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2. Nonlinear Anal. 75(1), 286–303 (2012)
Lundberg, A.: On iterated functions with asymptotic conditions at a fix point. Arkiv för Mat. 5, 193–206 (1963)
Ritt, J.: On the iteration of rational functions. Trans. Am. Math. Soc. 21(3), 348–356 (1920)
Romanovski, V.G., Shafer, D.S.: The Center and Cyclicity Problems: A Computational Algebra Approach. Birkhäuser, Boston (2009)
Sell, G.R.: Smooth linearization near a fixed point. Am. J. Math. 107, 1035–1091 (1985)
Stowe, D.: Linearization in two dimensions. J. Differ. Equ. 63, 183–226 (1986)
Targonski, Gy I.: Schröder’s equation and a generalization of the Chebyshev polynomials. Trans. N. Y. Acad. Sci. 27(2), 600–606 (1965)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003)
Yu, Z., Yang, L., Zhang, W.: Discussion on polynomials having polynomial iterative roots. J. Symb. Comput. 47(10), 1154–1162 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC #11771307, #11726623, #11521061 and #11701476, and Scientific Research Fund of Sichuan Provincial Education Department under Grant 18ZA0242. Corresponding to Zhiheng Yu (yuzhiheng9@163.com).
Rights and permissions
About this article
Cite this article
Liu, X., Yu, Z. & Zhang, W. Conjugation of Rational Functions to Power Functions and Applications to Iteration. Results Math 73, 31 (2018). https://doi.org/10.1007/s00025-018-0801-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0801-1