Dynamics of functions arising from Pisot and Salem polynomials

详细信息    查看全文

• 作者：Somjate Chaiya ; Aimo Hinkkanen
• 关键词：Primary 37F10 ; Secondary 11R06 ; Complex dynamics ; polynomials ; Salem numbers
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：17
• 期：2
• 页码：371-378
• 全文大小：516 KB
• 参考文献：1.A. F. Beardon, Iteration of Rational Functions. Grad. Texts in Math. 132, Springer, New York, 1991.
  2.M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J.-P. Schreiber, Pisot and Salem Numbers. Birkh?user, Basel, 1992.
  3.P. Blanchard, The dynamics of Newton’s method. In: Complex Dynamical Systems (Cincinnati, OH, 1994), Proc. Sympos. Appl. Math. 49, Amer. Math. Soc., Providence, RI, 1994, 139-54.
  4.Boyd D.W.: Small Salem numbers. Duke Math. J. 44, 315-28 (1977)MATH MathSciNet CrossRef
  5.Chaiya S., Hinkkanen A.: Location of the critical points of certain polynomials. Ann. Univ. Mariae Curie-Sk?odowska Sect. A 67, 1- (2013)MATH MathSciNet
  6.Mayer M., Schleicher D.: Immediate and virtual basins of Newton’s method for entire functions. Ann. Inst. Fourier (Grenoble) 56, 325-36 (2006)MATH MathSciNet CrossRef
  7.H.-O. Peitgen (ed.), Newton’s Method and Dynamical Systems. Springer, New York, 1989.
  8.F. Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps. In: Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ. 23, Polish Scientific Publishers, Warsaw, 1989, 229-35.
  9.Salem R.: A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan. Duke Math. J. 11, 103-08 (1944)MATH MathSciNet CrossRef
  10.Salem R.: Power series with integral coefficients. Duke Math. J. 12, 153-72 (1945)MATH MathSciNet CrossRef
  11.T. Sheil-Small, Complex Polynomials. Cambridge Stud. Adv. Math. 75, Cambridge University Press, Cambridge, 2002
  12.M. Shishikura, The connectivity of the Julia set and fixed points. In: Complex Dynamics, D. Schleicher (ed.), A. K. Peters, Wellesley, MA, 2009, 257-76.
  13.Siegel C.L.: Algebraic integers whose conjugates lie in the unit circle. Duke Math. J. 11, 597-02 (1944)MATH MathSciNet CrossRef
  14.Walsh J.L.: Note on the location of zeros of the derivative of a rational function whose zeros and poles are symmetric in a circle. Bull. Amer. Math. Soc. 45, 462-70 (1939)MathSciNet CrossRef
  
• 作者单位：Somjate Chaiya (1) (2)
  Aimo Hinkkanen (3)
  
  1. Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom, 73000, Thailand
  2. Centre of Excellence in Mathematics, Commission on Higher Education (CHE), Si Ayutthaya Road, Bangkok, 10400, Thailand
  3. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, IL, 61801, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746

文摘

Let Q(z) = z n P(z) ?z deg P P(z ?), where P is the minimal polynomial of a Pisot number. Boyd [Duke Math. J. 44 (1977), 315-28] showed that, for \({n > {\rm deg}\,P - 2 \frac{p'(1)}{P(1)}}\), Q is the product of cyclotomic polynomials and the minimal polynomial of a Salem number, say α. In this paper, we study the dynamics of the Newton map N = z ?Q/Q-induced by Q in the immediate basin U _α of α. We establish that N is a 2-fold covering map of U _α onto itself. Furthermore, there exists a conformal mapping φ of U _α onto the open unit disk \({\mathbb{D}}\) such that (\({\varphi \circ N \circ \varphi^{-1})(z) = z^{2}}\) for all \({z \in \mathbb{D}}\). Mathematics Subject Classification Primary 37F10 Secondary 11R06