A dynamical Mordell-Lang property on the disk
HTML articles powered by AMS MathViewer
- by Ming-Xi Wang PDF
- Trans. Amer. Math. Soc. 369 (2017), 2183-2204 Request permission
Abstract:
We prove that two finite endomorphisms of the unit disk with degree at least two have orbits with infinite intersections if and only if they have a common iterate.References
- Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911, DOI 10.1515/9781400874538
- Matthew Baker, A finiteness theorem for canonical heights attached to rational maps over function fields, J. Reine Angew. Math. 626 (2009), 205–233. MR 2492995, DOI 10.1515/CRELLE.2009.008
- Yuri F. Bilu and Robert F. Tichy, The Diophantine equation $f(x)=g(y)$, Acta Arith. 95 (2000), no. 3, 261–288. MR 1793164, DOI 10.4064/aa-95-3-261-288
- Gregory S. Call and Joseph H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205. MR 1255693
- A. B. Bogatyrëv, Chebyshev representation of rational functions, Mat. Sb. 201 (2010), no. 11, 19–40 (Russian, with Russian summary); English transl., Sb. Math. 201 (2010), no. 11-12, 1579–1598. MR 2768552, DOI 10.1070/SM2010v201n11ABEH004123
- K. Chandrasekharan, Elliptic functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396, DOI 10.1007/978-3-642-52244-4
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
- Dragos Ghioca, Thomas J. Tucker, and Michael E. Zieve, Intersections of polynomials orbits, and a dynamical Mordell-Lang conjecture, Invent. Math. 171 (2008), no. 2, 463–483. MR 2367026, DOI 10.1007/s00222-007-0087-5
- Dragos Ghioca, Thomas J. Tucker, and Michael E. Zieve, Linear relations between polynomial orbits, Duke Math. J. 161 (2012), no. 7, 1379–1410. MR 2922378, DOI 10.1215/00127094-1598098
- M. D. Lutovac, D. V. Tosic, and B. L. Evans, Filter Design for Signal Processing using MATLAB and Mathematica, Prentice Hall, New Jersey, 2001.
- Tuen Wai Ng and Ming-Xi Wang, Ritt’s theory on the unit disk, Forum Math. 25 (2013), no. 4, 821–851. MR 3089751, DOI 10.1515/form.2011.136
- J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), no. 1, 51–66. MR 1501189, DOI 10.1090/S0002-9947-1922-1501189-9
- Algebraic geometry. I, Encyclopaedia of Mathematical Sciences, vol. 23, Springer-Verlag, Berlin, 1994. Algebraic curves. Algebraic manifolds and schemes; A translation of Current problems in mathematics. Fundamental directions, Vol. 23 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988; Translation by D. Coray and V. N. Shokurov; Translation edited by I. R. Shafarevich. MR 1287418
- Carl L. Siegel, Über einige Anwendungen diophantischer Approximationen [reprint of Abhandlungen der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 1929, Nr. 1], On some applications of Diophantine approximations, Quad./Monogr., vol. 2, Ed. Norm., Pisa, 2014, pp. 81–138 (German). MR 3330350
- M. X. Wang, http://hub.hku.hk/handle/123456789/51854, MPhil Thesis (2008).
- M. X. Wang, Rational points and transcendental points, PhD thesis (2011).
- M. E. Zieve and P. Müller, On Ritt’s polynomial decomposition theorems, arXiv:0807.3578.
- E. I. Zolotarev, Application of elliptic functions to the question of the functions least and most deviating from zero, 1877.
Additional Information
- Ming-Xi Wang
- Affiliation: Department of Mathematics, University of Salzburg, Hellbrunnerstr. 34/I, 5020 Salzburg, Austria
- MR Author ID: 1031849
- Email: mingxi.waeng@gmail.com
- Received by editor(s): December 20, 2013
- Received by editor(s) in revised form: April 12, 2015, and June 16, 2015
- Published electronically: October 31, 2016
- Additional Notes: The author was partially supported by a scholarship of ZGSM and the SNF grant and Austrian Science Fund(FWF): P24574.
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 2183-2204
- MSC (2010): Primary 11Z05; Secondary 37P05
- DOI: https://doi.org/10.1090/tran/6775
- MathSciNet review: 3581231