Invariant varieties for polynomial dynamical systems. (English) Zbl 1347.37145

Medvedev, Alice; Scanlon, Thomas

Ann. Math. (2) 179, No. 1, 81-177 (2014).

This work is concerned with the algebraic dynamical systems obtained by iterating coordinatewise a finite sequence of univariate polynomials over the complex numbers. The aim is an explicit description of the algebraic varieties that are invariant under such maps. This is achieved by representing polynomials as composition of indecomposable polynomials, which are regrouped into ‘clusters’, determined by combinatorial properties. From this (nearly canonical) decomposition, compositional identities are easily derived.
By considering a weaker form of skew-invariance for varieties, the authors import techniques from the model theory of difference fields (which also provides motivations for this work). The original problem is thus reduced to the characterisation of the affine plane curves that are skew-invariant, in the case in which the polynomials involved are of a particular type (‘disintegrated’ polynomials).
This work may be viewed as a refinement of the classic work of J. F. Ritt [Trans. Am. Math. Soc. 23, 51–66 (1922; JFM 48.0079.01)] on compositional identities among polynomials, and of a related recent development by M. E. Zieve and P. Müller [“On Ritt’s polynomial decomposition theorems”, Preprint, arXiv:0807.3578]. Applications to algebraic dynamics include the proof of variants of two conjectures by S.-W. Zhang [in: Essays in geometry in memory of S. S. Chern. Somerville, MA: International Press. 381–430 (2006; Zbl 1207.37057)].

Reviewer: Franco Vivaldi (London)

MSC:

37P05	Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P15	Dynamical systems over global ground fields
37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
14J50	Automorphisms of surfaces and higher-dimensional varieties

Keywords:

algebraic dynamics; difference fields; polynomial decompositions

Citations:

JFM 48.0079.01; Zbl 1207.37057

References:

[1]	E. Amerik, F. Bogomolov, and M. Rovinsky, ”Remarks on endomorphisms and rational points,” Compos. Math., vol. 147, iss. 6, pp. 1819-1842, 2011. · Zbl 1231.14014
[2]	E. Amerik and F. Campana, ”Fibrations méromorphes sur certaines variétés à fibré canonique trivial,” Pure Appl. Math. Q., vol. 4, iss. 2, part 1, pp. 509-545, 2008. · Zbl 1143.14035
[3]	Z. Chatzidakis, ”Groups definable in ACFA,” in Algebraic Model Theory, Dordrecht: Kluwer Acad. Publ., 1997, vol. 496, pp. 25-52. · Zbl 0884.03038
[4]	Z. Chatzidakis and E. Hrushovski, ”Model theory of difference fields,” Trans. Amer. Math. Soc., vol. 351, iss. 8, pp. 2997-3071, 1999. · Zbl 0922.03054
[5]	Z. Chatzidakis and E. Hrushovski, ”Difference fields and descent in algebraic dynamics. I,” J. Inst. Math. Jussieu, vol. 7, iss. 4, pp. 653-686, 2008. · Zbl 1165.03014
[6]	Z. Chatzidakis, E. Hrushovski, and Y. Peterzil, ”Model theory of difference fields. II. Periodic ideals and the trichotomy in all characteristics,” Proc. London Math. Soc., vol. 85, iss. 2, pp. 257-311, 2002. · Zbl 1025.03026
[7]	D. Ghioca, T. J. Tucker, and S. Zhang, ”Towards a dynamical Manin-Mumford conjecture,” Int. Math. Res. Not., vol. 2011, iss. 22, pp. 5109-5122, 2011. · Zbl 1267.37110
[8]	M. Hindry, ”Autour d’une conjecture de Serge Lang,” Invent. Math., vol. 94, iss. 3, pp. 575-603, 1988. · Zbl 0638.14026
[9]	E. Hrushovski, ”The Manin-Mumford conjecture and the model theory of difference fields,” Ann. Pure Appl. Logic, vol. 112, iss. 1, pp. 43-115, 2001. · Zbl 0987.03036
[10]	E. Hrushovski and M. Itai, ”On model complete differential fields,” Trans. Amer. Math. Soc., vol. 355, iss. 11, pp. 4267-4296, 2003. · Zbl 1021.03024
[11]	A. Medvedev, Minimal sets in ACFA, ProQuest LLC, Ann Arbor, MI, 2007. · Zbl 1235.03069
[12]	A. Medvedev and T. Scanlon, Polynomial dynamics, 2009.
[13]	R. Pink and D. Roessler, ”On \(\psi\)-invariant subvarieties of semiabelian varieties and the Manin-Mumford conjecture,” J. Algebraic Geom., vol. 13, iss. 4, pp. 771-798, 2004. · Zbl 1072.14054
[14]	J. F. Ritt, ”On the iteration of rational functions,” Trans. Amer. Math. Soc., vol. 21, iss. 3, pp. 348-356, 1920. · JFM 47.0312.01
[15]	J. F. Ritt, ”Prime and composite polynomials,” Trans. Amer. Math. Soc., vol. 23, iss. 1, pp. 51-66, 1922. · JFM 48.0079.01
[16]	J. F. Ritt, ”Permutable rational functions,” Trans. Amer. Math. Soc., vol. 25, iss. 3, pp. 399-448, 1923. · JFM 49.0712.02
[17]	T. Scanlon, ”Analytic difference rings,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2007, pp. 71-92. · Zbl 1104.03029
[18]	T. Scanlon, ”Local André-Oort conjecture for the universal abelian variety,” Invent. Math., vol. 163, iss. 1, pp. 191-211, 2006. · Zbl 1086.14020
[19]	J. H. Silverman, The Arithmetic of Dynamical Systems, New York: Springer-Verlag, 2007, vol. 241. · Zbl 1130.37001
[20]	S. Zhang, ”Distributions in algebraic dynamics,” in Surveys in Differential Geometry. Vol. X, Int. Press, Somerville, MA, 2006, vol. 10, pp. 381-430. · Zbl 1207.37057
[21]	M. Zieve and P. Müller, On Ritt’s polynomial decomposition theorems, 2008.