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Degree-growth of monomial maps

Published online by Cambridge University Press:  01 October 2007

BORIS HASSELBLATT  and
JAMES PROPP
BORIS HASSELBLATT
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, USA (email: [email protected])
JAMES PROPP
Affiliation:
Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell,MA 01854, USA (email: [email protected])
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Abstract

For projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 that imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.

Type
SURVEY
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1375 - 1397
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Abarenkova, N., Anglès d’Auriac, J.-C., Bourkraa, S. and Maillard, J.-M.. Growth-complexity spectrum of some discrete dynamical systems. Phys. D 130 (1999), 27–42.CrossRefGoogle Scholar
[2]Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
[3]Arnold, V. I.. Dynamics of complexity of intersections. Bol. Soc. Brasil. Mat. 21 (1990), 1–10.CrossRefGoogle Scholar
[4]Bedford, E. and Kim, K.. On the degree growth of birational mappings in higher dimension. J. Geom. Anal. 14 (2004), 567–596.CrossRefGoogle Scholar
[5]Bellon, M. P.. Algebraic entropy of birational maps with invariant curves. Lett. Math. Phys. 50 (1999), 79–90.CrossRefGoogle Scholar
[6]Bellon, M. P. and Viallet, C.-M.. Algebraic entropy. Comm. Math. Phys. 204(2) (1999), 425–437 (chao-dyn/9805006).CrossRefGoogle Scholar
[7]Boucksom, S., Favre, C. and Jonsson, M.. Degree growth of meromorphic surface maps. Duke Math. J. to appear (math.DS/0608267).Google Scholar
[8]Bourkraa, S. and Maillard, J.-M.. Factorization properties of birational mappings. Phys. A 220 (1995), 403–470.CrossRefGoogle Scholar
[9]Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414; Erratum, 181 (1973), 509–510.CrossRefGoogle Scholar
[10]Diller, J. and Favre, C.. Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123 (2001), 1135–1169.CrossRefGoogle Scholar
[11]Dinaburg, E. I.. The relation between topological entropy and metric entropy. Sov. Math. Dokl. 11 (1970), 13–16.Google Scholar
[12]Dinh, T.-C. and Sibony, N.. Une borne supérieure pour l’entropie topologique d’une application rationnelle. Ann. of Math. (2) 161 (2005), 1637–1644.CrossRefGoogle Scholar
[13]Falqui, G. and Viallet, C.-M.. Singularity, complexity, and quasi-integrability of rational mappings. Comm. Math. Phys. 154 (1993), 111–125.CrossRefGoogle Scholar
[14]Favre, C.. Les applications monomiales en deux dimensions. Michigan Math. J. 51 (2003), 467–475.CrossRefGoogle Scholar
[15]Favre, C. and Jonsson, M.. Eigenvaluations. Preprint, 2004, math.DS/0410417.Google Scholar
[16]Fomin, S. and Reading, N.. Root systems and generalized associahedra. Preprint, 2005, math.CO/0505518.Google Scholar
[17]Fomin, S. and Zelevinsky, A.. The Laurent phenomenon. Adv. Appl. Math. 28(2) (2002), 119–144 (math.CO/0104241).CrossRefGoogle Scholar
[18]Friedland, S. and Milnor, J.. Dynamical properties of plane polynomial automorphisms. Ergod. Th. & Dynam. Sys. 9 (1989), 67–99.CrossRefGoogle Scholar
[19]Friedland, S.. Entropy of rational selfmaps of projective varieties. Proc. Int. Conf. on Dynamical Systems and Related Topics (Advanced Series on Dynamical Systems, 9). Ed. K. Shiraiwa. World Scientific, Singapore, 1990, pp. 128–140.Google Scholar
[20]Friedland, S.. Entropy of polynomial and rational maps. Ann. of Math. (2) 133 (1991), 359–368.CrossRefGoogle Scholar
[21]Friedland, S.. Entropy of algebraic maps. J. Fourier Anal. Appl. (Kahane Special Issue) (1995), 215–228.Google Scholar
[22]Friedland, S.. Entropy of holomorphic and rational maps: a survey. Dynamics, Ergodic Theory and Geometry. Cambridge University Press, Cambridge, 2006.Google Scholar
[23]Gale, D.. The strange and surprising saga of the Somos sequences. Math. Intelligencer 13 (1991), 40–42 and Somos Sequence Update, Math. Intelligencer 13, (1991), 49–50. (Republished in D. Gale. Tracking the Automatic Ant. Springer, Berlin, 1998, pp. 2–5, 22–24.)Google Scholar
[24]Grammaticos, B., Ramani, A. and Papageorgiou, V. G.. Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67 (1991), 1825–1826.CrossRefGoogle ScholarPubMed
[25]Guedj, V.. Entropie topologique des applications méromorphes. Ergod. Th. & Dynam. Sys. 25(6) (2005), 1847–1855.CrossRefGoogle Scholar
[26]Hartshorne, R.. Algebraic Geometry. Springer, Berlin, 1977.CrossRefGoogle Scholar
[27]Hasselblatt, B., Nitecki, Z. and Propp, J.. Topological entropy for nonuniformly continuous maps. Discrete Contin. Dyn. Syst. (Yakov Pesin Birthday Issue) to appear (math.DS/0511495).Google Scholar
[28]Hietarinta, J. and Viallet, C.-M.. Singularity confinement and chaos in discrete systems. Phys. Rev. Lett. 81 (1998), 325–328.CrossRefGoogle Scholar
[29]Hientarinta, J. and Viallet, C.. Singularity confinement and degree growth. Symmetries and Integrability of Difference Equations (SIDE III) (CRM Proceedings and Lecture Notes, 25). Eds. D. Levi and O. Ragnisco. American Mathematical Society, Providence, RI, 2000, pp. 209–216. Available from http://www.lpthe.jussieu.fr/∼viallet/.CrossRefGoogle Scholar
[30]Hone, A.. Diophantine nonintegrability of a third order recurrence with the Laurent property. J. Phys. A: Math. Gen. 39 (2006), L171–L177.CrossRefGoogle Scholar
[31]Hone, A.. Singularity confinement for maps with the Laurent property. Phys. Lett. A 361 (2007), 341–345 (nlin.SI/0602007).CrossRefGoogle Scholar
[32]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[33]Lafortune, S., Ramani, A., Grammaticos, B., Ohta, Y. and Tamizhmani, K. M.. Blending two discrete integrability criteria: singularity confinement and algebraic entropy. Bäcklund and Darboux Transformations. The Geometry of Solitons (Halifax, NS, 1999) (CRM Proc. Lecture Notes, 29). American Mathematical Society, Providence, RI, 2001, pp. 299–311 (nlin.SI/0104020).CrossRefGoogle Scholar
[34]Lind, D. and Ward, T.. Automorphisms of solenoids and p-adic entropy of dynamical systems. Ergod. Th. & Dynam. Sys. 8 (1988), 411–419.CrossRefGoogle Scholar
[35]Maegawa, K.. Quadratic polynomial automorphisms of dynamical degree golden ratio of C 3. Ergod. Th. & Dynam. Sys. 21 (2001), 823–832.CrossRefGoogle Scholar
[36]Mumford, D.. Algebraic Geometry, Vol. 1: Complex Projective Varieties. Springer, Berlin, 1995.Google Scholar
[37]Musiker, G. and Propp, J.. Combinatorial interpretations for rank-two cluster algebras of affine type. Electron. J. Combin. 14 (2007), paper 15 (math.CO/0602408).CrossRefGoogle Scholar
[38]Nguyén, V.. Algebraic degrees for iterates of meromorphic self-maps of . Publ. Mat. 50(2) (2006), 457–473 (math.CV/0603545).CrossRefGoogle Scholar
[39]Osin, D. V.. Algebraic entropy of elementary amenable groups. Geom. Dedicata 107 (2004), 133–151 (math.GR/0404075).CrossRefGoogle Scholar
[40]Ramani, A., Grammaticos, B., Lafortune, S. and Ohta, Y.. Linearisable mappings and the low-growth criterion. J. Phys. A: Math. Gen. 33(31) (2000), L287–L292 (nlin.SI/0104015).CrossRefGoogle Scholar
[41]Russakovskii, A. and Shiffman, B.. Value distribution for sequences of rational mappings and complex dynamics. Indiana Univ. Math. J. 46 (1997), 897–932.CrossRefGoogle Scholar
[42]Sherman, P. and Zelevinsky, A.. Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. Moscow Math. J. 4 (2004), 947–974.CrossRefGoogle Scholar
[43]Speyer, D. and Sturmfels, B.. Tropical mathematics. Preprint, 2004, math.CO/0408099 (Notes prepared for Clay Mathematics Institute Senior Scholar Lecture, by Bernd Sturmfels, Park City, UT, 22 July 2004).Google Scholar
[44]Stanley, R. P.. Enumerative Combinatorics Vol. 1. With a foreword by Gian-Carlo Rota. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986.Google Scholar
[45]Takenawa, T.. A geometric approach to singularity confinement and algebraic entropy. J. Phys. A: Math. Gen. 34(10) (2001), L95–L102 (nlin.SI/0011037).CrossRefGoogle Scholar
[46]Takenawa, T.. Algebraic entropy and the space of initial values for discrete dynamical systems. Symmetries and Integrability of Difference Equations (Tokyo, 2000). J. Phys. A: Math. Gen. 34(48) (2001), 10 533–10 545 (nlin.SI/0103011).CrossRefGoogle Scholar
[47]Takenawa, T., Eguchi, M., Grammaticos, B., Ohta, Y., Ramani, A. and Satsuma, J.. The space of initial conditions for linearisable mappings. Nonlinearity 16(2) (2003), 457–477 (nlin.SI/02040070).CrossRefGoogle Scholar
[48]Veselov, A. P.. Integrable maps. Russian Math. Surveys 46(5) (1991), 1–51. Russian original: Uspekhi Mat. Nauk. 46(5) (1991), 3–45.CrossRefGoogle Scholar
[49]Wielandt, H.. Unzerlegbare, nicht negative Matrizen (Engl.: Irreducible non-negative matrices). Math. Z. 52 (1950), 642–648.CrossRefGoogle Scholar
[50]Zelevinsky, A.. Semicanonical basis generators of the cluster algebra of type A 1(1). Preprint, 2006, math.RA/0606775.CrossRefGoogle Scholar