Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-14T12:03:55.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical invariants of monomial correspondences

Part of: Special varieties Projective and enumerative geometry Complex dynamical systems Curves

Published online by Cambridge University Press:  02 June 2020

NGUYEN-BAC DANG  and
ROHINI RAMADAS
NGUYEN-BAC DANG
Affiliation:
Stony Brook University, Institute of Mathematical Sciences, Stony Brook, NY 11794, USA email [email protected]
ROHINI RAMADAS
Affiliation:
Brown University, 219 Kassar House, 151 Thayer Street, Providence, RI 02912, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We focus on various dynamical invariants associated to monomial correspondences on toric varieties, using algebraic and arithmetic geometry. We find a formula for their dynamical degrees, relate the exponential growth of the degree sequences to a strict log-concavity condition on the dynamical degrees and compute the asymptotic rate of the growth of heights of points of such correspondences.

Keywords

arithmetic and algebraic dynamicsmonomial correspondancesdynamical degrees

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 14N99: None of the above, but in this section 14M99: None of the above, but in this section 37F05: Relations and correspondences
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2000 - 2015
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blanc, J. and Cantat, S.. Dynamics of birational transformations of projective surfaces. J. Amer. Math. Soc. 29 (2016), 415–471.CrossRefGoogle Scholar
Baker, M. and DeMarco, L.. Preperiodic points and unlikely intersections. Duke Math. J. 159(1) (2011), 1–29.CrossRefGoogle Scholar
Blanc, J. and Déserti, J.. Degree growth of birational maps of the plane. Ann. Sc. Norm. Supér. Pisa Cl. Sci. (5) 14(2) (2015), 507–533.Google Scholar
Bell, J. P., Diller, J. and Jonsson, M.. A transcendental dynamical degree. Preprint, 2019, arXiv:1907.00675.Google Scholar
Bedford, E. and Kim, K.. Linear recurrences in the degree sequences of monomial mappings. Ergod. Th. & Dynam. Sys. 28(5) (2008), 1369–1375.CrossRefGoogle Scholar
Bedford, E. and Smillie, J.. Polynomial diffeomorphisms of C 2 III. Ergodicity, exponents and entropy of the equilibrium measure. Math. Ann. 294(3) (1992), 395–420.CrossRefGoogle Scholar
Cantat, S.. Sur les groupes de transformations birationnelles des surfaces. Ann. of Math. (2) 174(1) (2011), 299–340.CrossRefGoogle Scholar
Cerveau, D. and Déserti, J.. Birational maps preserving the contact structure on ℙℂ 3 . J. Math. Soc. Japan 70(2) (2018), 573–615.CrossRefGoogle Scholar
Cantat, S. and Zeghib, A.. Holomorphic actions, Kummer examples, and Zimmer program. Ann. Sci. Éc. Norm. Supér. (4) 45(3) (2012), 447–489.CrossRefGoogle Scholar
Dang, N.-B.. Degrees of iterates of rational maps on normal projective varieties. Proc. London Math. Soc. to appear. Preprint, 2017, arXiv:1701.07760.Google Scholar
Dang, N.-B.. Degree growth for tame automorphisms of an affine quadric threefold. Preprint, 2018, arXiv:1810.09094.Google Scholar
Diller, J., Dujardin, R. and Guedj, V.. Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure. Comment. Math. Helv. 86(2) (2011), 277–316.CrossRefGoogle Scholar
DeMarco, L.. Bifurcations, intersections, and heights. Algebra Number Theory 10(5) (2016), 1031–1056.CrossRefGoogle Scholar
Diller, J. and Favre, C.. Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123(6) (2001), 1135–1169.CrossRefGoogle Scholar
Douady, A. and Hubbard, J. H.. A proof of Thurston’s topological characterization of rational functions. Acta Math. 171(2) (1993), 263–297.CrossRefGoogle Scholar
Déserti, J. and Han, F.. On cubic birational maps of ℙℂ 3 . Bull. Soc. Math. France 144(2) (2016), 217–249.CrossRefGoogle Scholar
Déserti, J. and Leguil, M.. Dynamics of a family of polynomial automorphisms of ℂ3 , a phase transition. J. Geom. Anal. 28(1) (2018), 190–224.CrossRefGoogle Scholar
DeMarco, L. and Mavraki, N. M.. Variation of canonical height and equidistribution. Preprint, 2017, arXiv:1701.07947.Google Scholar
Dinh, T.-C. and Sibony, N.. Groupes commutatifs d’automorphismes d’une variété kählérienne compacte. Duke Math. J. 123(2) (2004), 311–328.Google Scholar
Dinh, T.-C. and Sibony, N.. Dynamics of regular birational maps in ℙ k . J. Funct. Anal. 222(1) (2005), 202–216.CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N.. Une borne supérieure pour l’entropie topologique d’une application rationelle. Ann. of Math. (2) 161(3) (2005), 1637–1644.CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N.. Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv. 81(1) (2006), 221–258.CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N.. Upper bound for the topological entropy of a meromorphic correspondence. Israel J. Math. 163 (2008), 29–44.CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N.. Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings. Holomorphic Dynamical Systems (Lecture Notes in Mathematics, 1998) . Springer, Berlin, 2010.Google Scholar
De Thélin, H. and Vigny, G.. Entropy of meromorphic maps and dynamics of birational maps. Mém. Soc. Math. France (N.S.)(122) (2010).CrossRefGoogle Scholar
DeMarco, L., Wang, X. and Ye, H.. Torsion points and the Lattès family. Amer. J. Math. 138(3) (2016), 697–732.CrossRefGoogle Scholar
Dang, N.-B. and Xiao, J.. Positivity of valuations on convex bodies and invariant valuations by linear actions. Preprint, 2017, arXiv:1709.08304.Google Scholar
info[issue]2 Favre, C. and Gauthier, T.. Classification of special curves in the space of cubic polynomials. Int. Math. Res. Not. IMRN 2018 (2018), 362–411.Google Scholar
Favre, C. and Jonsson, M.. Dynamical compactifications of C 2 . Ann. of Math. (2) 173(1) (2011), 211–249.CrossRefGoogle Scholar
Fulton, W.. Intersection Theory (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge [Results in Mathematics and Related Areas. 3rd Series]. A Series of Modern Surveys in Mathematics, 2) , 2nd edn. Springer, Berlin, 1998.CrossRefGoogle Scholar
Favre, C. and Wulcan, E.. Degree growth of monomial maps and McMullen’s polytope algebra. Indiana Univ. Math. J. 61(2) (2012), 493–524.CrossRefGoogle Scholar
Gizatullin, M. K.. Rational G-surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 44(1) (1980), 110–144, 239.Google Scholar
Gromov, M.. On the entropy of holomorphic maps. Enseign. Math. 49 (2003), 217–235.Google Scholar
Guedj, V.. Ergodic properties of rational mappings with large topological degree. Ann. of Math. (2) 161(3) (2005), 1589–1607.Google Scholar
Hasselblatt, B. and Propp, J.. Degree-growth of monomial maps. Ergod. Th. & Dynam. Sys. 27(5) (2007), 1375–1397.Google Scholar
Jonsson, M. and Wulcan, E.. Stabilization of monomial maps. Michigan Math. J. 60(3) (2011), 629–660.CrossRefGoogle Scholar
Koch, S.. Teichmüller theory and critically finite endomorphisms. Adv. Math. 248 (2013), 573–617.CrossRefGoogle Scholar
Koch, S. and Roeder, R. K. W.. Computing dynamical degrees of rational maps on moduli space. Ergod. Th. & Dynam. Sys. 36(8) (2016), 2538–2579.CrossRefGoogle Scholar
Kawaguchi, S. and Silverman, J. H.. On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. J. Reine Angew. Math. 713 (2016), 21–48.Google Scholar
Lo Bianco, F.. On the primitivity of birational transformations of irreducible holomorphic symplectic manifolds. Int. Math. Res. Not. IMRN 2019(1) (2019), 1–32.CrossRefGoogle Scholar
Lin, J.-L.. Pulling back cohomology classes and dynamical degrees of monomial maps. Bull. Soc. Math. France 140(4) (2012), 533–549 (2013).CrossRefGoogle Scholar
Lin, J.-L. and Wulcan, E.. Stabilization of monomial maps in higher codimension. Ann. Inst. Fourier. Grenoble 64(5) (2014), 2127–2146.CrossRefGoogle Scholar
Masser, D. and Zannier, U.. Torsion anomalous points and families of elliptic curves. Amer. J. Math. 132(6) (2010), 1677–1691.Google Scholar
Néron, A.. Quasi-fonctions et hauteurs sur les variétés abéliennes. Ann. of Math. (2) 82 (1965), 249–331.CrossRefGoogle Scholar
Ramadas, R.. Hurwitz correspondences on compactifications of 𝓜 0, N . Adv. Math. 323 (2018), 622–667.CrossRefGoogle Scholar
Ramadas, R.. Dynamical degrees of Hurwitz correspondences. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2018.125. Published online 4 December 2018.CrossRefGoogle Scholar
Russakovskii, A. and Shiffman, B.. Value distribution for sequences of rational mappings and complex dynamics. Indiana Univ. Math. J. 46(3) (1997), 897–932.CrossRefGoogle Scholar
Schanuel, S. H.. Heights in number fields. Bull. Soc. Math. France 107(4) (1979), 433–449.CrossRefGoogle Scholar
Serre, J.-P.. A Course in Arithmetic. Springer, New York, 1973.CrossRefGoogle Scholar
Sibony, N.. Dynamique et géométrie complexes (Lyon 1997) (Panorama et Synthèses, 8) . Société Mathématique de France, Paris, 1999.Google Scholar
Silverman, J. H.. Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space. Ergod. Th. & Dynam. Sys. 34(2) (2014), 647–678.CrossRefGoogle Scholar
Szpiro, L., Ullmo, E. and Zhang, S.. Équirépartition des petits points. Invent. Math. 127(2) (1997), 337–347.CrossRefGoogle Scholar
Tate, J. T.. The arithmetic of elliptic curves. Invent. Math. 23 (1974), 179–206.CrossRefGoogle Scholar
Truong, T. T.. Relative dynamical degrees of correspondences over a field of arbitrary characteristic. J. Reine Angew. Math. (Crelles J.) 2020(758) (2020), 139–182.CrossRefGoogle Scholar
Yomdin, Y.. Volume growth and entropy. Israel J. Math. 57(3) (1987), 285–300.CrossRefGoogle Scholar
Yuan, X.. Big line bundles over arithmetic varieties. Invent. Math. 173(3) (2008), 603–649.CrossRefGoogle Scholar
Zhang, D.-Q.. Compact Kähler manifolds with automorphism groups of maximal rank. Trans. Amer. Math. Soc. 366(7) (2014), 3675–3692.CrossRefGoogle Scholar