Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets
HTML articles powered by AMS MathViewer
- by Nils Hemmingsson;
- Conform. Geom. Dyn. 28 (2024), 97-114
- DOI: https://doi.org/10.1090/ecgd/394
- Published electronically: August 30, 2024
- HTML | PDF | Request permission
Abstract:
In this paper, we analyze a certain family of holomorphic correspondences on $\hat {\mathbb {C}}\times \hat {\mathbb {C}}$ and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map $F$ is such that for any $a\in \mathbb {C}$, we have that $(F^n)_*(\delta _a)$ converges to a probability measure $\mu _F$ for which $F_*(\mu _F)=\mu _F d$ where $d$ is the degree of $F$. This result is used to show that the minimal Hutchinson invariant set, introduced by P. Alexandersson, P. Brändén, and B. Shapiro [An inverse problem in Pólya–Schur theory. I. Non-degenerate and degenerate operators, preprint, 2024], of a large class of operators and for sufficiently large $n$ exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, the minimal Hutchinson-invariant set is a Cantor set.References
- P. Alexandersson, P. Brändén, and B. Shapiro, An inverse problem in Pólya–Schur theory. II. Exactly solvable operators and complex dynamics, Preprint, 2020.
- P. Alexandersson, P. Brändén, and B. Shapiro, An inverse problem in Pólya–Schur theory. I. Non-degenerate and degenerate operators, Preprint, arXiv:2404.14365, 2024.
- Per Alexandersson, Nils Hemmingsson, Dmitry Novikov, Boris Shapiro, and Guillaume Tahar, Linear first order differential operators and their Hutchinson invariant sets, J. Differential Equations 391 (2024), 265–320. MR 4703364, DOI 10.1016/j.jde.2024.01.018
- Julius Borcea and Petter Brändén, Pólya-Schur master theorems for circular domains and their boundaries, Ann. of Math. (2) 170 (2009), no. 1, 465–492. MR 2521123, DOI 10.4007/annals.2009.170.465
- Petter Brändén and Matthew Chasse, Classification theorems for operators preserving zeros in a strip, J. Anal. Math. 132 (2017), 177–215. MR 3666810, DOI 10.1007/s11854-017-0018-3
- Shaun Bullett and Luna Lomonaco, Mating quadratic maps with the modular group II, Invent. Math. 220 (2020), no. 1, 185–210. MR 4071411, DOI 10.1007/s00222-019-00927-9
- Shaun Bullett, Luna Lomonaco, and Carlos Siqueira, Correspondences in complex dynamics, arXiv:1710.03385, 2017.
- David Boyd, An invariant measure for finitely generated rational semigroups, Complex Variables Theory Appl. 39 (1999), no. 3, 229–254. MR 1717572, DOI 10.1080/17476939908815193
- Shaun Bullett and Christopher Penrose, A gallery of iterated correspondences, Experiment. Math. 3 (1994), no. 2, 85–105. MR 1313875, DOI 10.1080/10586458.1994.10504282
- Shaun Bullett and Christopher Penrose, Mating quadratic maps with the modular group, Invent. Math. 115 (1994), no. 3, 483–511. MR 1262941, DOI 10.1007/BF01231770
- Gautam Bharali and Shrihari Sridharan, The dynamics of holomorphic correspondences of $\Bbb P^1$: invariant measures and the normality set, Complex Var. Elliptic Equ. 61 (2016), no. 12, 1587–1613. MR 3520247, DOI 10.1080/17476933.2016.1185419
- Shaun Bullett, Dynamics of quadratic correspondences, Nonlinearity 1 (1988), no. 1, 27–50. MR 928947, DOI 10.1088/0951-7715/1/1/002
- Shaun Bullett, Matings in holomorphic dynamics, Geometry of Riemann surfaces, London Math. Soc. Lecture Note Ser., vol. 368, Cambridge Univ. Press, Cambridge, 2010, pp. 88–119. MR 2665006
- Thomas Craven and George Csordas, Composition theorems, multiplier sequences and complex zero decreasing sequences, Value distribution theory and related topics, Adv. Complex Anal. Appl., vol. 3, Kluwer Acad. Publ., Boston, MA, 2004, pp. 131–166. MR 2173299, DOI 10.1007/1-4020-7951-6_{6}
- Tien-Cuong Dinh, Lucas Kaufmann, and Hao Wu, Dynamics of holomorphic correspondences on Riemann surfaces, Internat. J. Math. 31 (2020), no. 5, 2050036, 21. MR 4104599, DOI 10.1142/S0129167X20500366
- Tien-Cuong Dinh and Nessim Sibony, Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv. 81 (2006), no. 1, 221–258 (French, with English summary). MR 2208805, DOI 10.4171/CMH/50
- M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 351–385. MR 741393, DOI 10.1017/S0143385700002030
- Seung-Yeop Lee, Mikhail Lyubich, Nikolai G. Makarov, and Sabyasachi Mukherjee, Schwarz reflections and anti-holomorphic correspondences, Adv. Math. 385 (2021), Paper No. 107766, 88. MR 4250610, DOI 10.1016/j.aim.2021.107766
- Mayuresh Londhe, Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroups, Indiana Univ. Math. J. 71 (2022), no. 3, 1131–1154. MR 4448581, DOI 10.1512/iumj.2022.71.9753
- H. F. Münzner and H.-M. Rasch, Iterated algebraic functions and functional equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1 (1991), no. 4, 803–822. MR 1157045, DOI 10.1142/S0218127491000592
- Carlos Siqueira, Dynamics of hyperbolic correspondences, Ergodic Theory Dynam. Systems 42 (2022), no. 8, 2661–2692. MR 4448402, DOI 10.1017/etds.2021.49
Bibliographic Information
- Nils Hemmingsson
- Affiliation: Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden
- MR Author ID: 1597972
- ORCID: 0000-0001-7744-3713
- Email: nils.hemmingsson@math.su.se
- Received by editor(s): September 13, 2023
- Received by editor(s) in revised form: April 10, 2024
- Published electronically: August 30, 2024
- © Copyright 2024 American Mathematical Society
- Journal: Conform. Geom. Dyn. 28 (2024), 97-114
- MSC (2020): Primary 37F05, 37A05; Secondary 32H50
- DOI: https://doi.org/10.1090/ecgd/394