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Exponential mixing property for Hénon–Sibony maps of $\mathbb {C}^k$

Published online by Cambridge University Press:  17 September 2021

HAO WU
HAO WU*
Affiliation:
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore119076
*
(e-mail: [email protected])
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Abstract

Let f be a Hénon–Sibony map, also known as a regular polynomial automorphism of $\mathbb {C}^k$ , and let $\mu $ be the equilibrium measure of f. In this paper we prove that $\mu $ is exponentially mixing for plurisubharmonic observables.

Keywords

equilibrium measureexponential mixingpositive closed current
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 12 , December 2022 , pp. 3818 - 3830
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Bedford, E., Lyubich, M. and Smillie, J.. Polynomial diffeomorphisms of ${\mathbb{C}}^2$ . IV. The measure of maximal entropy and laminar currents. Invent. Math. 112(1) (1993), 77–125.CrossRefGoogle Scholar
Bedford, E. and Smillie, J.. Polynomial diffeomorphisms of ${\mathbb{C}}^2$ . III. Ergodicity, exponents and entropy of the equilibrium measure. Math. Ann. 294(3) (1992), 395–420.CrossRefGoogle Scholar
Demailly, J.-P.. Complex Analytic and Differential Geometry. Available at http://www-fourier.ujf- grenoble.fr/~demailly/.Google Scholar
Dinh, T.-C.. Decay of correlations for Hénon maps. Acta Math. 195 (2005), 253–264.CrossRefGoogle Scholar
Dinh, T.-C., Nguyên, V.-A. and Sibony, N.. Exponential estimates for plurisubharmonic functions and stochastic dynamics. J. Differential Geom. 84(3) (2010), 465–488.CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N.. Dynamique des applications d’allure polynomiale. J. Math. Pures Appl. (9) 82(4) (2003), 367–423.CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N.. Dynamics of regular birational maps in ${\mathbb{P}}^k$ . J. Funct. Anal. 222(1) (2005), 202–216.CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N.. Super-potentials of positive closed currents, intersection theory and dynamics. Acta Math. 203(1) (2009), 1–82.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, pp. 165–294.CrossRefGoogle Scholar
Dinh, T.-C. and Sibony, N.. Rigidity of Julia sets for Hénon type maps. J. Mod. Dyn. 8(3–4) (2014), 499–548.Google Scholar
Fornæss, J. E.. Dynamics in Several Complex Variables (CBMS Regional Conference Series in Mathematics, 87). American Mathematical Society, Providence, RI, 1996. Published for the Conference Board of the Mathematical Sciences, Washington, DC.Google Scholar
Fornæss, J. E. and Sibony, N.. Complex Hénon mappings in ${\mathbb{C}}^2$ and Fatou–Bieberbach domains. Duke Math. J. 65(2) (1992), 345–380.CrossRefGoogle Scholar
Friedland, S. and Milnor, J.. Dynamical properties of plane polynomial automorphisms. Ergod. Th. & Dynam. Sys. 9(1) (1989), 67–99.CrossRefGoogle Scholar
Sibony, N.. Dynamique des applications rationnelles de ${\mathbb{P}}^k$ . Dynamique et Géométrie Complexes (Lyon, 1997) (Panoramas et Synthèses, 8). Société Mathématique de France, Paris, 1999, pp. 97–185.Google Scholar
Vigny, G.. Exponential decay of correlations for generic regular birational maps of ${\mathbb{P}}^k$ . Math. Ann. 362(3–4) (2015), 1033–1054.CrossRefGoogle Scholar