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Codimension-one attracting sets in $\mathbb{P}^{k}(\mathbb{C})$

Published online by Cambridge University Press:  28 July 2016

SANDRINE DAURAT  and
JOHAN TAFLIN
SANDRINE DAURAT
Affiliation:
University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA email daurat@umich.edu
JOHAN TAFLIN
Affiliation:
Université Bourgogne-Franche-Comté, IMB, UMR 5584, 21078 Dijon Cedex, France email johan.taflin@u-bourgogne.fr
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Abstract

We are interested in attracting sets of $\mathbb{P}^{k}(\mathbb{C})$ which are of small topological degree and of codimension one. We first show that there exists a large family of examples. Then we study their ergodic and pluripotential theoretic properties.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 63 - 80
Copyright
© Cambridge University Press, 2016 

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References

Brin, M. and Katok, A.. On local entropy. Geometric Dynamics (Lecture Notes in Mathematics, 1007) . Springer, Berlin, 1983, pp. 30–38.Google Scholar
Bedford, E., Lyubich, M. and Smillie, J.. Polynomial diffeomorphisms of ℂ2 . IV. The measure of maximal entropy and laminar currents. Invent. Math. 112(1) (1993), 77–125.Google Scholar
Daurat, S.. On the size of attractors in ℙ k . Math. Z. 277(3) (2014), 629–650.Google Scholar
Demailly, J.-P.. Complex analytic and differential geometry. www-fourier.ujf-grenoble.fr/∼demailly/books.html (2012).Google Scholar
de Thélin, H.. Sur les exposants de Lyapounov des applications méromorphes. Invent. Math. 172(1) (2008), 89–116.Google Scholar
Dinh, T.-C.. Attracting current and equilibrium measure for attractors on ℙ k . J. Geom. Anal. 17(2) (2007), 227–244.Google Scholar
Dinh, T.-C. and Sibony, N.. Equidistribution towards the Green current for holomorphic maps. Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 307–336.Google 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.Google Scholar
Dupont, C.. Large entropy measures for endomorphisms of ℂℙ k . Israel J. Math. 192(2) (2012), 505–533.Google Scholar
Fakhruddin, N.. Questions on self maps of algebraic varieties. J. Ramanujan Math. Soc. 18(2) (2003), 109–122.Google Scholar
Fornæss, J. E. and Sibony, N.. Dynamics of ℙ2 (examples). Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998) (Contemporary Mathematics, 269) . American Mathematical Society, Providence, RI, 2001, pp. 47–85.Google Scholar
Fornæss, J. E. and Weickert, B.. Attractors in ℙ2 . Several Complex Variables (Berkeley, CA, 1995–1996) (Mathematical Sciences Research Institute Publications, 37) . Cambridge University Press, Cambridge, 1999, pp. 297–307.Google Scholar
Hörmander, L.. An introduction to complex analysis in several variables (North-Holland Mathematical Library, 7), 3rd edn. North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
Jonsson, M. and Weickert, B.. A nonalgebraic attractor in ℙ2 . Proc. Amer. Math. Soc. 128(10) (2000), 2999–3002.Google Scholar
Rong, F.. Non-algebraic attractors on ℙ k . Discrete Contin. Dyn. Syst. 32(3) (2012), 977–989.CrossRefGoogle Scholar
Taflin, J.. Speed of convergence towards attracting sets for endomorphisms of ℙ k . Indiana Univ. Math. J. 62 (2013), 33–44.Google Scholar
Triebel, H.. Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Heidelberg, 1995.Google Scholar