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Holomorphic Dynamical Systems pp 165–294Cite as

Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1998))

Abstract

The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic (p.s.h.) functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. Nervertheless, we choose to show their effectiveness and to describe the theory for two large families of maps: the endomorphisms of projective spaces and the polynomial-like mappings. The first section deals with holomorphic endomorphisms of the projective space \({\mathbb{P}}^{k}\). We establish the first properties and give several constructions for the Green currents T p and the equilibrium measure μ = T k. The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems. We show the existence of a proper algebraic set E, totally invariant, i.e. \({f}^{-1}(\mathit{E}) = f(\mathit{E}) = \mathit{E}\), such that when a ∉ E, the probability measures, equidistributed on the fibers f − n(a), converge towards the equilibrium measure μ, as n goes to infinity. A similar result holds for the restriction of f to invariant subvarieties. We survey the equidistribution problem when points are replaced with varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of μ: K-mixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space \(\mathrm{DSH}({\mathbb{P}}^{k})\) of differences of quasi-p.s.h. functions. In particular, we show that the measure μ is moderate, i.e. ⟨μ, e α | φ | ⟩ ≤ c, on bounded sets of φ in \(\mathrm{DSH}({\mathbb{P}}^{k})\), for suitable positive constants α, c. Finally, we study the entropy, the Lyapounov exponents and the dimension of μ. The second section develops the theory of polynomial-like maps, i.e. proper holomorphic maps f : U → V where U, V are open subsets of ℂ k with V convex and U⋐V. We introduce the dynamical degrees for such maps and construct the equilibrium measure μ of maximal entropy. Then, under a natural assumption on the dynamical degrees, we prove equidistribution properties of points and various statistical properties of the measure μ. The assumption is stable under small pertubations on the map. We also study the dimension of μ, the Lyapounov exponents and their variation. Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.

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Authors and Affiliations

  1. UMR 7586, Institut de Mathématiques de Jussieu, UPMC Univ Paris 06, F-75005, Paris, France

    Tien-Cuong Dinh

  2. Université Paris-Sud 11, Mathématique Bâtiment 425, 91405, Orsay, France

    Nessim Sibony

Authors
  1. Tien-Cuong Dinh
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  2. Nessim Sibony
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Correspondence to Tien-Cuong Dinh .

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Editors and Affiliations

  1. Department of Mathematics "U.Dini", University of Florence, viale Morgagni 67/A, Firenze, 50134, Italy

    Graziano Gentili

  2. Dipartimento di Strutture, Università della Calabria, Ponte Pietro Bucci 39, Arcavacata di Rende (CS), 87036, Italy

    Jacques Guenot

  3. Dipartimento di Matematica "U.Dini", Università di Firenze, viale Morgagni 67/A, Firenze, 50134, Italy

    Giorgio Patrizio

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Dinh, TC., Sibony, N. (2010). Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings. In: Gentili, G., Guenot, J., Patrizio, G. (eds) Holomorphic Dynamical Systems. Lecture Notes in Mathematics(), vol 1998. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13171-4_4

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