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

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Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-Like Mappings Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings Tien-Cuong Dinh and Nessim Sibony Abstract The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness prop- erties 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 familiesofmaps:theendomorphismsofprojective spaces and the polynomial-like mappings. The first section deals with holomorphic endomorphisms of the projective space Pk.Weestablishthefirstpropertiesandgive 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 1 invariant, i.e. f − (E )=f (E )=E ,suchthatwhena E ,theprobabilitymeasures, n "∈ equidistributed on the fibers f − (a),convergetowardstheequilibriummeasureµ, as n goes to infinity. A similar result holds for the restriction of f to invariant sub- varieties. 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,exponentialdecayofcorrela- tions for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space DSH(Pk) of differences of quasi-p.s.h. functions. In particular, we show that the measure µ is moderate, i.e. α ϕ k µ,e | | c,onboundedsetsofϕ in DSH(P ),forsuitablepositiveconstants $ %≤ α,c.Finally,westudytheentropy,theLyapounovexponentsandthedimension of µ.Thesecondsectiondevelopsthetheoryofpolynomial-likemaps,i.e.proper T.-C. Dinh UPMC Univ Paris 06, UMR7586,Institut de Math´ematiques de Jussieu, F-75005 Paris, France e-mail: [email protected] N. Sibony Universit´eParis-Sud11,Math´ematique - Bˆatiment 425, 91405 Orsay, France e-mail: [email protected] G. Gentili et al. (eds.), Holomorphic Dynamical Systems,165 Lecture Notes in Mathematics 1998, DOI 10.1007/978-3-642-13171-4 4, c Springer-Verlag Berlin Heidelberg 2010 ! 166 Tien-Cuong Dinh and Nessim Sibony k holomorphic maps f : U V where U,V are open subsets of C with V convex → and U ! V .Weintroducethedynamicaldegreesforsuchmapsandconstructthe 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.Theassumptionisstableundersmallpertu- bations on the map. We also study the dimension of µ,theLyapounovexponents and their variation. Our aim is to get a self-contained text that requires only a min- imal 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. Introduction These notes are based on a series of lectures given by the authors at IHP in 2003, Luminy in 2007, Cetraro in 2008 and Bedlewo 2008. The purpose is to provide an introduction to some developments in dynamics of several complex variables. We have chosen to treat here only two sections of the theory: the dynamics of endomor- phisms of the projective space Pk and the dynamics of polynomial-like mappings in higher dimension. Besides the basic notions and results, we describe the recent developments and the new tools introduced in the theory. These tools are useful in other settings. We tried to give a complete picture of the theory for the above families of dynamical systems. Meromorphic maps on compact K¨ahler manifolds, in particular polynomial automorphisms of Ck,willbestudiedinaforthcoming survey. Let us comment on how complex dynamics fits in the general theory of dynamical systems. The abstract ergodic theory is well-developed with remark- able achievements like the Oseledec-Pesin theory. It is however difficult to show in concrete examples that an invariant measure satisfies exponential decay of corre- lations for smooth observables or is hyperbolic, i.e. has only non-zero Lyapounov exponents, see e.g. Benedicks-Carleson [BC], Viana [V], L.S. Young [Y, Y1]. One of our goals is to show that holomorphic dynamics in several variables provides remarkable examples of non-uniformly hyperbolic systems where the abstract the- ory can be applied. Powerful tools from the theory of several complex variables permit to avoid delicate combinatorial estimates. Complex dynamics also require a development of new tools like the calculus on currents and the introduction of new spaces of observables, which are of independent interest. 1 Complex dynamics in dimension one, i.e. dynamics of rational maps on P , is well-developed and has in some sense reached maturity. The main tools there are Montel’s theorem on normal families, the Riemann measurable mapping theo- rem and the theory of quasi-conformal maps, see e.g. Beardon, Carleson-Gamelin [BE, CG]. When dealing with maps in several variables such tools are not available: the Kobayashi hyperbolicity of a manifold and the possibility to apply normal fam- ily arguments, are more difficult to check. Holomorphic maps in several variables are not conformal and there is no Riemann measurable mapping theorem. Dynamics in Several Complex variables 167 The theory in higher dimension is developed using mostly pluripotential theory, i.e. the theory of plurisubharmonic (p.s.h. for short) functions and positive closed currents. The Montel’s compactness property is replaced with the compactness properties of p.s.h. or quasi-p.s.h. functions. Another crucial tool is the use of good estimates for the ddc-equation in various settings. One of the main ideas is: in order to study the statistical behavior of orbits of a holomorphic map, we consider its ac- tion on some appropriate functional spaces. We then decompose the action into the “harmonic” part and the “non-harmonic” one. This is done solving a ddc-equation with estimates. The non-harmonic part of the dynamical action may be controled thanks to good estimates for the solutions of a ddc-equation. The harmonic part can be treated using either Harnack’s inequality in the local setting or the linear action of maps on cohomology groups in the case of dynamics on compact K¨ahler manifolds. This approach has permitted to give a satisfactory theory of the ergodic properties of holomorphic and meromorphic dynamical systems: construction of the measure of maximal entropy, decay of correlations, central limit theorem, large deviations theorem, etc. with respect to that measure. In order to use the pluripotential methods, we are led to develop the calculus on positive closed currents. Readers not familiar with these theories may start with the appendix at the end of these notes where we have gathered some notions and results on currents and pluripotential theory. A large part in the appendix is classical but there are also some recent results, mostly on new spaces of currents and on the notion of super-potential associated to positive closed currents in higher bidegree. Since we only deal here with projective spaces and open sets in Ck,thisiseasier and the background is limited. The main problem in the dynamical study of a map is to understand the behavior of the orbits of points under the action of the map. Simple examples show that in general there is a set (Julia set) where the dynamics is unstable: the orbits may di- verge exponentially. Moreover, the geometry of the Julia set is in general very wild. In order to study complex dynamical systems, we follow the classical concepts. We introduce and establish basic properties of some invariants associated to the system, like the topological entropy and the dynamical degrees which are the analogues of volume growth indicators in the real dynamical setting. These invariants give a rough classification of the system. The remarkable fact in complex dynamics is that they can be computed or estimated in many non-trivial situations. Acentralquestionindynamicsistoconstructinterestinginvariantmeasures, in particular, measures with positive entropy. Metric entropy is an indicator of the complexity of the system with respect to an invariant measure. We focus our study on the measure of maximal entropy. Its support is in some sense the most chaotic part of the system. For the maps we consider here, measures of maximal entropy are constructed using pluripotential methods. For endomorphisms in Pk,theycan be obtained as self-intersections of some invariant positive closed (1,1)-currents (Green currents). We give estimates on the Hausdorff dimension and on Lyapounov exponents of these measures. The results give the behavior on the most chaotic part. Lyapounov exponents are shown to be strictly positive. This means in some sense that the system is expansive in all directions, despite of the existence of a critical set. 168 Tien-Cuong Dinh and Nessim Sibony Once, the measure of maximal entropy is constructed,we study its fine dynamical properties. Typical orbits can be observed using test functions. Under the action of the map, each observable provides a sequence of functions that can be seen as dependent random variables. The aim is to show that the dependence is weak and then to establish stochastic properties which are known for independent random variables in probability theory. Mixing, decay of correlations, central limit theorem, large deviations theorems, etc. are proved for the measure of maximal entropy. It is crucial here that the Green currents and the measures of maximal entropy are obtained using an iterative process with estimates; we can then bound the speed of convergence. Another problem, we consider in these notes, is the equidistribution of periodic points or of preimages of points with respect to the measure of maximal entropy.
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