Discrete Holomorphic Local Dynamical Systems

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. For
endomorphisms of P k , we also study the equidistribution of varieties with respect to
the Green currents. Results in this direction give some informations on the rigidity
of the system and also some strong ergodic properties that the Green currents or the
measure of maximal entropy satisfy. The results we obtain are in spirit similar to a
second main theorem in value distribution theory and should be useful in order to
study the arithmetic analogues. We give complete proofs for most results, but we
only survey the equidistribution of hypersurfaces and results using super-potentials,
in particular, the equidistribution of subvarieties of higher codimension.
The text is organized as follows. In the first section, we study holomorphic endomorphisms
of P k . We introduce several methods in order to construct and to study
the Green currents and the Green measure, i.e. equilibrium measure or measure of
maximal entropy. These methods were not originally introduced in this setting but
here they are simple and very effective. The reader will find a description and the
references of the earlier approach in the ten years old survey by the second author
[S3]. The second section deals with a very large family of maps: polynomial-like
maps. In this case, f : U → V is proper and defined on an open set U, strictly contained
in a convex domain V of C k . Holomorphic endomorphisms of P k can be
lifted to a polynomial-like maps on some open set in C k+1 . So, we can consider
polynomial-like maps as a semi-local version of the endomorphisms studied in the
first section. They can appear in the study of meromorphic maps or in the dynamics
of transcendental maps. The reader will find in the end of these notes an appendix
on the theory of currents and an extensive bibliography. We have given exercises,
basically in each section, some of them are not straightforward.
1 Endomorphisms of Projective Spaces
In this section, we give the main results on the dynamics of holomorphic maps
on the projective space P k . Several results are recent and some of them are new
even in dimension 1. The reader will find here an introduction to methods that can
be developed in other situations, in particular, in the study of meromorphic maps
on arbitrary compact Kähler manifolds. The main references for this section are
[BD1, BD2, DNS, DS9, DS10, FS1, S3].