Discrete Holomorphic Local Dynamical Systems

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 dd c -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 action
on some appropriate functional spaces. We then decompose the action into the
“harmonic” part and the “non-harmonic” one. This is done solving a dd c -equation
with estimates. The non-harmonic part of the dynamical action may be controled
thanks to good estimates for the solutions of a dd c -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 Kähler 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 C k , this is easier
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 diverge
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.
A central question in dynamics is to construct interesting invariant measures,
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 P k , they can
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.