Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 233
measure fills out its support. The following result was obtained by Binder-DeMarco
[BI] and Dinh-Dupont [DD]. The fact that μ has positive dimension has been proved
in [S3]; indeed, a lower bound is given in terms of the Hölder continuity exponent
of the Green function g.
Theorem 1.121. Let f be an endomorphism of algebraic degree d ≥ 2 of P k and
μ its equilibrium measure. Let χ1,...,χk denote the Lyapounov exponents of μ
ordered by χ1 ≥···≥ χk and Σ their sum. Then
k logd
2Σ − k logd
≤ dimH(μ) ≤ 2k − ·
χ1
The proof is quite technical. It is based on a delicate study of the inverse branches
of balls along a generic negative orbit. We will not give the proof here. A better
estimate in dimension 2, probably the sharp one, was recently obtained by Dupont.
Indeed, Binder-DeMarco conjecture that the Hausdorff dimension of μ satisfies
χ1
dimH(μ)= logd
+ ···+ logd
·
Dupont gives in [DP3] results in this direction.
χ1
Exercise 1.122. Let X be an analytic subvariety of pure dimension p in P k .Let f
be an endomorphism of algebraic degree d ≥ 2ofP k . Show that ht( f ,X) ≤ plogd.
Exercise 1.123. Let f : X → X be a smooth map and K an invariant compact subset
of X. Assume that K is hyperbolic, i.e. there is a continuously varying decomposition
TX |K = E ⊕ F of the tangent bundle of X restricted to K, into the sum of two
invariant vector bundles such that �Df� < 1onE and �(Df) −1 � < 1onF for some
smooth metric near K. Show that f admits a hyperbolic ergodic invariant measure
supported on K.
Exercise 1.124. Let f : X → X be a holomorphic map on a compact complex manifold
and let ν be an ergodic invariant measure. Show that in Theorem 1.119 applied
to the action of f on the complex tangent bundle, the spaces Ei(x) are complex.
Exercise 1.125. Let α > 0 be a constant. Show that there is an endomorphism f of
P k such that the Hausdorff dimension of the equilibrium measure of f is smaller
than α. Show that there is an endomorphism f such that its Green function g is not
α-Hölder continuous.
Notes. We do not give here results on local dynamics near a fixed point. If this point is non-critical
attractive or repelling, a theorem of Poincaré says that the map is locally conjugated to a polynomial
map [ST]. Maps which are tangent to the identity or are semi-attractive at a fixed point, were
studied by Abate and Hakim [A,H1,H2,H3], see also Abate, Bracci and Tovena [ABT]. Dynamics
near a super-attractive fixed point in dimension k = 2 was studied by Favre-Jonsson using a theory
of valuations in [FJ1].
The study of the dynamical system outside the support of the equilibrium measure is not yet
developped. Some results on attracting sets, attracting currents, etc. were obtained by de Thélin,
Dinh, Fornæss, Jonsson, Sibony, Weickert [DT1,DT2,D3,FS6,FW,JW], see also Bonifant, Dabija,
Milnor and Taflin [BO, T], Mihailescu and Urbański [MI, MIH].
χk