Discrete Holomorphic Local Dynamical Systems

226 Tien-Cuong Dinh and Nessim Sibony
If g is the Green function of f ,wehave
( f n1 ) ∗ (ωFS)=d n1 T − dd c (g ◦ f n1 ).
The above integral is equal to the sum of the following integrals
d n1
�
T p−r+1 ∧ ( f n2 ∗
) (ωFS) ∧ ...∧ ( f ns ∗
) (ωFS) ∧ χω k−p+r−s
FS
and
�
−
T p−r ∧ dd c (g ◦ f n1 n2 ∗
) ∧ ( f ) (ωFS) ∧ ...∧ ( f ns ∗
) (ωFS) ∧ χω k−p+r−s
FS .
Using the case of (r − 1,s − 1) we can bound the first integral by cdn(r−1) .Stokes’
theorem implies that the second integral is equal to
�
− T p−r ∧ ( f n2 ∗
) (ωFS) ∧ ...∧ ( f ns ∗
) (ωFS) ∧ (g ◦ f n1 c k−p+r−s
)dd χ ∧ ωFS which is bounded by
�g�∞�χ� C 2�T p−r ∧ ( f n2 ) ∗ (ωFS) ∧ ...∧ ( f ns ) ∗ (ωFS)�Wr,s−1
since χ has support in Wr,s−1. We obtain the result using the (r,s − 1) case. ⊓⊔
The above result suggests a local indicator of volume growth. Define for a ∈ P k
lov( f ,a) := inf
r>0 limsup
1
n
n→∞
logvolume(Π −1
0 (Br) ∩Γn),
where Br is the ball of center a and of radius r. We can show that if a ∈ Jp \ Jp+1
and if Br does not intersect Jp+1, the above limsup is in fact a limit and is equal
to plogd. One can also consider the graph of f n instead of Γn. The notion can be
extended to meromorphic maps and its sub-level sets are analogues of Julia sets.
We discuss now the metric entropy, i.e. the entropy of an invariant measure, a
notion due to Kolgomorov-Sinai. Let g : X → X be map on a space X which is measurable
with respect to a σ-algebra F .Letν be an invariant probability measure
for g. Letξ = {ξ1,...,ξm} be a measurable partition of X. The entropy of ν with
respect to ξ is a measurement of the information we gain when we know that a
point x belongs to a member of the partition generated by g −i (ξ ) with 0 ≤ i ≤ n −1.
The information we gain when we know that a point x belongs to ξi is a positive
function I(x) which depends only on ν(ξi), i.e.I(x) =ϕ(ν(ξi)). The information
given by independent events should be additive. In other words, we have
ϕ(ν(ξi)ν(ξ j)) = ϕ(ν(ξi)) + ϕ(ν(ξ j))
for i �= j. Hence, ϕ(t) =−clogt with c > 0. With the normalization c = 1, the
information function for the partition ξ is defined by
I ξ (x) := ∑−logν(ξi)1 ξi (x).