We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that
$|f(z)|\ge \exp (|z|^\alpha )$
for some
$\alpha>0$
and all z outside a set of finite Lebesgue measure.
Let $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$
g
(
z
)
=
∫
0
z
p
(
t
)
exp
(
q
(
t
)
)
d
t
+
c
where p, q are polynomials and $$c\in {\mathbb {C}}$$
c
∈
C
, and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$
g
′
′
the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$
f
n
(
z
)
converges to zeros of g almost everywhere in $${\mathbb {C}}$$
C
if this is the case for each zero of $$g''$$
g
′
′
that is not a zero of g or $$g'$$
g
′
. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.
We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f have finite Lebesgue measure. Essentially, these conditions are designed such that |f (z)| ≥ exp(|z| α ) for some α > 0 and all z outside a set of finite Lebesgue measure.
Let g(z) = z 0 p(t) exp(q(t)) dt + c where p, q are polynomials and c ∈ C, and let f be the function from Newton's method for g. We show that under suitable assumptions the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that f n (z) converges to zeros of g almost everywhere in C if this is the case for each zero of g . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.
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