The Lebesgue Measure of the Julia Sets of Permutable Transcendental Entire Functions ()
1. Introduction
Let
be a transcendental entire function. We write
, and
,
for the nth iterates of
. The Fatou set
of
consists of all z in the complex plane
that has a neighborhood U such that the family
is a normal family. The Julia set
of
is defined by
. The Julia set
can be characterized as the closure of the repelling periodic points of
[1]. The set
and
are completely invariant of
. For fundamental results in the iteration theory of rational and entire functions, we refer to the original papers of Fatou [2] [3] [4] [5] and Julia [6] and the books of Beardon [7], Carleson and Gamelin [8], Milnor [9], Ren [10], Zheng [11], and Qiao [12].
Two functions
and
are called permutable if
holds for all values of z. In 1922-23, Julia [13] and Fatou [4] independently proved that rational functions
and
of degree at least 2 such that
and
are permutable, then
. It is natural to consider the following open problem which was first posed in [14] by Baker.
Problem Let
and
be nonlinear entire functions. If
and
are permutable, is
?
In [14], Baker proved the following result:
Theorem A (Baker [14]) Suppose that
and
are transcendental entire functions such that
, where a and b are complex numbers. If
permute with
, then
.
Langley [15] showed that if
and
are permutable functions of finite order with no wandering domains, then
.
Theorem B (Langley [15]) Suppose that
and
are permutable transcendental entire functions. If both
and
have no wandering domains, then
.
At the same time, Bergweiler and Hinkkanen [16] introduced the so-called fast escaping set
and used it to prove a result that includes the following.
Theorem C (Bergweiler and Hinkkanen [16]) If
and
are permutable transcendental entire functions such that
and
, then
. In particular, this holds if
and
have no wandering domains.
For a long time, there have been many results about the problem of permutable transcendental entire functions, see [17] - [23]. However, until now, the problem has not been completely solved. In order to study the problem of permutable transcendental entire functions, by the properties of permutable transcendental entire functions, we prove that if
and
are permutable entire functions, then
. Moreover, we give some results about the zero measure of the Julia sets of the permutable transcendental entire functions family.
2. Main Results
Write
for the plane Lebesgue measure of a set E. Recently, various authors have studied the Lebesgue measure of Julia sets. Results on Julia sets of positive Lebesgue measures are treated in [24] [25] [26] [27]. Julia sets of Lebesgue measure zero are given in [28]. We consider the Lebesgue measure of the permutable transcendental entire functions and give some results about the zero measure of the Julia sets of the permutable transcendental entire functions family. Firstly we prove the following result.
Theorem 1. If
and
are permutable transcendental entire functions, then
Let
be entire functions. Put
, and
For
, we define
as following
We also define the inverse of
as following:
A point
is said to be a normal point of
. If there exists a neighborhood U of z such that
is a normal family on U for each
. The set of normal points is called the Fatou set of
, denoted by
, and its complement in
, denoted by
, is called the Julia set of
. The Fatou set
is open and forward invariant and Julia set
is closed and backward invariant. More information about the random dynamical system can be found in [10] [29] [30].
In this paper, we study the random dynamics of entire functions family of which the orbits of singularity stay away from the Julia set. Let
and
. If
, put
McMullen [31] proved the following theorem.
Theorem D (McMullen [31]) If
, then for
we have
Let
and
We prove the following result.
Theorem 2. If
,
, then for any
,
, as
.
McMullen [31] gave the following notion. A plane set E is called thin at
, if its density is bounded away from 1 in all sufficiently large discs, that is, if there exist positive R and
such that all complex z and every discs
of center z and radius
.
In [31], McMullen proved the following result.
Theorem E (McMullen [31]) If
, E is a measurable completely invariant subset of
such that E is thin at
, then
.
We consider the entire function family in C, and show the following results.
Theorem 3. If
, E is a measurable completely invariant set of
, and
such that E is thin at
, then
.
For the permutable transcendental entire functions family, we prove the following result.
Theorem 4. If
,
, for
and exits
such that
, then
, for any
.
Remark. By using theorem 1, we can remove the special condition of the transcendental entire functions family in theorem 4. Let
be a permutable transcendental entire functions family and exits a
such that
, then
, for any
.
3. Proofs of Theorems 1, 2, 3 and 4
The following well-known result is needed in the proof of theorems (see [32] Lemma 4.1).
Lemma 1 (Baker [32]) If
and
are permutable transcendental entire functions, then
3.1. Proof of Theorem 1
Since
are permutable transcendental entire functions, Lemma 1 imply that
, and hence that
. By the complete invariance of
we have
(1)
So
(2)
Similarly we have
then
(3)
So
(4)
If
, we have a contradiction with (1) and (2);
If
, we have a contradiction with (3) and (4).
So
.
3.2. Proof of Theorem 2
Since
, hence
Since
then
so
have uniform expansion, that is, for all
, exist a number
such that
, where
. Since
, for any
,
3.3. Proof of Theorem 3
If
, for some
, by the theorem E, we have
.
If
, for any
, put
. If
, then
and
. By the completely invariant of
and E, we have
and
, so that,
. By the definition of
,
, then
. On the other hand, by the definition of
and E,
are completely invariant sets, then
So
. Similarly
So
. Thus
So
,
are all completely invariant sets of
. Since E is thin at
, then
is thin at
. By theorem E and
we have
.
Since
and
so
Therefore
3.4. Proof of Theorem 4
If
is not thin at
, then from the definition of E is thin at
, for any
and
, exists
and
, such that
So that
Then
Since for any
,
by Lemma 1,
hence
(5)
and
(6)
(5) and (6) contradiction with
and
, so
is thin at
. By Theorem 3, we have
.
Acknowledgements
The authors express sincere gratitude to the reviewer for his valuable and constructive comments. This research was supported by the National Natural Science Foundation of China (Grant No.11861005).