1 Introduction
Given a rational function
$f \in {\mathbb C}(X)$
and an initial point
$\alpha \in {\mathbb C}$
, we consider the orbit
$$ \begin{align*} {\mathcal O}_f(\alpha) = (f^{n}(\alpha): n =0,1, \ldots), \end{align*} $$
where we set
$f^{0}(X) = X$
and then define recursively
$$ \begin{align*} f^{n}(X)=f(f^{n-1}(X)), \quad n=1,2 \ldots. \end{align*} $$
We also assume that the orbit
${\mathcal O}_f(\alpha )$
terminates if
$\alpha $
is a pole of
$f^{n}(X)$
for some n.
We emphasise that we treat orbits as naturally ordered sequences, rather than as sets. If this sequence becomes eventually periodic, we call
$\alpha $
preperiodic.
There is a vast amount of work studying the distribution of orbits
${\mathcal O}_f(\alpha )$
in various subsets
${\mathcal S} \subseteq {\mathbb C}$
, under various algebraic and arithmetic constraints on the function f, the initial point
$\alpha $
, and the set
${\mathcal S}$
(see [Reference Bell, Chen and Hossain1, Reference Chen2, Reference Ferraguti, Ostafe and Zannier4, Reference Mello5, Reference Ostafe7–Reference Ostafe and Young10] and the references therein).
Here, we use a recent result of Orevkov and Pakovich [Reference Orevkov and Pakovich6] to study an apparently new question and estimate the frequency of the elements in an orbit
${\mathcal O}_f(\alpha )$
, which fall on the unit circle
$$ \begin{align*} {\mathbb U} = \{z\in {\mathbb C}:~|z| = 1\}. \end{align*} $$
Some arithmetic analogues of this question have been considered by Ostafe [Reference Ostafe7], where a rational function
$f \in {\mathbb K}(X)$
is defined over a number field
${\mathbb K}$
and the unit circle is replaced by the set
${\mathcal U}_{\mathbb K}$
of roots of unity of
${\mathbb K}$
. In particular, in this setting, using [Reference Ostafe7, Theorem 1.2], one can derive the finiteness of the sets
${\mathcal O}_f(\alpha ) \cap {\mathcal U}_{\mathbb K}$
(under some natural conditions on
$\alpha $
and f).
2 Main result
Given an integer N, we define
$$ \begin{align*} T_f(\alpha, N) = \# \{n: f^{n}(\alpha) \in {\mathbb U}, 1 \leqslant n \leqslant N\}. \end{align*} $$
We say that
$f(z) \in {\mathbb C}(z)$
is special if
$f^{n}({\mathbb U})\subseteq {\mathbb U}$
for some n, that is,
$f^{n}$
maps
${\mathbb U}$
to
${\mathbb U}$
.
For example, if
$f(z)$
is a polynomial, than as follows from the discussion in [Reference Orevkov and Pakovich6, Section 1], in particular, [Reference Orevkov and Pakovich6, (3)], this is possible only if
$f^{n}(z)$
is a monomial, that is,
$f^{n}(z) = z^k$
for some integer
$k\geqslant 0$
and thus
$f(z)$
is also a monomial.
Theorem 2.1. For any nonspecial rational function
$f(z) \in {\mathbb C}(z)$
and an initial point
$\alpha \in {\mathbb C}$
, which is not preperiodic, we have
$$ \begin{align*} T_f(\alpha, N) = O( N/\log N), \end{align*} $$
where the implied constant depends only on
$\deg f$
.
As we have mentioned, nonmonomial polynomials are always nonspecial. However, it is not immediately clear how to characterise nonspecial rational functions, which we pose as an open question.
3 Preliminaries
The following result is a special case of [Reference Orevkov and Pakovich6, Theorem 1.1], which improves a similar bound from [Reference Pakovich and Shparlinski11] on the number of intersections of lemniscates.
Lemma 3.1. Let
$F(z) \in {\mathbb C}(z)$
be a rational function of degree D such that
$F({\mathbb U}) \not \subseteq {\mathbb U}$
. Then,
$ \# \{z \in {\mathbb U}: F(z) \in {\mathbb U}\} \leqslant 2D. $
We also need the following simple combinatorial statement given by [Reference D’Andrea, Ostafe, Shparlinski and Sombra3, Lemma 5.7].
Lemma 3.2. Let
$2 \leqslant M<N/2$
. For any sequence
$ 0 \leqslant n_1< \cdots < n_M\leqslant N, $
there exists
$r\leqslant 2N/(M-1)$
such that
$n_{i+1} - n_i = r$
for at least
$(M-1)^2/4N$
values of
$i \in \{1, \ldots , M-1\}$
.
4 Proof of Theorem 2.1
We fix some
$\alpha \in {\mathbb C}$
, which is not preperiodic, and define
$z_n = f^{n}(\alpha )$
,
$n =0, 1, \ldots $
, which are obviously pairwise distinct.
If
$T_f(\alpha , N) =0$
, there is nothing to prove. Thus, we now assume that
$T_f(\alpha , N) \geqslant 1$
.
For
$t = T_f(\alpha , N) - 1$
, let
$1 \leqslant n_1< \ldots < n_{t+1} \leqslant N$
be such that
$z_{n_i} \in {\mathbb U}$
. Consider the gaps
$d_i = n_{i+1} - n_i$
,
$i =1, \ldots , t$
.
By Lemma 3.2, there is some
$r \leqslant 2N/t$
such that
$d_i = r$
for at least
$I \geqslant t^2/(4N)$
values of
$i \in \{1, \ldots , t\}$
. Hence, there are at least I solutions to the equation
$$ \begin{align*} |f^{r}(z)| = |z| = 1 \end{align*} $$
given by
$z_i = f^{n_i}(\alpha )$
for each i such that
$d_i = r$
. Note that since
$\alpha \in {\mathbb C}$
is not preperiodic, all these points are distinct.
Since
$\deg f^{r} \leqslant (\deg f)^r$
and f is nonspecial, by Lemma 3.1,
$$ \begin{align} \frac{t^2}{4N} \leqslant I \leqslant 2 (\deg f)^r \leqslant 2 (\deg f)^{2N/t}. \end{align} $$
If
$$ \begin{align} 2 (\deg f)^{2N/t} \leqslant N^{1/2}, \end{align} $$
then from (4.1), we have
$t \leqslant 2N^{3/4}$
, which is much stronger that the desired bound. Otherwise, that is, if (4.2) is false,
$$ \begin{align*} \frac{2N}{t} \log(\deg f) + \log 2> \frac{1}{2} \log N , \end{align*} $$
which implies the result.
Acknowledgement
The author is grateful to the referee for the careful reading of the manuscript.
Open access
