Abstract

The prime orbit theorem and Mertens’ theorem are proved for a shift dynamical system of infinite type called the Dyck shift. Different and more direct methods are used in the proof without any complicated theoretical discussion.

1. Introduction and Preliminaries

In number theory, recall that, for functions , one says means that is bounded with respect to for all sufficiently large . That is, there exist constants and such that and is asymptotic to , denoted by , if The prime-counting function that gives the number of primes less than or equal to , for any real number , is denoted by . The prime number theorem (PNT) is the statement

However, the asymptotic formula for primes , where is Euler’s constant and , is known as Mertens’ theorem of analytic number theory. The logarithmic equivalent of Mertens’ theorem is and Mertens’ constant .

Let be a nonempty set and a map. The pair is said to be a dynamical system. In techniques, a dynamical system is an abstract mathematical model describing the time dependence of point’s position in its space. This is conventionally modeled by the map whose iterates denote the passage of time. Many dynamical questions involve counting the number of closed orbits or the periodic points under iteration of a map. A closed (periodic) orbit of length for a continuous map is a set of the form where for some and .

Let be a map, and define which are the set of points of least period under , the set of points of period under , and the set of closed orbits of length under , respectively. It is well known that the Möbius function, of , is

Indeed, for any natural number . Moreover, the sum over all positive divisors of of the Möbius function is zero except when . The Möbius Inversion Formula [1] is defined as follows. If are two arithmetic functions satisfying then

Let

It follows that Consequently and, hence, by Möbius Inversion Formula, Following the analogy between closed orbits and prime numbers, the asymptotic behavior of expressions like may be viewed as a dynamical analogue of the prime number theorem and a dynamical analogue of Mertens’ theorem concerns asymptotic estimates for expressions like where denotes the topological entropy of the map .

Parry in [2] initiated a line of research which uses ideas and techniques of analytic number theory to attack problems of this nature. When has a metric structure with respect to which is hyperbolic or shift of finite type (which will be given in detail in the subsequent section), results of Parry and Pollicott [3] and Noorani [4] have shown similar analogy between the number of closed orbits and the prime number theorem. It has been shown that Sharp in [5] also obtained an analogy between the number of closed orbits and Mertens’ theorem for hyperbolic maps as follows for some constant .

Several orbit-counting results on the asymptotic behavior of both (14) and (15) for other maps like quasihyperbolic toral automorphism (ergodic but not hyperbolic) can be found, for example, in [6–9].

In this paper, analogies between the number of closed orbits of a shift of infinite type called the Dyck shift and both (3) and (4) have been obtained. This paper is organized as follows. In the first section some introduction and preliminaries are given. In the second section the Dyck shift is introduced. The prime orbit theorem and Mertens’ theorem are proved in Section 3.

2. The Dyck Shift

Let be a finite alphabet. On there acts the shift that sends the point into the point . The dynamical systems that are given by the closed shift invariant subsets of , with the restriction of the shift acting on them, are called subshifts. These are studied in symbolic dynamics. An element of will be known as a word, or a block of length . A word of length is called an empty word and denoted by . The set of all finite words with letters taken from is the set . A word is called admissible for the subshift if it appears somewhere in a point of . Let and let denote the set of all admissible words of length in . Then the language of is the collection . The topological entropy of a subshift is given by

Subshifts can be also defined using the notion of forbidden sets. Let be a collection of words over , that is, , which is called the forbidden set. For any such , define to be the subset of sequences in which do not contain any word in . Then, the subshift is a subset of a full shift such that for some collection of forbidden blocks over . If furthermore is finite, then we call a subshift of finite type. The golden mean shift which is defined as the shift system over the alphabet having a forbidden set is a subshift of finite type.

The subshift comprises a lot of SFTs (shifts of finite type) in the shift space, which is said to have property A. A class of nonsofic systems, known as the Dyck systems, first suggested by Krieger [10] and named after an early contributor to the study of free groups and formal languages, codifies the rules of matching parentheses, which is one of these shift spaces. The Dyck shift which comes from language theory is defined to be the shift system over an alphabet that consists of negative symbols and positive symbols. For an in the full shift , is in if and only if every finite block appearing in has a nonzero reduced form. Therefore, the constraint for cannot be bounded. A beautiful way to describe the Dyck shift is in terms of its syntactic monoid.

Let . The alphabet consists of pairs of matching left and right delimiters or symbols. Let be a monoid (with zero) with generators , , and . The relations on the monoid are

We use a mapping : such that, for

Definition 1. The Dyck shift [10] is defined by where .

When , is the full shift on two symbols; we will tacitly assume that . The topological entropy of the Dyck shift is already computed as in [10].

Theorem 2 (see [11]). The number of points in the Dyck shift having period is given by

3. Counting Closed Orbits

In this section, we prove two theorems that involve the counting of orbits for the Dyck shift, where the first one is the prime orbit theorem and the second one is Mertens’ orbit theorem. However, in order to prove these theorems, we firstly prove the following lemma which plays an important role in proving our main results.

Lemma 3. There exist constants and such that the following inequality holds for all :

Proof. Assume that is even. The case where is odd can be proved analogously.
Note first that Therefore, we obtain Also Estimating the term using the binomial theorem we obtain We set using the definition of the binomial coefficient it is easy to prove that and also utilizing (29) and (30) we obtain Applying inequality (31) to (27), we get hence, inequality (26) implies that Thus, if we take and , then we are done.

Theorem 4 (prime orbit theorem). Let be the Dyck shift on a set of pairs and a closed orbit for . Let be the number of closed orbits of not exceeding and the number of points having period under as given by Theorem 2. Then

Proof. Using Lemma 3, we get Similarly, by combining (35) and (36), we obtain Thus, by Möbius Inversion Formula we have Subtracting the dominant terms, we obtain since , , then , so To estimate the dominant terms, let . Then Finally in fact then is a geometric series with as a first term; then we have since .
Therefore, it can be seen that from (40), (41), and (45) Thus since , then Using Lemma 3, we obtain Thus, letting implies that So Therefore which is equivalent to as required.