Research Article Periodic Points of Asymmetric Bernoulli Shifts

It is well-known that Sharkovskii’s theorem gives a complete structure of periodic order for a continuous self-map on a closed bounded interval. As a further study, a natural problem is how to determine the location and number of periodic points for a speciﬁc map. This paper considers the periodic points of asymmetric Bernoulli shift, which is a piecewise linear chaotic map.


Introduction
In 1964, Sharkovskii [1] firstly introduced a special ordering on the set of positive integers. is ordering implies that if p ⊲ q and a continuous self-map of a closed bounded interval has a point of period p; then it has a point of period q. e least number with respect to this ordering is 3. us, if a map has a point of period 3, then it has points of any periods. In 1975, the latter result was rediscovered by Li and Yorke [2]. en numerous papers are devoted to the study of interval maps (see e.g., [3][4][5] and references therein).
Bifurcation points of some interval maps were studied in [6], and the limit behavior of orbits and probabilistic some problems were considered in [7,8]. Recently, Ivanov in [9] considered an exact lower bound for the number of orbits of a given period for a self-map of a closed bounded interval.
Specially, when a � 1/2, it is the Bernoulli Shift or the binary transformation, also known as doubling map or the binary transformation. Conjugacies between asymmetric Bernoulli shifts are constructed in [10].
Given a positive integer n, one interesting question is how to find all n-periodic points of F. e other is how many n-periodic points of F.
In this paper, we study periodic orbits of F. In the next section, we present dynamics of jumps of F n . Section 3 recalls the real number representation, i.e, F-expansion. In Section 4, we use the F-expansion to give explicit formulas of F n (x) for n ∈ N, explicit formulas of jumps of F n (x), explicit formulae of fixed points of F n (x), and explicit formulas of all n-periodic points of F(x). e last section gives the number h(n) of periodic orbits of a given period n for F and the limit behavior of h(n).

Dynamics of Jumps of F n
A point c ∈ (0, 1) is called a jump of F if the one-sided limits, F(c− ) and F(c+), exist and are finite, but are not equal. e set of jumps of F is denoted by J(F). One can see that Each element of J(F n+1 )\J(F n ) must be a preimage under F of a point from J(F n ). More precisely, (3) e map F has the unique jump a. Put x 1,0 : � 0, x 1,1 : � a, and x 1,2 : � 1. Let I denote the unit interval [0, 1], I 1,1 : � (x 1,0 , x 1,1 ), and I 1,2 : � (x 1,1 , x 1,2 ). One can see that F n has 2 n − 1 jumps for n ≥ 2 by induction. For i, j ∈ N + , let x i,0 : � 0, x i,2 i : � 1, and x i,j denote the j th jumps of F i in the following order:

Lemma 1.
For n ≥ 1, the jumps of F n and F n− 1 have the following relationship: Proof. We first claim that a is a jump of F n for every n ≥ 1. In fact, since a is a jump of F(x), a is also a jump of F n (x) for n ≥ 2. Moreover, it is easy to check that F n (a) � 1 for n ≥ 2. Next, we prove (i) and (ii) by induction. It is clear that these results holds for n � 2.
Assume that these results hold for n � m ≥ 2, i.e., Now we shall prove these results hold for n � m + 1.
Further, by the definition of jump, x m+1,2 m +k is a jump of en for 1 ≤ k ≤ 2 n− i and 1 ≤ i ≤ n − 1, is completes the proof.

F-Expansion
In this section, we will introduce a new real number representation.
In fact, the itinerary of x ∈ [0, 1] with respect to F and a ∈ (0, 1) is just the F-expansion of a real x ∈ [0, 1]. According to [10], or these two classic papers [11,12], we have an expansion for x in powers of the numbers a and 1 − a: where s 0 � 0 and s k : � k j�1 ε j for k ≥ 1. us, every x ∈ [0, 1] can be represented through its digit sequence ε k k∈N +. In this situation, write x � [ε 1 , ε 2 , . . . , ε k , . . .] for short. One can see that every infinite F-expansion is unique, whereas each x ∈ (0, 1) with a finite F-expansion can be expanded in exactly two ways, namely, one immediately verifies that In the following, we employ a convention in which finite fractions such as  Proof. It follows from a property of the asymmetric Bernoulli shift F(x) that x ≤ a provided that ε 1 � 0 in x � [ε 1 , ε 2 , . . . , ε k , . . .]. One then finds On the other hand, one has a < x ≤ 1 if ε 1 � 1 and hence □ is shows that, from the perspective of symbolic dynamics, F corresponds to the shift map on the space 0, 1 { } N + , at least for those points with an infinite F-expansion.
One easily finds that the periodicity of the orbits is related to recurring F-expansions. For example, is a recurring F-expansion with the recurring unit of the length 5, and hence, it is a 5-periodic point of F.

The Explicit Formula of F n
Since F n is a piecewise linear map, and F n is strictly increasing on each subinterval I n,k . One can obtain the explicit formula of F n .
ε j a n− j− s n +s j − 1 (1 − a) s n − s j +1 , for n ∈ N + . (17) Proof. We prove this result by mathematical induction.

Journal of Mathematics
erefore, the result holds for n � m + 1. e proof is completed.

□
As a corollary, we present the exact formulas of these jumps of F n .

Corollary 1. All jumps of F n are given by
where ε j � 0 or 1, and not all are ε j equal to 0.
Proof. If all ε j are zero, then x 1,0 � 0, and it is not a jump. From eorem 2, solving F n (x) � 0, we can obtain all these jumps of F n . (24) e smallest positive integer n satisfying the above is called the prime period or least period of the point x, the point x is called an n-periodic point of f, and the sequence x, f(x), . . . , f n− 1 (x) is called an n-periodic orbit.
In particularly, an 1-periodic point is called a fixed point. e following corollary presents the exact formulas of all fixed points of F n .

Corollary 2.
All fixed points of F n are given by where ε j � 0 or 1.
Proof. Since the curve of y � F n (x) intersects the line of y � x at 2 n points, F n (x) has 2 n fixed points. Solving In general, the intersections of y � F n (x) and y � x have 2 n periodic points. If x is a p-periodic point of F, then p | n. Let h(p) denote the number of the p-periodic points. en, p|n h(p) � 2 n , for every integer n ≥ 1, where the sum extends over all positive divisors p of n.
In order to obtain the exact number h(n) of n-periodic points of F, we need to introduce the Möbius function and Möbius inversion formula (see, for example, [13,14]).
Lemma 3 (Möbius inversion formula). If h and g are arithmetic functions, i.e., from N to C, satisfying