Abstract
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.
1. Introduction
In 1964, Sharkovskii [1] firstly introduced a special ordering on the set of positive integers. This ordering implies that if and a continuous self-map of a closed bounded interval has a point of period ; then it has a point of period . The least number with respect to this ordering is 3. Thus, 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]. Then numerous papers are devoted to the study of interval maps (see e.g., [3–5] 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.
Consider the asymmetric Bernoulli shift with a parameter , defined by
Specially, when , 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 , one interesting question is how to find all -periodic points of . The other is how many -periodic points of .
In this paper, we study periodic orbits of . In the next section, we present dynamics of jumps of . Section 3 recalls the real number representation, i.e, -expansion. In Section 4, we use the -expansion to give explicit formulas of for , explicit formulas of jumps of , explicit formulae of fixed points of , and explicit formulas of all -periodic points of . The last section gives the number of periodic orbits of a given period for and the limit behavior of .
2. Dynamics of Jumps of
For , let denote the -th iterate of , which is recursively defined by and for .
A point is called a jump of if the one-sided limits, and , exist and are finite, but are not equal. The set of jumps of is denoted by . One can see that
Each element of must be a preimage under of a point from . More precisely,
The map has the unique jump . Put , , and . Let denote the unit interval , , and . One can see that has jumps for by induction. For , let , and denote the th jumps of in the following order:
Put for every . It is clear that is the -th monotonic interval of .
Lemma 1. For , the jumps of and have the following relationship:(i)for(ii)(iii)forand
Proof. We first claim that is a jump of for every . In fact, since is a jump of , is also a jump of for . Moreover, it is easy to check that for .
Next, we prove (i) and (ii) by induction. It is clear that these results holds for .
Assume that these results hold for , i.e.,(i) for (ii)Now we shall prove these results hold for . Denote jumps of bySince is strictly increasing on the subinterval and , for each , there exists the unique point, denoted by , in such that . Since is strictly increasing on , one can see thatFurther, by the definition of jump, is a jump of for each .
Similarly, since is strictly increasing on the subinterval and , for each , there exists the unique point, denoted by , in such that . Since is strictly increasing on , one can see thatFurther, by the definition of jump, is a jump of for each . Let denote . Therefore,(i) for (ii)It follows from (i) that for ,Then for and ,This completes the proof.
3. -Expansion
In this section, we will introduce a new real number representation.
Definition 1. A sequence of 0 and 1 is called the itinerary of with respect to the asymmetric Bernoulli shift and , if, for ,
In fact, the itinerary of with respect to and is just the -expansion of a real . According to [10], or these two classic papers [11, 12], we have an expansion for in powers of the numbers and :where and for .
Thus, every can be represented through its digit sequence . In this situation, write for short. One can see that every infinite -expansion is unique, whereas each with a finite -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 asare represented as finite fractions with infinite zeros, as or , unless otherwise stated.
Lemma 2. If , then .
Proof. It follows from a property of the asymmetric Bernoulli shift that provided that in . One then findsOn the other hand, one has if and hence
This shows that, from the perspective of symbolic dynamics, corresponds to the shift map on the space , at least for those points with an infinite -expansion.
One easily finds that the periodicity of the orbits is related to recurring -expansions. For example,is a recurring -expansion with the recurring unit of the length 5, and hence, it is a 5-periodic point of .
4. The Explicit Formula of
Since is a piecewise linear map, and is strictly increasing on each subinterval . One can obtain the explicit formula of .
Theorem 1. If , then
Proof. We prove this result by mathematical induction.
We firstly consider the trivial case . If , then and . Thus,If , then and . Thus,Therefore, the result holds for .
Assume that the result holds for , i.e.,Now we shall prove that the result holds for . If ; then and . Thus,If ; then and . Thus,Therefore, the result holds for . The proof is completed.
As a corollary, we present the exact formulas of these jumps of .
Corollary 1. All jumps of are given bywhere or 1, and not all are equal to 0.
Proof. If all are zero, then , and it is not a jump.
From Theorem 2, solving , we can obtain all these jumps of .
Definition 2. A point in is called a periodic point of a self-mapping if there exists an positive integer such thatThe smallest positive integer satisfying the above is called the prime period or least period of the point , the point is called an -periodic point of , and the sequence is called an -periodic orbit.
In particularly, an 1-periodic point is called a fixed point.
The following corollary presents the exact formulas of all fixed points of .
Corollary 2. All fixed points of are given bywhere or 1.
Proof. Since the curve of intersects the line of at points, has fixed points. Solving , we have
5. The Number of -Periodic Points
The fixed points of are the intersections of and , namely, two points . The intersections of and have four points where there are two 2-periodic points, namely, two points
The other two intersections and 1 are the fixed points. The intersections of and have eight points where there are six 3-periodic points and two fixed points.
In general, the intersections of and have periodic points. If is a -periodic point of , then . Let denote the number of the -periodic points. Then,where the sum extends over all positive divisors of .
In order to obtain the exact number of -periodic points of , we need to introduce the Möbius function and Möbius inversion formula (see, for example, [13, 14]).
Define the Möblius function by
Thus, if , then , and for any , there holds
Lemma 3 (Möbius inversion formula). If and are arithmetic functions, i.e., from to , satisfying
Then,where is the Möbius function and the sums extend over all positive divisors of .
In effect, the original can be determined given by using the inversion formula.
Corollary 3. The number of -periodic points of is given by,
Let denote the number of -periodic orbits of . Then,
The first several arewhile for larger are
If the values just above are compared toone finds that the ratio of and approaches 1 as .
Now we shall prove that the ratio of and approaches 1 as .
Theorem 2. Let be the number of the -periodic points of . Then,where is the Möbius function.
Proof. On one hand,On the other hand,Consequently,By the squeeze theorem,
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.