Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 2, April 2022
Page(s) 93 - 98
DOI https://doi.org/10.1051/wujns/2022272093
Published online 20 May 2022

© Wuhan University 2022

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

For any real number , the Luroth map is defined by(1)where and [x] denotes the greatest integer not exceeding x. We define the integer sequence by(2)where denotes the k-th iterate of T. By (1) and (2), for any , this map can generate a series expansion of x, i.e.,where are positive integer for any . We call the digits of the Luroth expansion of x, and write the above representation as for simplicity. Such a series expansion was first studied by Luroth[1] in 1883.

Luroth series expansion plays an important role in the representation theory of real numbers and dynamical systems. It is well-known that every irrational number has a unique infinite expansion and each rational number has either a finite or a periodic expansion (see Galambos[2]). For dynamical properties, the transformation T is invariant and ergodic with respect to Lebesgue measure[3-5]. In other direction, Fan et al [6] obtained the Hausdorff dimension of sets of real numbers with prescribed digit frequencies in Luroth expansion. Barreira and Iommi[7] considered the Hausdorff dimension of a class of sets defined in terms of the frequencies of digits in Luroth expansion. For more details, we can refer the reader to Refs. [8-10].

Asymptotic behavior of the orbits is one of the most important theme in dynamical systems. The first mathematical treatment of chaotic behavior of dynamical system appeared in the work of Li and Yorke in 1975[11]. A dynamical system is a pair (X,f), where X is a compact metric space with a metric d and f being a continuous map . A subset , containing at least two points, is a scrambled set for f if every pair of distinct points in S is a scrambled pair, i.e.,and

If X contains an uncountable scrambled set, then the dynamical system (X,f) is called chaotic in the sense of Li-Yorke. It is well-known that the surjection continuous transformation on compact metric space with positive topological entropy is chaotic in the sense of Li-Yorke[12]. Note that the topological entropy of the Luroth map is infinite, the result in Ref. [12] can not be applied since it is not continuous.

In the following we will show that the scrambled set for Luroth map is small in the sense of Lebesgue measure and large in the sense of Haudsdorff dimension, not just uncountable.

Theorem 1   Let T be the Luroth map on (0, 1), then all scrambled sets for T have null Lebesgue measure.

Theorem 2   Let T be the Luroth map on (0, 1), then there exists a scrambled set in (0, 1) with full Hausdorff dimension.

Corollary 1   The Luroth dynamical system ((0, 1), T) is chaotic in the sense of Li-Yorke.

We use N to denote the set of positive integers, Λ the set of points whose Luroth expansion is finite, λ the Lebesgue measure and dimH the Hausdorff dimension.

1 Preliminaries

In this section, we present some elementary results in the theory of Luroth expansion and some fundamental concepts in symbolic dynamics. For more details, we refer to monographs of Dajani and Kraaikamp[8] and Kurka[13].

Lemma 1   [3] The digits are independent identically distributed.

For any , we call a rank-n basic interval. Denote by the rank-n basic interval containing x. Write |I| for the length of an interval I. The next proposition concerns the length of rank-n basic intervals.

Proposition 1   [2] For any with ,the rank-n basic interval is the interval with the endpointsand

As a result,(3)

For any integer M≥2, let EM be the set of points in (0, 1) whose digits in the Luroth expansion do not exceed M. That is, We can see that EM is a self-similar set, and then the Hausdorff dimension of EM is the unique positive root of

Lemma 2   [14] For any integer M≥2. Let , then .

Now we give some notations for symbolic spaces. we denotes by A n the set of words of A with the length of n. A n is the set of infinite words and, the concatenation of words is written as uv. For any M≥3, let or , we denote the symbolic space of one-sided infinite sequence over A by . The symbolic xi is called the i-th coordinate of x. Let i,j be positive integers with i<j, write . For any , we define the metricThe shift map σ is defined by

2 Proofs of Theorem 1 and Theorem 2

In this section, we first prove that all scrambled sets of Luroth map have null Lebesgue measure. To deal with Theorem 2, inspired by Xiong[15], Liu and Li[16], we will construct a scrambled set in and then establish a continue and bijective map between and (0,1) such that the projection of the scrambled set in has full Hausdorff dimension.

Proof of Theorem 1   Suppose that there exists a scrambled set A for Luroth map T with , let k be the smallest positive integer such that . Let be all rank-k basic intervals from left to right, put , then for any , we haveThus there exist two positive integers ij such that . As a result, we have , where and . Hence for all nk, we have , which contradicts the definition of scrambled pair.

Let and ΔM be the maps dependent on the symbol M(M≥3), we write to emphasize the dependence of M.

1) For any and k≥1, we define byOne can see that(4)

2) For any , is given by(5)

3) Let be a sequence of positive integers, for any and , we define by(6)

4) For any , we define such that(7)Then we have

Remark 1  

1) The mappings are continuous and injective, the mapping ΔM is a continuous bijective from to .

2) For any , and .

For any , let and , we have the following lemma.

Lemma 3   For any , if , then .

Proof   Without loss of generality, we assume that . It is worth nothing that and .

Hence, by the definition of and , for any and , it is easy to see that M1 appears infinitely often in , but not appear in , and thus .

Lemma 4   The set S is a scrambled set of shift σ on .

Proof   Let and such that uv. We shall prove that (u,v) is a scrambled pair for shift σ.

Case (i)   M1=M2. Let , then there exist two different points such that and . Since xy, there exists k≥1 such that . Notice that the symbols xk and yk appear infinitely often in the same location of and respectively. Using the same method, we obtain that xk and yk appear infinitely often in the same location of and , respectively. As a result, there exists an increasing sequence such that .

On the other hand, by (4) and (5), there exits an increasing sequence such that

By the definition of , there exists an increasing sequence such that

Then we have and thus .

Case (ii)   M1M2. Let and . Then there exist and such that and . From 1) of Remark 1 we have

By the definition of ΔM, there exists an increasing sequence such that and .

It follows that , thus .

The proof of lower limits is similar to the case M1=M2. In fact, there exists an increasing sequence such that , with the same method, we getand thus .

It is the fact that the Luroth expansion of is infinite and unique. Then for any , we define a continuous bijective map Ф from to by , then we have .

Lemma 5   The set Ф (S) is a scrambled set of T on (0,1).

Proof   For any , uv. Let and , by the definition of scrambled set, we shall prove that is a scrambled pair of T.

Lower limits   By Lemma 3, we have , thus there exists an increasing sequence such that . Recall that the map Ф is continuous and . Then we have

It is easy to see thatand thus .

Upper limits   We divide the proof into two cases.

Case (i)   M1=M2. Let , then for any i≥1, . By Lemma 4, we have , thus there exists an increasing sequence such that . Without loss of generality, let , we obtain

Hence, we get .

Case (ii)   M1M2. Let and . From 2) of Remark 1, there exists an increasing sequence such that and . Suppose that M1<M2, with the same method, we have

Therefore, .

In order to estimate the Hausdorff dimension of , we shall make use of a kind of symbolic space described as follow: for any n≥1, set

For any n≥1 and , letwhere the union is taken over all dn+1 such that . It is obvious that

Recall that . For any n≥1 and , let t(n) be the number of k such that and . Let be the block obtained by eliminating the terms in , then the length of is n-t(n). For simplicity, set(8)

Then we have , where .

By the definition of t(n), it is easy to check that for large enough n there exist two positive constants c1,c2 such that .

Lemma 5   For any , there exists N1 such that for any , . We have .

Proof   Let , by (1) and (8), we have(9)

By (9) and Lemma 1, it is easy to see that

For any , without loss of generality, we assume that x<y. Notice that the points x and y can not be contained in the same for any and large enough k. Thus there exists a greatest integer n such that x,y are contained in the same basic interval of rank n, that is to say, there exists such that , and , . Sinceand

We have , . As a consequence, the distance y-x is not less than the gap between and .

Lemma 6  

Proof   Let denote the right endpoint of and the left endpoint of , respectively, then

So y-x is greater than the distance between δ1 and δ2, then

Proof of Theorem 2   Consider a map fined as follows: For any , let .

For any ε>0, by Lemma 4, when satisfywhere N1 is the same as in Lemma 5, we have

As a result, by Ref.[17] Proposition 2.3 and , we obtain(10)

Recall that , then . Combining with (10), we have

Let and then , we conclude that =1.

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