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 
Mathematics
CLC number: O173
Some Metric Properties of Sets Related to Luroth Expansion
^{1}
School of Mathematics and Statistics, Hubei University of Education, Wuhan
430205, Hubei, China
^{2}
School of Mathematics and Statistics, Wuhan University, Wuhan
430072, Hubei, China
Received:
21
November
2021
This paper is mainly concerned with scrambled sets for Luroth map. It is shown that all scrambled sets have null Lebesgue measure and there exists a scrambled set with full Hausdorff dimension.
Key words: Luroth expansion / Hausdorff dimension / LiYorke chaotic
Biography: DENG Jiang, male, Master candidate, research direction: metric number theory. Email: stxydj@hue.edu.cn
Foundation item: Supported by the Excellent Young and MiddleAged Science and Technology Innovation Team Project in Higher Education Institutions of Hubei Province (T201520)
© Wuhan University 2022
This 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 kth 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 wellknown 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^{[35]}. 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. [810].
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 LiYorke. It is wellknown that the surjection continuous transformation on compact metric space with positive topological entropy is chaotic in the sense of LiYorke^{[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 LiYorke.
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 dim_{H} 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 rankn basic interval. Denote by the rankn basic interval containing x. Write I for the length of an interval I. The next proposition concerns the length of rankn basic intervals.
Proposition 1 ^{[2]} For any with ,the rankn basic interval is the interval with the endpointsand
For any integer M≥2, let E_{M} be the set of points in (0, 1) whose digits in the Luroth expansion do not exceed M. That is, We can see that E_{M} is a selfsimilar set, and then the Hausdorff dimension of E_{M} 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 onesided infinite sequence over A by . The symbolic x_{i} is called the ith 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 rankk basic intervals from left to right, put , then for any , we haveThus there exist two positive integers i≠j such that . As a result, we have , where and . Hence for all n≥k, we have , which contradicts the definition of scrambled pair.
Let and Δ_{M} 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)
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 Δ_{M} 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 M_{1} 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 u≠v. We shall prove that (u,v) is a scrambled pair for shift σ.
Case (i) M_{1}=M_{2}. Let , then there exist two different points such that and . Since x≠y, there exists k≥1 such that . Notice that the symbols x_{k} and y_{k} appear infinitely often in the same location of and respectively. Using the same method, we obtain that x_{k} and y_{k} 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) M_{1}≠M_{2}. Let and . Then there exist and such that and . From 1) of Remark 1 we have
By the definition of Δ_{M}, there exists an increasing sequence such that and .
It follows that , thus .
The proof of lower limits is similar to the case M_{1}=M_{2}. 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 , u≠v. 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) M_{1}=M_{2}. 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) M_{1}≠M_{2}. Let and . From 2) of Remark 1, there exists an increasing sequence such that and . Suppose that M_{1}<M_{2}, 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 d_{n+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 nt(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 c_{1},c_{2} such that .
Lemma 5 For any , there exists N_{1} 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 yx 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 yx is greater than the distance between δ_{1} and δ_{2}, then
Proof of Theorem 2 Consider a map fined as follows: For any , let .
For any ε>0, by Lemma 4, when satisfywhere N_{1} 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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