Wuhan Univ. J. Nat. Sci.
Volume 27, Number 2, April 2022
|Page(s)||93 - 98|
|Published online||20 May 2022|
CLC number: O173
Some Metric Properties of Sets Related to Luroth Expansion
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
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 / Li-Yorke chaotic
Biography: DENG Jiang, male, Master candidate, research direction: metric number theory. E-mail: firstname.lastname@example.org
Foundation item: Supported by the Excellent Young and Middle-Aged 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.
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 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). For dynamical properties, the transformation T is invariant and ergodic with respect to Lebesgue measure[3-5]. In other direction, Fan et al  obtained the Hausdorff dimension of sets of real numbers with prescribed digit frequencies in Luroth expansion. Barreira and Iommi 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. 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. Note that the topological entropy of the Luroth map is infinite, the result in Ref.  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.
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 and Kurka.
Lemma 1  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  For any with ,the rank-n basic interval is the interval with the endpointsand
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  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
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, Liu and Li, 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 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 ΔM be the maps dependent on the symbol M(M≥3), we write to emphasize the dependence of M.
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 u≠v. 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 x≠y, 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 .
Then we have and thus .
By the definition of ΔM, there exists an increasing sequence such that and .
It follows that , 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.
Upper limits We divide the proof into two cases.
Hence, we get .
Case (ii) M1≠M2. 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
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 .
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 .
Proof of Theorem 2 Consider a map fined as follows: For any , let .
As a result, by Ref. Proposition 2.3 and , we obtain(10)
Let and then , we conclude that =1.
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