Minimal complexity of equidistributed infinite permutations
aa r X i v : . [ m a t h . C O ] D ec Minimal complexity of equidistributed infinitepermutations
S. V. AVGUSTINOVICH ∗ , A. E. FRID † and S. PUZYNINA ‡ August 31, 2018
Abstract An infinite permutatation is a linear ordering of the set of naturalnumbers. An infinite permutation can be defined by a sequence of realnumbers where only the order of elements is taken into account. Inthe paper we investigate a new class of equidistributed infinite permu-tations, that is, infinite permutations which can be defined by equidis-tributed sequences. Similarly to infinite words, a complexity p ( n ) ofan infinite permutation is defined as a function counting the number ofits subpermutations of length n . For infinite words, a classical result ofMorse and Hedlund, 1938, states that if the complexity of an infiniteword satisfies p ( n ) ≤ n for some n , then the word is ultimately peri-odic. Hence minimal complexity of aperiodic words is equal to n + 1,and words with such complexity are called Sturmian. For infinite per-mutations this does not hold: There exist aperiodic permutations withcomplexity functions growing arbitrarily slowly, and hence there areno permutations of minimal complexity. We show that, unlike forpermutations in general, the minimal complexity of an equidistributedpermutation α is p α ( n ) = n . The class of equidistributed permutationsof minimal complexity coincides with the class of so-called Sturmianpermutations, directly related to Sturmian words. Infinite permutations can be defined as equivalence classes of real sequenceswith distinct elements, such that only the order of elements is taken into ∗ Sobolev Institute of Mathematics, Novosibirsk, Russia, [email protected] † Aix-Marseille Universit´e, France, [email protected] ‡ Universit´e Paris Diderot, France, and Sobolev Institute of Mathematics, Novosibirsk,Russia, [email protected] N .An infinite permutation can be considered as an object close to an infiniteword where instead of symbols we have transitive relations < or > betweeneach pair of elements. So, many properties of such permutations can beconsidered from a symbolic dynamical point of view.Infinite permutations in the considered sense were introduced in [10]; seealso a very similar approach coming from dynamics [7] and summarised in[2]. Since then, they were studied in two main directions: first, permuta-tions directly constructed with the use of words are studied to reveal newproperties of words used for their construction [9, 17, 18, 19, 21, 22, 23]. Inthe other approach, properties of infinite permutations are studied in com-parison with those of infinite words, showing some resemblance and somedifference.In particular, both for words and permutations, the (factor) complexityis bounded if and only if the word or the permutation is ultimately periodic[10, 20]. However, for minimal complexity in the aperiodic case the situa-tions are different: The minimal complexity of an aperiodic word is n + 1,and the words of this complexity are well-studied Sturmian words [16, 20].As for the permutations, there is no “minimal” complexity function for theaperiodic case: for any unbounded non-decreasing function, we can con-struct an aperiodic infinite permutation of complexity ultimately less thanthis function [10]. The situation is different for the maximal pattern com-plexity [13, 14]: there is a minimal complexity for both aperiodic words andpermutations, but for permutations, unlike for words, the cases of minimalcomplexity are characterised [3]. All the permutations of lowest maximalpattern complexity are closely related to Sturmian words, whereas wordsmay have lowest maximal pattern complexity even if they have a differentstructure [14].Other results on the comparison of words and permutations include dis-cussions of automatic permutations [12] and of the Fine and Wilf theorem[11], and a study of square-free permutations [6].In this paper we introduce a new class of equidistributed infinite permu-tations and study their complexity. An equidistributed permutation then isa permutation which can be defined by an equidistributed sequence of dis-tinct numbers from [0 ,
1] with the natural order; and we show that this classof permutations is natural and wide. Some of equidistributed permutationscan be defined using uniquely ergodic infinite words, or, equivalently, sym-bolic dynamical systems. A very similar approach directly relating uniquelyergodic symbolic dynamical systems and specific dynamical systems on [0 , n . Moreover, equidistributed permutations of minimalcomplexity are exactly Sturmian permutations in the sense of [19].The paper is organized as follows. After general basic definitions anda section on the properties of Sturmian words (and permutations), we in-troduce equidistributed permutations and study their basic properties. Themain result of the paper, Theorem 5.1, characterising equidistributed per-mutations of minimal complexity, is proved in Section 5.Some of the results of this paper, for a much more restrictive definitionof an ergodic permutation, were presented at the conference DLT 2015 [5]. In this paper, we consider three following types of infinite objects. First,we need infinite words over a finite, often binary, alphabet: an infinite wordis denoted by u = u [0] u [1] . . . u [ n ] . . . , where u [ i ] are letters of the alphabet.Then, we make use of infinite sequences of reals, denoted by a = ( a [ n ]) ∞ n =0 .We say that two infinite sequences ( a [ n ]) ∞ n =0 and ( b [ n ]) ∞ n =0 of pairwise dis-tinct reals are equivalent , denoted by ( a [ n ]) ∞ n =0 ∼ ( b [ n ]) ∞ n =0 , if for all i, j the conditions a [ i ] < a [ j ] and b [ i ] < b [ j ] are equivalent. Since we consideronly sequences of pairwise distinct real numbers, the same condition can bedefined by substituting ( < ) by ( > ): a [ i ] > a [ j ] if and only if b [ i ] > b [ j ]. Atlast, we consider infinite permutations defined as follows. Definition 2.1. An infinite permutation is an equivalence class of infinitesequences of pairwise distinct reals under the equivalence ∼ .So, an infinite permutation is a linear ordering of the set N = { , . . . , n, . . . } , and a sequence of reals from the equivalence class definingthe permutation is called a representative of a permutation. We denotean infinite permutation by α = ( α [ n ]) ∞ n =0 , where α [ i ] are abstract elementsequipped by an order: α [ i ] < α [ j ] if and only if a [ i ] < a [ j ] for a representative( a [ n ]) of α . So, one of the simplest ways to define an infinite permutationis by a representative, which can be any sequence of distinct real numbers. Example 2.2.
Both sequences ( a [ n ]) = (1 , − / , / , . . . ) with a [ n ] =( − / n and ( b [ n ]) with b [ n ] = 1000 + ( − / n are representatives of thesame permutation α = α [0] , α [1] , . . . defined by α [2 n ] > α [2 n + 2] > α [2 k + 3] > α [2 k + 1]3 .. Figure 1: A graphic illustration of the permutation from Example 2.2for all n, k ≥
0. So, the sequence of elements with even indices is decreasing,the sequence of elements with odd indices is increasing, and every elementwith an even index is greater than any element with an odd index. A way torepresent the permutation α as a chart is given in Fig. 1; here the elementswhich are bigger are higher on the image.A factor of an infinite word (resp., sequence, permutation) is any finitesequence of its consecutive letters (resp., elements). For j ≥ i , the factor u [ i ] · · · u [ j ] of an infinite word u = u [0] u [1] · · · u [ n ] · · · is denoted by u [ i..j ],and we use similar notation for sequences and permutations. The length ofsuch a factor f , denoted by | f | , is j − i + 1. Factors are considered as newobjects unrelated to their position in the bigger object, so, a factor of aninfinite word is just a finite word, and a factor of an infinite permutation canbe interpreted as a usual finite permutation. In particular, for the exampleabove for any even i we have α [ i ] > α [ i + 2] > α [ i + 3] > α [ i + 1] and thus canwrite α [ i..i + 3] = (cid:18) (cid:19) . However, in general infinite permutationscannot be defined as permutations of N . For instance, the permutationfrom Fig. 1 has a maximal element.An infinite word u is called ultimately ( | w | )- periodic if u = · · · = vw ω for some finite words v, w , where w is non-empty. An infinite per-mutation α is called ultimately ( t )- periodic if for all sufficiently large i, j theconditions α [ i ] < α [ j ] and α [ i + t ] < α [ j + t ] are equivalent. The permutationfrom Fig. 1 is ultimately 2-periodic, as well as the word 0010101 · · · = 0(01) ω .A word or a permutation which is not ultimately periodic is called aperiodic .The complexity p u ( n ) (resp., p α ( n )) of an infinite word u (resp., permu-tation α ) is a function counting the number of its factors of length n . Bothfor infinite words [20] and for infinite permutations [10], the complexity is anon-decreasing function, and the bounded complexity is equivalent to peri-odicity. However, for words, a stronger result holds: The complexity of anaperiodic word u satisfies p u ( n ) ≥ n + 1 [20]. The words of complexity n + 1are called Sturmian and are discussed in Section 4.4s it was proved in [10], contrary to words, we cannot distinguish per-mutations of “minimal” complexity: for each unbounded non-decreasingfunction f ( n ) with integer values, we can find a permutation α on N suchthat for n large enough, p α ( n ) < f ( n ). The required permutation can bedefined by the inequalities α [2 n − < α [2 n + 1] and α [2 n ] < α [2 n + 2] forall n ≥
1, and α [2 n k − < α [2 k − < α [2 n k ] for a sequence { n k } ∞ k =1 whichgrows sufficiently fast (see [10] for further details).In this paper, we introduce a new natural notion of an equidistributed permutation and prove that the minimal complexity of an equidistributedpermutation is n . First, a sequence ( a [ n ]) ∞ n =0 of reals from [ a, b ] is called equidistributed if for each t ∈ [ a, b ] the following limit exists and is equal to t − ab − a : lim n →∞ ♯ { a [ i ] | a [ i ] < t, ≤ i < n } n = t − ab − a . In particular, in an equidistributed sequence the fraction of elements froman interval from [0 ,
1] is equal to the length of the interval.
Definition 2.3.