Persistence of Wandering Intervals in Self-Similar Affine Interval Exchange Transformations
aa r X i v : . [ m a t h . D S ] J a n PERSISTENCE OF WANDERING INTERVALS IN SELF-SIMILARAFFINE INTERVAL EXCHANGE TRANSFORMATIONS
XAVIER BRESSAUD, PASCAL HUBERT, AND ALEJANDRO MAASS
Abstract.
In this article we prove that given a self-similar interval exchangetransformation T ( λ,π ) , whose associated matrix verifies a quite general alge-braic condition, there exists an affine interval exchange transformation withwandering intervals that is semi-conjugated to it. That is, in this context theexistence of Denjoy counterexamples occurs very often, generalizing the resultof M. Cobo in [C]. Introduction
Since the work of Denjoy [D] it is known that every C -diffeomorphism of thecircle such that the logarithm of its derivative is a function of bounded variationhas no wandering intervals. There is no analogous result for interval exchangetransformations. Levitt in [L] found an example of a non-uniquely ergodic affineinterval exchange transformation with wandering intervals. Latter, Camelier andGutierrez [CG], using Rauzy induction technique exhibited a uniquely ergodic affineinterval exchange transformation with wandering intervals. Moreover, this exampleis semi-conjugated to a self-similar interval exchange transformation. In geometriclanguage, it means that this inter- val exchange transformation is induced by apseudo-Anosov diffeomorphism. In combinatorial terms, the symbolic system isgenerated by a substitutionAn interval exchange transformation (IET) is defined by the length of the intervals λ = ( λ , . . . , λ r ) and a permutation π . It is denoted by T ( λ,π ) . To define an affineinterval exchange transformation (AIET) one additional information is needed; theslope of the map on each interval. This is a vector ( w , . . . , w r ) with w i > i = 1 , . . . , r . Camelier and Gutierrez remarked that a necessary condition for anAIET to be conjugated to the interval exchange transformation T ( λ,π ) is that thevector log( w ) = (log( w ) , . . . , log( w r )) is orthogonal to λ .The conjugacy of an affine interval exchange transformation with an interval ex-change transformation was studied in details by Cobo [C]. He proved that theregularity of the conjugacy depends on the position of the vector log( w ) in theflag of the lyapunov exponents of the Rauzy-Veech-Zorich induction. In particular,assume that T ( λ,π ) is self-similar, which means that λ is an eigenvector of a positive r × r matrix R obtained by applying Rauzy induction a finite number of times.Cobo proves that if log( w ) belongs to the contracting space of t R then f is C conjugated to T ( λ,π ) . If log( w ) is orthogonal to λ and is not in the contracting Date : July 7, 2007.1991
Mathematics Subject Classification.
Primary: 37C15; Secondary: 37B10.
Key words and phrases. interval exchange transformations, substitutive systems, wanderingsets. space of t R then any conjugacy between f and T ( λ,π ) is not an absolutely contin-uous function. Moreover, Camelier and Guttierez example shows that conjugacybetween f and T ( λ,π ) does not always exist.In this paper, we prove the following result: Theorem 1.
Let T ( λ,π ) be a self-similar interval exchange transformation and R the associated matrix obtained by Rauzy induction. Let θ be the Perron-Frobeniuseigenvalue of R . Assume that R has an eigenvalue θ such that (1) θ is a conjugate of θ , (2) θ is a real number, (3) 1 < θ ( < θ ) .Then there exists an affine interval exchange transformation f with wandering in-tervals that is semi-conjugated to T ( λ,π ) . This result means that Denjoy counterexamples occur very often (see section 5).1.1.
Reader’s guide.
Camelier-Gutierrez [CG] and Cobo [C] developed an strat-egy to prove the existence of a wandering interval in an affine interval exchangetransformation f which is semi-conjugated with a given IET. We explain it in sec-tion 4. This strategy allowed them to achieve a first concrete example. Here weexplore the limits of this method in order to consider a large (and in some senseabstract) family of IET. Let T ( λ,π ) be a self-similar interval exchange transforma-tion with associated matrix R . Let γ = ( γ , . . . , γ r ) be the vector of the logarithmof the slopes of the affine interval exchange transformation f . If f admits a wan-dering interval I , the length | f ( I ) | is equal to e γ j | I | if I is contained in interval j . Roughly speaking, to create a wandering interval from the interval exchangetransformation T ( λ,π ) , one blows up an orbit of T ( λ,π ) . The difficulty is to insurethat the total length remains finite. More precisely, if the symbolic coding of theorbit is x = ( x n ) n ∈ Z , we have to check that the series(1.1) X n ≥ e − γ ( x ) − ... − γ ( x n − ) and X n ≥ e γ ( x − n )+ ... + γ ( x − ) converge. This is certainly not true for a generic point x of the symbolic systemassociated to T ( λ,π ) . Let ℓ ( x ) be the broken line with vertices ( n, γ ( x ) + . . . + γ ( x n − )) n ∈ N and ( n, γ ( x − n ) + . . . + γ ( x − )) n ≥ . Since γ is orthogonal to λ , for ageneric point x , the line ℓ ( x ) oscillates around 0 as predicted by H´alasz’s Theorem([Ha]). If the vector γ is not in the contracting space of t R the amplitude of theoscillations tends to infinity with speed n log( θ ) / log( θ ) . It is hoped that the series (1.1) converge if the y -coordinate of the broken line ℓ ( x )is always positive and tends to infinity fast enough as n tends to ±∞ . Points withthis property are called minimal points . Those are the main tool of the paper.This analysis applies to a very large class of substitutions and not only to substitu-tions arising from interval exchange transformations. Section 3 gives an algorithmto construct minimal points. We prove that the prefix-suffix decomposition of anyminimal point is ultimately periodic. From this analysis, we deduce that for anyminimal point x one has(1.2) lim inf n →∞ γ ( x ) + . . . + γ ( x n ) n log( θ θ > n →∞ − γ ( x − n ) − . . . − γ ( x − ) n log( θ θ > ersistence of wandering intervals in self-similar AIET 3 Formulas in (1.2) imply immediately the convergence of the series in (1.1). More-over, formulas in (1.2) has its own interest. It is a strengthening of a result byAdamczewski [Ad] about discrepancy of substitutive systems.Even if the fractal curves studied by Dumont and Thomas in [DT1], [DT2] are notconsidered explicitly in the article, they were a source of inspiration for the authors.These curves correspond to the renormalization of the broken lines ℓ ( x ) and appearin subsection 3.3 in another language.In section 5, we discuss the hypothesis of the main result in a geometric language.We exhibit many examples where our hypothesis on the matrix R are fulfilled.2. Preliminaries
Words and sequences.
Let A be a finite set. One calls it an alphabet andits elements symbols . A word is a finite sequence of symbols in A , w = w . . . w ℓ − .The length of w is denoted | w | = ℓ . One also defines the empty word ε . Theset of words in the alphabet A is denoted A ∗ and A + = A ∗ \ { ε } . We will needto consider words indexed by integer numbers, that is, w = w − m . . . w − .w . . . w ℓ where ℓ, m ∈ N and the dot separates negative and non-negative coordinates. Ifnecessary we call them dotted words .The set of one-sided infinite sequences x = ( x i ) i ∈ N in A is denoted by A N . Analo-gously, A Z is the set of two-sided infinite sequences x = ( x i ) i ∈ Z .Given a sequence x in A + , A N or A Z one denotes x [ i, j ] the sub-word of x appearingbetween indexes i and j . Similarly one defines x ( −∞ , i ] and x [ i, ∞ ). Let w = w − m . . . w − .w . . . w ℓ be a (dotted) word in A . One defines the cylinder set [ w ] as { x ∈ A Z : x [ − m, ℓ ] = w } .The shift map T : A Z → A Z or T : A N → A N is given by T ( x ) = ( x i +1 ) i ∈ N for x = ( x i ) i ∈ N . A subshift is any shift invariant and closed (for the product topology)subset of A Z or A N . A subshift is minimal if all of its orbits by the shift are dense.In what follows we will use the shift map in several contexts, in particular restrictedto a subshift. To simplify notations we keep the name T all the time.2.2. Substitutions and minimal points.
We refer to [Qu] and [F] and referencestherein for the general theory of substitutions.A substitution is a map σ : A → A + . It naturally extends to A + , A N and A Z ; for x = ( x i ) i ∈ Z ∈ A Z the extension is given by σ ( x ) = . . . σ ( x − ) σ ( x − ) .σ ( x ) σ ( x ) . . . where the central dot separates negative and non-negative coordinates of x . Afurther natural convention is that the image of the empty word ε is ε .Let M be the matrix with indices in A such that M ab is the number of times letter b appears in σ ( a ) for any a, b ∈ A . The substitution is primitive if there is N > a ∈ A , σ N ( a ) contains any other letter of A (here σ N means N consecutive iterations of σ ). Under primitivity one can assume without loss ofgenerality that M > X σ ⊆ A Z be the subshift defined from σ . That is, x ∈ X σ if and only if anysubword of x is a subword of σ N ( a ) for some N ∈ N and a ∈ A .Assume σ is primitive. Given a point x ∈ X σ there exists a unique sequence( p i , c i , s i ) i ∈ N ∈ ( A ∗ × A × A ∗ ) N such that for each i ∈ N : σ ( c i +1 ) = p i c i s i and . . . σ ( p ) σ ( p ) σ ( p ) p .c s σ ( s ) σ ( s ) σ ( s ) . . . Xavier Bressaud, Pascal Hubert, Alejandro Maass is the central part of x , where the dot separates negative and non-negative coordi-nates. This sequence is called the prefix-suffix decomposition of x (see for instance[CS]).If only finitely many suffixes s i are nonempty, then there exists a ∈ A and non-negative integers ℓ and q such that x [0 , ∞ ) = c s σ ( s ) . . . σ ℓ ( s ℓ ) lim n →∞ σ nq ( a )Analogously, if only finitely many p i are non empty, then x ( −∞ , −
1] = lim n →∞ σ np ( b ) σ m ( p m ) . . . σ ( p ) p for some b ∈ A and non-negative integers p and m .Let θ be the Perron-Frobenius eigenvalue of M . Let λ = ( λ ( a ) : a ∈ A ) t be astrictly positive right eigenvector of M associated to θ . We will also assume thefollowing algebraic property that we call (AH): M has an eigenvalue θ which is aconjugate of θ . Notice that this property coincides with hypothesis (1) of Theorem1.The following lemma are important consequences of the algebraic property (AH). Lemma 2.
Let η : Q [ θ ] → Q [ θ ] be the field homomorphism that sends θ to θ .The vector γ = η ( λ ) = ( η ( λ ( a )) : a ∈ A ) t is an eigenvector of M associated to θ .Proof. The field homomorphism η naturally extends to Q [ θ ] | A | . Since λ belongsto Q [ θ ] | A | (up to normalization), then one deduces that M η ( λ ) = θ η ( λ ). Thus, η ( λ ) is an eigenvector of M associated to θ . (cid:3) Lemma 3.
Let γ be the eigenvector of M associated to θ as in Lemma 2. Thenfor any | A | -tuple of non-negative integers ( n a : a ∈ A ) , P a ∈ A n a γ ( a ) = 0 implies n a = 0 for any a ∈ A .Proof. Assume P a ∈ A n a γ ( a ) = 0. Since γ = η ( λ ), applying η − one gets that P a ∈ A n a λ ( a ) = 0. This equality implies that n a = 0 for every a ∈ A because thecoordinates of λ are positive. (cid:3) Let γ = η ( λ ) as in Lemma 2. For w = w . . . w l − ∈ A + denote γ ( w ) = γ ( w ) + . . . + γ ( w l − ).Let x ∈ X σ . Define γ ( x ) = 0, γ n ( x ) = P n − i =0 γ ( x i ) for n > γ n ( x ) = P − i = n γ ( x i ) for n <
0. Put Γ( x ) = { γ n ( x ) : n ∈ Z } . In a similar way, given a(dotted) word w = w − m . . . w . . . w l − one defines γ ( w ) = 0, γ n ( w ) = P n − i =0 γ ( w i )for 0 < n ≤ l , γ n ( w ) = P − i = n γ ( w i ) for − m ≤ n < w ).The best occurrence of a symbol a ∈ A in w is − m ≤ i < l such that w i = a and γ i +1 ( w ) = min { γ j +1 ( w ) : − m ≤ j < l, w j = a } . By Lemma 3, under hypotheses(AH) this number is well defined and unique.One says x is minimal if γ n ( x ) ≥ n ∈ Z . The set of minimal points for σ is denoted by M ( σ ). It is important to mention that if x is a minimal point ofa substitution satisfying hypothesis (AH) then, by Lemma 3, γ n ( x ) > n = 0.2.3. Affine interval exchange transformations.