Persistence of Wandering Intervals in Self-Similar Affine Interval Exchange Transformations

PERSISTENCE OF W ANDERING INTER V ALS IN SELF-SIMILAR AFFINE INTER V AL EX CHANGE TRANSFORMA TIONS XA VIER BRESSAUD, P ASCAL HUB E R T, AND ALEJANDRO MAASS Abstract. In this article we pro ve that given a s elf-simi lar in terv al exc hange transformation T ( λ,π ) , whose associated matrix verifies a quite general alge- braic condition, there exists an affine interv al exc hange transformation with wa ndering i n terv als that is semi-conjugated to it. That is, in this contex t the existence of Denjoy coun terexamples occurs very often, generalizing the result of M. C ob o i n [C]. 1. Introduction Since the w or k of Denjoy [D] it is known that e very C 1 -diffeomorphism o f the circle such tha t the logar ithm o f its deriv ative is a function of b ounded v ariation has no w andering interv als. There is no analo gous result for in terv al exchange transformatio ns. Levitt in [L] found an ex ample o f a non-uniquely e rgo dic a ffine int erv al exc hange transforma tion with wandering interv a ls. Latter, Camelier a nd Gutierrez [CG], using Rauzy induction technique exhibited a uniquely ergodic affine int erv al exchange trans formation with wandering interv als. Moreover, this example is semi-conjuga ted to a self-similar interv al exchange tr ansformatio n. In geometric language, it means tha t this int er- v a l exchange tra ns formation is induced b y a pseudo-Anosov diffeomorphism. In combinatorial terms, the sym bo lic system is generated by a substitution An interv al exc hange transfo r mation (IET) is defined by the length of the in terv als λ = ( λ 1 , . . . , λ r ) a nd a p ermutation π . It is deno ted by T ( λ,π ) . T o define an a ffine int erv al exc hange transfor mation (AIET) one additional information is nee de d; the slop e of the map on eac h interv a l. This is a vector ( w 1 , . . . , w r ) with w i > 0 for i = 1 , . . . , r . Ca melier and Gutierrez r emarked that a neces sary condition for an AIET to b e conjugated to the interv al exchange transfor mation T ( λ,π ) is that the vector log( w ) = (log( w 1 ) , . . . , log( w r )) is orthogo nal to λ . The co njugacy of an affine in terv al exc hange tra nsformation with an in terv al ex- change transforma tion w as studied in details by Cobo [C]. He proved that the regular ity of the conjugacy depends on the positio n of the vector lo g( w ) in the flag o f the lyapuno v exp onents of the Rauzy-V eech-Zoric h induction. In pa rticular, assume that T ( λ,π ) is self-simila r, which mea ns that λ is an eigenv ector of a p ositive r × r matrix R obtained by a pplying Rauzy induction a finite n um ber of times. Cob o proves that if log ( w ) b elongs to the contracting spac e of t R then f is C 1 conjugated to T ( λ,π ) . If lo g( w ) is orthogonal to λ and is not in the con tracting Date : July 7, 2007. 1991 Mathematics Subje ct Class ific ation. Primary: 37C15; Secondary: 37B10. Key wor ds and phr ases. int erv al exc hange transformations, substitutiv e systems, w andering sets. 1 2 Xavier Bressaud, Pascal Hub ert, Alejand ro Maass space of t R then any conjugacy b etw een f and T ( λ,π ) is not an a bs olutely contin- uous function. Moreov er, Camelie r and Guttierez exa mple shows that conjugacy betw een f and T ( λ,π ) do es not alwa ys exist. In this pap er , we prov e the following result: Theorem 1. L et T ( λ,π ) b e a self-similar interval exchange tr ansformation and R the asso ciate d matrix obtaine d by R auzy induction. L et θ 1 b e t he Perr on-F r ob enius eigenvalue of R . Assume t hat R has an eigenvalue θ 2 such that (1) θ 2 is a c onjugate of θ 1 , (2) θ 2 is a r e al n umb er, (3) 1 < θ 2 ( < θ 1 ) . Then ther e exists an affine int erval exchange tr ansformation f with wandering in- tervals that is semi-c onjugate d to T ( λ,π ) . This result means that Denjo y counterexamples o ccur very often (see section 5). 1.1. Reader’s guide. Ca melie r -Gutierrez [CG] and Cob o [C] develop ed an stra t- egy to prov e the existence of a wandering in ter v al in an affine interv al exchange transformatio n f which is semi-conjugated with a given IET. W e expla in it in sec- tion 4. This strategy allow ed them to a chiev e a first concrete example. Here we explore the limits of this method in order to consider a large (and in some sense abstract) family of IET. Let T ( λ,π ) be a self-similar interv al ex change transforma - tion with asso cia ted matrix R . Let γ = ( γ 1 , . . . , γ r ) b e the vector of the logar ithm of the slop es o f the affine interv al exchange trans formation f . If f a dmits a wan- dering int erv al I , the length | f ( I ) | is equal to e γ j | I | if I is contained in in terv al j . Roughly spea king, to cr eate a w andering interv al from the in terv al exc hange transformatio n T ( λ,π ) , o ne blo ws up an orbit of T ( λ,π ) . The difficulty is to ins ur e that the total leng th r emains finite. More precisely , if the symbo lic co ding of the orbit is x = ( x n ) n ∈ Z , we hav e to chec k tha t the s eries (1.1) X n ≥ 1 e − γ ( x 0 ) − ... − γ ( x n − 1 ) and X n ≥ 1 e γ ( x − n )+ ... + γ ( x − 1 ) conv e r ge. This is certainly no t true for a ge ne r ic p oint x of the symbolic sys tem asso ciated to T ( λ,π ) . Let ℓ ( x ) b e the broken line with vertices ( n, γ ( x 0 ) + . . . + γ ( x n − 1 )) n ∈ N and ( n, γ ( x − n ) + . . . + γ ( x − 1 )) n ≥ 1 . Since γ is orthog o nal to λ , for a generic point x , the line ℓ ( x ) oscilla tes around 0 as predicted by H´ alasz’s Theorem ([Ha]). If the v ector γ is not in the c o ntracting spac e of t R the amplitude o f the oscillations tends to infinity with sp eed n log( θ 1 ) / log( θ 2 ) . It is hop ed that the series (1.1) c o nv er ge if the y -co or dinate of the broken line ℓ ( x ) is alwa ys p ositive and tends to infinit y fast enough as n tends to ±∞ . Poin ts with this prop erty are called m inimal p oints . Those ar e the main to o l of the pap er. This ana lysis applies to a v ery lar g e class o f substitutions and no t o nly to substitu- tions arising from in terv al ex change tra nsformations. Section 3 gives an alg orithm to construct minimal p oints. W e prove that the prefix-suffix decomp osition of any minimal p oint is ultimately p erio dic. F r om this a nalysis, we deduce that for any minimal p oint x one ha s (1.2) lim inf n →∞ γ ( x 0 ) + . . . + γ ( x n ) n log( θ 2 ) log( θ 1 ) > 0 and lim inf n →∞ − γ ( x − n ) − . . . − γ ( x − 1 ) n log( θ 2 ) log( θ 1 ) > 0 Persistence of wandering interv als in self -similar AIET 3 F ormulas in (1.2) imply immediately the co nv er g ence of the s eries in (1.1). More- ov er , for m ulas in (1.2) has its own int erest. It is a s trengthening of a result by Adamczewski [Ad] a b out discrepancy of substitutive systems. Even if the fractal curv es studied b y Dumon t and Thomas in [DT1], [DT2] are not considered explic itly in the article, they were a sourc e of inspir ation for the autho rs. These curves cor r esp ond to the renorma lization of the broken lines ℓ ( x ) and app ear in subsection 3.3 in ano ther languag e. In section 5, we discuss the hyp o thesis of the main r e sult in a geometr ic language. W e exhibit many examples where o ur hypo thesis o n the matrix R ar e fulfilled. 2. Preliminaries 2.1. W ords and sequences. Let A b e a finite set. One calls it an alphab et a nd its elemen ts symb ols . A wor d is a finite sequence of symbols in A , w = w 0 . . . w ℓ − 1 . The length of w is denoted | w | = ℓ . One also defines the empt y word ε . The set o f words in the alphab et A is denoted A ∗ and A + = A ∗ \ { ε } . W e will need to consider words indexed by integer num ber s , that is, w = w − m . . . w − 1 .w 0 . . . w ℓ where ℓ, m ∈ N a nd the dot separ ates nega tive and non-negative co ordinates . If necessary we call them dotte d wor ds . The set of o ne-sided infinite seq uences x = ( x i ) i ∈ N in A is denoted by A N . Analo - gously , A Z is the set of t wo-sided infinit e seq uences x = ( x i ) i ∈ Z . Given a seq ue nc e x in A + , A N or A Z one deno tes x [ i, j ] the sub-word o f x app ea ring betw een indexes i and j . Similarly one defines x ( −∞ , i ] and x [ i, ∞ ). Let w = w − m . . . w − 1 .w 0 . . . w ℓ be a (dotted) word in A . O ne 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 in v ariant a nd c lo sed (for the pro duct to p o logy) subset of A Z or A N . A subshift is minimal if all of its orbits b y the shift are dense. In what follows w e will use the shift map in several contexts, in particular restricted to a subshift. T o s implify notations we k ee p the name T all the time. 2.2. Substitutions and minimal p oi n ts. W e refer to [Q u] and [F] and refer e nces therein for the genera l theor y of substitutions. A substitu tion is a ma p σ : A → A + . It naturally extends to A + , A N and A Z ; fo r x = ( x i ) i ∈ Z ∈ A Z the extension is given b y σ ( x ) = . . . σ ( x − 2 ) σ ( x − 1 ) .σ ( x 0 ) σ ( x 1 ) . . . where the central dot s eparates negative a nd non-negative co or dina tes of x . A further natural c onv ention is that the ima ge of the empty w ord ε is ε . Let M b e the matrix with indices in A s uch that M ab is the num ber of times letter b app ears in σ ( a ) for any a, b ∈ A . The substitution is primitive if there is N > 0 such tha t for any a ∈ A , σ N ( a ) contains any other letter of A (here σ N means N consec utive itera tio ns of σ ). Under pr imitivit y one can a ssume without loss of generality that M > 0. Let X σ ⊆ A Z be the subshift defined from σ . Tha t is, x ∈ X σ if and only if any subw or d of x is a subw o rd of σ N ( a ) for s o me N ∈ N and a ∈ A . Assume σ is primitive. Given a po int 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 3 ( p 3 ) σ 2 ( p 2 ) σ 1 ( p 1 ) p 0 .c 0 s 0 σ 1 ( s 1 ) σ 2 ( s 2 ) σ 3 ( s 3 ) . . . 4 Xavier Bressaud , Pascal Hub e rt, Alejand ro Maass is the central part of x , where the do t separates negative and non-negative c o ordi- nates. This sequence is called the prefix-s uffix dec o mpo sition of x (see for instance [CS]). If only finitely many suffixes s i are nonempt y , then there exists a ∈ A and no n- negative in teger s ℓ and q such that x [0 , ∞ ) = c 0 s 0 σ 1 ( s 1 ) . . . 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 1 ( p 1 ) p 0 for some b ∈ A and non-neg ative integers p a nd m . Let θ 1 be the P erron-F r ob enius eigen v alue of M . Let λ = ( λ ( a ) : a ∈ A ) t be a strictly p os itive right eigenv ector of M asso ciated to θ 1 . W e will also assume the following algebraic prop erty that we call (AH): M has an eigenv alue θ 2 which is a conjugate of θ 1 . Notice that this prop erty coincides with hyp o thesis (1) of Theorem 1. The following lemma are imp orta nt co nsequences of the algebr aic prop erty (A H). Lemma 2. L et η : Q [ θ 1 ] → Q [ θ 2 ] b e the field homomo rphism that sends θ 1 to θ 2 . The ve ctor γ = η ( λ ) = ( η ( λ ( a )) : a ∈ A ) t is an eigenve ctor of M asso ciate d to θ 2 . Pr o of. The field homomorphism η natura lly extends to Q [ θ 1 ] | A | . Since λ be lo ngs to Q [ θ 1 ] | A | (up to normaliza tio n), then one deduces that M η ( λ ) = θ 2 η ( λ ). Thus, η ( λ ) is an eigenv ector of M asso ciated to θ 2 .  Lemma 3. L et γ b e the eigenve ctor of M asso ciate d to θ 2 as in L emma 2 . Then for any | A | -tuple of non-ne gative inte gers ( n a : a ∈ A ) , P a ∈ A n a γ ( a ) = 0 impl ies n a = 0 for any a ∈ A . Pr o of. Assume P a ∈ A n a γ ( a ) = 0. Since γ = η ( λ ), applying η − 1 one gets that P a ∈ A n a λ ( a ) = 0. This equality implies that n a = 0 for every a ∈ A b ecause the co ordinates of λ are p ositive.  Let γ = η ( λ ) as in Le mma 2. F or w = w 0 . . . w l − 1 ∈ A + denote γ ( w ) = γ ( w 0 ) + . . . + γ ( w l − 1 ). Let x ∈ X σ . Define γ 0 ( x ) = 0, γ n ( x ) = P n − 1 i =0 γ ( x i ) for n > 0 and γ n ( x ) = P − 1 i = n γ ( x i ) for n < 0. Put Γ( x ) = { γ n ( x ) : n ∈ Z } . In a simila r wa y , g iven a (dotted) word w = w − m . . . w 0 . . . w l − 1 one defines γ 0 ( w ) = 0, γ n ( w ) = P n − 1 i =0 γ ( w i ) for 0 < n ≤ l , γ n ( w ) = P − 1 i = n γ ( w i ) for − m ≤ n < 0 and the set Γ( w ). The b est o ccurrence o f a symbol a ∈ A in w is − m ≤ i < l such that w i = a a nd γ i +1 ( w ) = min { γ j +1 ( w ) : − m ≤ j < l , w j = a } . By Lemma 3 , under hypotheses (AH) this num ber is well defined and unique. One says x is minimal if γ n ( x ) ≥ 0 fo r any n ∈ Z . The set of minimal p oints fo r σ is denoted by M ( σ ). It is imp ortant to mention that if x is a minimal p o in t o f a substitution sa tisfying hypothesis (AH) then, b y Lemma 3, γ n ( x ) > 0 whenever n 6 = 0 . 2.3. Affine interv al exc hange transformations. Let 0 = a 0 < a 1 < . . . < a r − 1 < a r = 1 and A = { 1 , . . . , r } . An affine interval ex change tr ansformation (AIET) is a bijective ma p f : [0 , 1 ) → [0 , 1) of the form f ( t ) = w i t + v i if t ∈ [ a i − 1 , a i ) for i ∈ A . The vector w = Persistence of wandering interv als in self -similar AIET 5 ( w 1 , . . . , w r ) is calle d the slope of f . W e assume furthermo r e the slop e is s trictly po sitive. An interval ex change t r ansformation (IET) is an AIET with slo p e w = (1 , . . . , 1). Commonly an IET is given b y a vector λ = ( λ 1 , . . . , λ r ) such that λ i = | a i − a i − 1 | for i ∈ A and a p er mutation π of A w hich indicates the w ay interv als [ a i − 1 , a i )’s are r earra nged by the IET. Clearly , a i = P i j =1 λ j . W e use T ( λ,π ) to r efer to the IET asso ciated to λ and π . One says the AIE T f is s emi-conjugated with the IET T ( λ,π ) if there is a monotonic, surjective and contin uous ma p h : [0 , 1) → [0 , 1) such that h ◦ f = T ( λ,π ) ◦ h . Let T ( λ,π ) be a n interv al exc hange tr ansformation. There is a natural sym bo lic co ding of the orbit of an y p oint t ∈ [0 , 1) by T ( λ,π ) . C o nsider the pa rtition α = { [0 , a 1 ) , . . . , [ a i − 1 , a i ) , . . . , [ a r − 1 , 1) } and define φ ( t ) = ( x i ) i ∈ Z ∈ A Z by t i = j if and only if T i ( λ,π ) ( t ) ∈ [ a j − 1 , a j ). The set φ ([0 , 1)) is in v ariant for the shift but it is not necessarily clo sed, then one considers its clo sure X = φ ([0 , 1)). This pro cedure pro duces a semi-conjugacy (factor map) ϕ : ( X , T ) → ([0 , 1) , T ( λ,π ) ). If t is not in the orbit of the extreme p oints 0 , a 1 , . . . , 1, then it has a unique preimage by ϕ . If not, it has at most tw o preimages cor r esp onding to the co ding of ( T i ( λ,π ) (lim s → t − s )) i ∈ Z . W e use fr eely co ncepts re la ted to Rauzy-Zo rich-V eech induction. Rauzy induction was defined in [Ra], extended to zippere d rectangles by V eech [V e], a nd accelerated by Z orich [Zo]. F or a complete description ab out the Rauzy-V eech-Zorich induction see also the exp ository pap ers by Zorich [Z o2] and Y o ccoz [Y o]. An IET T ( λ,π ) is self-similar if it can b e recov ered fr o m itself after finitely man y steps of Ra uz y inductions (up to normalizatio n). More precise ly , there exists a lo o p in the Rauzy diagr am and an asso c ia ted Perr on-F r ob enius matrix R such that θ 1 λ = Rλ with θ 1 the dominant eigenv alue o f R . F or a self-similar IET T ( λ,π ) there is a direct relation betw een the subshift X and the matrix R asso ciated to T ( λ,π ) . Indeed, there exists a substitution σ : A → A + with asso ciated matr ix M = t R suc h that X σ = X (s e e [CG] and r eferences ther ein). If the IET T ( λ,π ) is minimal then the subshift X σ is minimal to o. In the sequel, we will use the fact that the substitution σ is primitive which implies that X σ is minimal. Nevertheless, no specific prop erty of substitutions obtained fr o m T ( λ,π ) will b e needed for our purp ose. The relation betw ee n self-similar IET and pseudo -Anosov diffeomorphisms is ex- plained in [V e ]. 3. Construction of minimal points Let σ : A → A + be a primitive substitution with asso ciated matrix M > 0. Let θ 1 , θ 2 , λ and γ be as in subsectio n 2.2. In addition, a ssume θ 2 verifies the hypotheses of Theorem 1. B y Perron-F ro be nius theo r em, γ has nega tive and p ositive co ordinates . The main ob jective of the section is to give a co mbinatorial constructio n of minimal po int s in this case. 6 Xavier Bressaud , Pascal Hub e rt, Alejand ro Maass 3.1. Existence of mi nimal p oints. Lemma 4. L et a ∈ A such that γ ( a ) > 0 and n ∈ N . Write σ n ( a ) = p n s n wher e the minimum of Γ( σ n ( a )) is attaine d at γ i ( σ n ( a )) and i = | p n | . Then γ ( s n ) ≥ θ n 2 γ ( a ) . In p articular | s n | gr ows ex p onential ly fast with n . Pr o of. Observe that γ ( p n ) + γ ( s n ) = θ n 2 γ ( a ) and γ ( p n ) ≤ 0.  Lemma 5. M ( σ ) 6 = ∅ Pr o of. Since γ ha s p os itive and neg ative co ordinates and X σ is minimal, then there exist b, c ∈ A suc h that bc is a subw ord o f a point in X σ and γ ( b ) < 0 , γ ( c ) > 0 holds. Let n ≥ 0 a nd define u n = σ n ( b ) .σ n ( c ). The s equence Γ( u n ) attains its minimum at some N n ∈ {−| σ n ( b ) | , . . . , − 1 , 0 , . . . , | σ n ( c ) |} . Define the (dotted)w ord v n = u n [ −| σ n ( b ) | , N n − 1] .u n [ N n , | σ n ( c ) | − 1] = v − n .v + n . The minimum of Γ( v n ) is attained at co ordina te 0, and is equal to 0 . By Lemma 4 ther e is a subsequence ( n i ) i ∈ N such that lim i →∞ | v − n i | = lim i →∞ | v + n i | = ∞ By co mpactness a nd even tually taking once a gain a subse q uence there exists x ∈ X such that for an y m ∈ N there is i ∈ N with n i ≥ m and x ∈ [ v − n i .v + n i ]. Thu s Γ( x [ − m, m ]) ⊆ R + and its minimum is zero at zero co o r dinate. This implies x ∈ M ( σ ).  3.2. The b est str ate gy algori thm. In wha t follows we develop a pro cedure to c o n- struct minimal po int s that will b ecome useful in next subsections. The following t wo lemma follow directly from equality M γ = θ 2 γ . Their simple pro ofs are left to the reader . Lemma 6. L et m ∈ N and w ∈ A + . Then γ ( σ m ( w )) = θ m 2 γ ( w ) . Lemma 7. L et w = w 0 . . . w l − 1 ∈ A + . Write σ ( w ) = σ ( w 0 ) . . . σ ( w l − 1 ) . The minimum of Γ( σ ( w )) is attaine d in a c o or dinate c orr esp onding to s ome σ ( w i ) , wher e w i is the b est o c curr enc e of this symb ol in w . 3.2.1. The b asic pr o c e dur e. The following pro c e dure will allow to construct the prefix-suffix decomp osition o f a minimal p oint. Step 0: F or each a ∈ A write σ ( a ) = p a, 0 0 c a, 0 0 s a, 0 0 where Γ( σ ( a )) attains its minimum at γ | p 0 ( a ) | ( σ ( a )). Step 1: Let a ∈ A . By Lemma 7, the minimum of Γ( σ 2 ( a )) co mes from σ ( b ) for some b ∈ A in its b est o ccurre nc e in σ ( a ). W r ite σ ( a ) = p a, 1 1 c a, 1 1 s a, 1 1 where c a, 1 1 = b is the b est o ccurrence of b in σ ( a ). P ut w 1 ( a ) = σ ( p a, 1 1 ) p b, 0 0 .c b, 0 0 s b, 0 0 σ ( s a, 1 1 ), where the dot s e parates nega tive and non-negative co ordina tes . Let p a, 1 0 = p b, 0 0 , c a, 1 0 = c b, 0 0 and s a, 1 0 = s b, 0 0 . The sequence ( p a, 1 i , c a, 1 i , s a, 1 i ) 1 i =0 is called the b est strateg y for symbol a at step 1. By co ns truction Γ( w 1 ( a )) ⊆ R + and the minimum is equal to zero at c o ordinate zero. Step n+1: assume in pr e vious step we hav e cons tructed for each s ymbol a ∈ A the b est strategy ( p a,n i , c a,n i , s a,n i ) n i =0 . This sequence verifies: (i) for 0 ≤ i ≤ n , σ ( c a,n i +1 ) = p a,n i c a,n i s a,n i (here c a,n n +1 = a ). Moreov er, each c a,n i is the bes t o ccurr ence of this symbo l in σ ( c a,n i +1 ). Persistence of wandering interv als in self -similar AIET 7 (ii) Γ( w n ( a )) ⊆ R + and its minimum is ze r o at zer o co ordinate, wher e w n ( a ) = σ n +1 ( a ) = σ n ( p a,n n ) . . . σ ( p a,n 1 ) p a,n 0 .c a,n 0 s a,n 0 σ ( s a,n 1 ) . . . σ n ( s a,n n ) Now we pro c eed as in step 1. Consider a ∈ A . By Lemma 7, the minimum of Γ( σ n +2 ( a )) co mes from σ n +1 ( b ) for some b ∈ A in its b est o ccur rence in σ ( a ). W r ite σ ( a ) = p a,n +1 n +1 c a,n +1 n +1 s a,n +1 n +1 where c a,n +1 n +1 = b is the b e st o ccur r ence of b in σ ( a ). The finite sequence ( p a,n +1 i , c a,n +1 i , s a,n +1 i ) n +1 i =0 where ( p a,n +1 i , c a,n +1 i , s a,n +1 i ) = ( p b,n i , c b,n i , s b,n i ) for 0 ≤ i ≤ n is a b est stra tegy for a at step n + 1 and verifies conditions (i) a nd (ii) by construction. 3.2.2. Finitely many minimal p oints. F or each a ∈ A and n ∈ N consider the cylinder set C a,n = [ w n ( a )], where w n ( a ) is the do tted word defined in previous subsection. It is clea r from the basic pro cedure tha t for an y a ∈ A and n ∈ N there exists a unique b ∈ A such that C a,n +1 ⊆ C b,n . Thus, by compactness, ther e exis t at most | A | infinite decreasing sequences of the form ( C a n ,n ) n ∈ N . Let C 1 , . . . , C ℓ with ℓ ≤ | A | be the collection of intersections of suc h sequence s. Remark that suc h sets are finite. Given a minimal p oint x ∈ X with prefix- suffix decomp o sition ( p i , c i , s i ) i ∈ N and n ∈ N , there is a n ∈ A such that ( p i , c i , s i ) = ( p a n ,n i , c a n ,n i , s a n ,n i ) for 0 ≤ i ≤ n . Therefore, x ∈ C i = T n ∈ N C a n ,n for some 1 ≤ i ≤ ℓ . The following prop osition is pla in. Prop ositi o n 8. Ther e ar e fin itely many minimal p oints. W e will see later tha t minimal p oints hav e ultimately perio dic pr efix-suffixe decom- po sition. This fact yields to an alterna tive pro o f of prev ious prop osition. 3.3. Serie asso ciated to a minim al p oi n t. Define S = { ( p i , c i , s i ) i ∈ N : ∀ i > 0 , σ ( c i ) = p i − 1 c i − 1 s i − 1 } and S = { ( p i , c i , s i ) i ∈ N : ∀ i ≥ 0 , σ ( c i ) = p i +1 c i +1 s i +1 } . Observe that finite sequences taken from sequences in S and S coincide once re- versed. Let a ∈ A a nd n ≥ 1. Then σ n ( a ) ca n b e deco mp os ed as σ n ( a ) = σ n − 1 ( p 1 ) . . . σ ( p n − 1 ) p n c n s n σ ( s n − 1 ) . . . σ n − 1 ( s 1 ) where for all 1 ≤ i ≤ n , σ ( c i − 1 ) = p i c i s i (w e ha ve co ns idered c 0 = a ). This de c om- po sition is not unique. T o a and the finite sequence ( p i , c i , s i ) n i =1 one as so ciates the finite sum: v ( a ; ( p i , c i , s i ) n i =1 ) = n X i =1 θ − i 2 γ ( p i ) Clearly , given x = ( p x i , c x i , s x i ) i ∈ N ∈ S with c x 0 = a , the s e ries v ( a ; x ) = lim n →∞ v ( a ; ( p x i , c x i , s x i ) n i =1 ) = X i ≥ 1 θ − i 2 γ ( p x i ) exists. Let v ( a ) = min { v ( a ; x ) : x ∈ S with c x 0 = a } . A sequence x ∈ S with c x 0 = a s uch that v ( a ; x ) = v ( a ) is said to b e m inimal for a . The b est strategy for sy m bo l a at step n ≥ 1 given by the a lgorithm pro duces a finite sequence ( p a,n i , c a,n i , s a,n i ) n i =0 . Set v n ( a ) = P n i =0 θ − n + i − 1 γ ( p a,n i ). It follows that v n ( a ) = v ( a ; ( p a,n n − i , c a,n n − i , s a,n n − i ) n i =0 ). 8 Xavier Bressaud , Pascal Hub e rt, Alejand ro Maass Lemma 9. F or every a ∈ A and n ≥ 1 , v n ( a ) is minimal among the v ( a ; ( p i , c i , s i ) n +1 i =1 ) and v ( a ) = lim n →∞ v n ( a ) . Pr o of. The firs t fact is a nalogous to say that ( p a,n i , c a,n i , s a,n i ) n i =0 is the b est str a tegy . Moreov er, | v n ( a ) − v ( a ) | ≤ K θ − n 2 for some co nstant K > 0. This implies the desired result.  Lemma 10. L et a ∈ A . Assume ther e is a finite s e quenc e ( ¯ p j , ¯ c j , ¯ s j ) l j =1 such that for infinitely many n ∈ N , ( p a,n n − j +1 , c a,n n − j +1 , s a,n n − j +1 ) l j =1 = ( ¯ p j , ¯ c j , ¯ s j ) l j =1 . Then, ther e exists y = ( p y i , c y i , s y i ) i ∈ N ∈ S su ch that ( y j ) l j =1 = ( ¯ p j , ¯ c j , ¯ s j ) l j =1 , c y 0 = a and v ( a ) = v ( c y 0 ; y ) . Pr o of. F o r any n ∈ N wher e the prop erty of the lemma ho lds consider the p oint y ( n ) = y ( n ) 0 . . . y ( n ) n = ( p, a, s )( p a,n n , c a,n n , s a,n n ) . . . ( p a,n 0 , c a,n 0 , s a,n 0 ) where σ ( b ) = pa s for some b ∈ A . Let y = ( p y i , c y i , s y i ) i ∈ N be the limit of a subse quence ( y ( n i ) ) i ∈ N . It follows by construction that ( y j ) l j =1 = ( ¯ p j , ¯ c j , ¯ s j ) l j =1 , c y 0 = a and σ ( c y i ) = p y i +1 c y i +1 s y i +1 for any i ≥ 0. Also, c y i +1 is the b est o ccurre nce of this sy m b o l in σ ( c y i ). Let ǫ > 0 and i 0 ∈ N such that | v ( a ) − v n i ( a ) | ≤ ǫ/ 2 for i ≥ i 0 . Let L ∈ N be such that θ − L 2 ≤ ǫ/ 4 C where C > 0 is s uch that | γ ( p c,n i ) | / ( θ 2 − 1) ≤ C for any c ∈ A and n ∈ N . Thus for i enough la rge, ( p y j , c y j , s y j ) = ( p n i − j +1 , c n i − j +1 , s n i − j +1 ) for 0 ≤ j < L and | v ( a ) − P i ≥ 1 θ − i 2 γ ( p y i ) | ≤ ǫ . Since ǫ is a rbitrary one concludes v ( c y 0 ) = v ( a ) = P i ≥ 1 θ − i 2 γ ( p y i ).  One says that a p oint y = ( p y i , c y i , s y i ) i ∈ N ∈ S verifies the c ontinuation pr op erty if v ( c y i ) = v ( c y i ; T i ( y )) fo r all i ≥ 0, wher e T is the shift map. It is clear that T i ( y ) has the co n tin uation prop erty to o, for any i ∈ N . In fact to sa tis fy the con tin uation prop erty it is enough to b e minimal for c y 0 . Lemma 11. If y = ( p y i , c y i , s y i ) i ∈ N ∈ S is minimal for c y 0 (that is, v ( c y 0 ) = v ( c y 0 ; y ) ) then y verifies the c ontinuation pr op erty. Pr o of. Let b = c y 1 and z = ( p z i , c z i , s z i ) i ∈ N ∈ S with c z 0 = b and v ( b ; z ) = v ( b ) given b y Lemma 10 (considering l = 0). The sequence w = y 0 y 1 T ( z ) b elongs to S and verifies v ( a ; w ) = θ − 1 2 γ ( p y 1 ) + θ − 1 2 v ( b ). Thus, if v ( b ; T ( y )) > v ( b ), from v ( a ) = v ( a ; y ) = θ − 1 2 γ ( p y 1 ) + θ − 1 2 v ( b ; T ( y )), one deduces that v ( a ; w ) < v ( a ) which is a contradiction.  This lemma pr ov es that sequences y constructed in Lemma 10 verifies the contin- uation prop er ty . 3.4. Minimal p o i n ts are ultimately p erio dic. In this se ction we pr ov e that any minimal p oint x ∈ X σ has ultimately pe rio dic prefix-suffix decomp os ition. That is, if ¯ x = ( p i , c i , s i ) i ∈ N is the prefix-suffix decomp osition of x , then T p + q ( ¯ x ) = T q ¯ x for some p > q ≥ 0. If q = 0 one s ays x is a p erio dic minimal p o int. Lemma 12. F or every a ∈ A ther e exists a ultimately p erio dic p oint x ( a ) = ( p x ( a ) i , c x ( a ) i , s x ( a ) i ) i ∈ N ∈ S with c x ( a ) 0 = a and v ( a ; x ( a )) = v ( a ) (so, x ( a ) has the c ontinuation pr op erty). Persistence of wandering interv als in self -similar AIET 9 Pr o of. Let a ∈ A a nd y = ( p y i , c y i , s y i ) i ∈ N ∈ S with c y 0 = a and v ( a ; y ) = v ( a ) given by Le mma 10 (considering l = 0). W e are go ing to construct a nother one with ultimately p erio dic decomp osition. Let 0 < q < p b e s uch that y q = y p and c y q − 1 = c y p − 1 = b . The pr ep erio dic sequence x = y 0 . . . y q − 1 y q . . . y p − 1 y q . . . y p − 1 . . . ∈ S since σ ( c y p − 1 ) = p y q c y q s y q by hypothesis. W e a r e going to pr ov e that v ( a ; x ) = v ( a ). Observe that, b y Lemma 10, v ( b ) = X i ≥ q θ − ( i − q +1) 2 γ ( p y i ) and v ( b ) = X i ≥ p θ − ( i − p +1) 2 γ ( p y i ) . Thu s, v ( b ) = P p − 1 i = q θ − ( i − q +1) 2 γ ( p y i )+ P i ≥ p θ − ( i − q +1) 2 γ ( p y i ) = P p − 1 i = q θ − ( i − q +1) 2 γ ( p y i )+ θ − ( p − q ) 2 v ( b ). If we denote B = P p − 1 i = q θ − ( i − q +1) 2 γ ( p y i ), then v ( b ) = B P i ≥ 0 θ − ( p − q ) i 2 . Consequently , v ( a ) = q − 1 X i =1 θ − i 2 γ ( p y i ) + θ − ( q − 1) 2 B X i ≥ 0 θ − ( p − q ) i 2 On the o ther hand, a direct computation yields to v ( a ; x ) = q − 1 X i =1 θ − i 2 γ ( p y i ) + θ − ( q − 1) 2 ( X i ≥ 0 θ − ( p − q ) i 2 B ) , which implies , v ( a ; x ) = v ( a ).  T o each prep erio dic sequence x ( a ) constructed in previous lemma one can asso cia te a p oint x in the symbolic spa ce X σ with p er io dic pre fix -suffix deco mpo sition o f per io d ( p 0 , c 0 , s 0 ) , . . . , ( p p − q , c p − q , s p − q ) = ( p x ( a ) p − 1 , c x ( a ) p − 1 , s x ( a ) p − 1 ) , . . . , ( p x ( a ) q , c x ( a ) q , s x ( a ) q ) . Even if, by constr uction, this p oint is asso ciated to the minimal v alue v ( b ), there is no reaso n for it to b e a minimal p oint. Without los s of generality w e will do the following simplification. B y iterating σ enough times one can assume that all ultimately p erio dic s equences constr ucted in Lemma 12 are of p erio d 1 and o f prep er io d 1. That is, for each letter a ∈ A , c x ( a ) 0 = a and x i = ( p ( a ) , ˆ a, s ( a ) ) for all i ≥ 1. The letter a ∈ A is p erio dic if ˆ a = a and one deno tes ˆ A the subse t o f p erio dic letters . Since , the co nstruction of Lemma 12 implies that v ( c x ( a ) i ) = v ( a ; T i ( x ( a ))) for 0 ≤ i ≤ p − 1, then under this simplification v ( ˆ a ) = v ( ˆ a ; T ( x ( a ))). Lemma 13. L et y ∈ S verifying t he c ontinuation pr op ert y. Then, for any i ≥ 1 the p oint y ( i ) = y 0 . . . y i T ( x ( c y i )) has t he c ontinu ation pr op erty to o. Pr o of. Let i ≥ 1 and 1 ≤ j ≤ i . F r om the contin ua tion pro p erty one deduces that v ( c y j ) = P i − j k =1 θ − k 2 γ ( p y k + j ) + θ − ( i − j ) 2 v ( c y i ). But, v ( c y i ) = v ( c y i ; x ( c y i )) and v ( ˆ c y i ) = v ( ˆ c y i ; T ( x ( c y i ))), then y ( i ) = y 0 . . . y i − 1 T ( x ( c y i )) has the contin ua tion property too.  Lemma 14. L et x , y ∈ S such that ( x i ) i ≥ l +1 = ( y i ) i ≥ l +1 and c x 0 = c y 0 = a . If v ( a ; x ) = v ( a ; y ) then ( x i ) i ≥ 1 = ( y i ) i ≥ 1 . 10 Xavier Bressaud , Pascal Hub e rt, Alejand ro Maass Pr o of. Let x = ( p x i , c x i , s x i ) i ∈ N and y = ( p y i , c x i , s y i ) i ∈ N . F rom the h yp othesis one deduces that l X i =1 θ − i 2 γ ( p x i ) = l X i =1 θ − i 2 γ ( p y i ) and consequently γ ( σ l − 1 ( p x 1 ) . . . p x l ) = γ ( σ l − 1 ( p y 1 ) . . . p y l ) . But words σ l − 1 ( p x 1 ) . . . p x l and σ l − 1 ( p y 1 ) . . . p y l are pr efixes of σ l ( a ). Then, by the algebraic conditio n (Lemma 3) they must b e the sa me. This implies ( p x i , c x i , s x i ) = ( p y i , c y i , s y i ) for 1 ≤ i ≤ l .  Theorem 15. The pr efix-suffix de c omp osition of any minimal p oint is ultimately p erio dic. Pr o of. Let x ∈ X σ be minimal p oint with prefix-suffixe decomp osition ( p i , c i , s i ) i ∈ N . There exists a finite seq uence ( ¯ p j , ¯ c j , ¯ s j ) l j =0 such that ( ¯ p 0 , ¯ c 0 , ¯ s 0 ) = ( ¯ p l , ¯ c l , ¯ s l ) and for infinitely ma ny i ∈ N , ( p i − j , c i − j , s i − j ) l j =0 = ( ¯ p j , ¯ c j , ¯ s j ) l j =0 . Let a = ¯ c 0 = ¯ c l . By Lemma 10, there is a po int y ∈ S verifying the contin uatio n prop erty suc h that ( y j ) l j =0 = ( ¯ p j , ¯ c j , ¯ s j ) l j =0 . In particular, v ( a ; y ) = v ( a ) and v ( a ; T l ( y )) = v ( a ). Since, v ( a ) = v ( a ; x ( a )), then by Lemma 13 the s equence z = y 0 . . . y l T ( x ( a )) ha s the cont inu ation prop er ty and v ( a ) = v ( a ; z ) holds. The r efore, by Lemma 14, one concludes tha t ( x ( a )) i ≥ 1 = ( z i ) i ≥ 1 . W e hav e prov ed that a ∈ ˆ A , that is a = ˆ a , and that the word ( p ( a ) , a, s ( a ) )( p ( a ) , a, s ( a ) ) app ears infinitely many times in the prefix-suffixe decomp osition of x . Now w e prov e that ( p i , c i , s i ) i ∈ N is ultimately p erio dic with p erio d ( p ( a ) , a, s ( a ) ). Assume this r e sult do es no t hold. Then there is b 6 = a in A such that ( p i , c i , s i )( p i − 1 , c i − 1 , s i − 1 )( p i − 2 , c i − 2 , s i − 2 ) = ( p ( a ) , a, s ( a ) )( p ( a ) , a, s ( a ) )( p, b, s ) for infinitely ma ny i ∈ N . By Lemma 10, there is a p oint w ∈ S verifying the con tinuation pro pe r ty and such that w 0 w 1 w 2 = ( p ( a ) , a, s ( a ) )( p ( a ) , a, s ( a ) )( p, b, s ). Since v ( b ) = v ( b ; T 2 ( w )) and v ( b ) = v ( b ; x ( b )), b y Lemma 13, the po int s u = w 0 w 1 w 2 T ( x ( b )) and v = x ( a ) 0 x ( b ) hav e the contin uation prop erty . Since u and v ar e ultimately equal, then, by Lemma 14, one co ncludes a = b w hich is a contradiction. This proves the theorem.  W e stress the fact that it is p o ssible to construct exa mples with minimal p oints having ultima tely p erio dic but no t p er io dic pre fix-suffix decomp ositio n. 3.5. Con v ergen ce of series asso ciated to m inimal p oints. Lemma 16. L et y ∈ S such that c y 0 = a ∈ ˆ A and v ( a ; y ) = v ( a ) . Then, y 1 = ( p ( a ) , a, s ( a ) ) . Pr o of. Put c y 0 = a . First we pro v e tha t v ( c y 1 ) = v ( c y 1 ; T ( y )). Let z = y 0 y 1 T ( x ( c y 1 )) ∈ S . If the asser tion is not true then v ( a ) = θ − 1 2 ( γ ( p y 1 ) + v ( c y 1 ; T ( y ))) > θ − 1 2 ( γ ( p y 1 ) + v ( c y 1 )) = v ( a ; z ) ≥ v ( a ) which is a contradiction. Thus, v ( c y 1 ) = v ( c y 1 ; T ( y )) a nd furthermor e v ( a ) = v ( a ; z ). Then, the p oint w = ( p ( a ) , a, s ( a ) )( p ( a ) , a, s ( a ) ) T ( z ) verifies v ( a ) = v ( a ; w ). But w and x ( a ) are ultimately eq ua l, then by Lemma 14, y 1 = ( p ( a ) , a, s ( a ) ).  Persistence of wandering interv als in self -similar AIET 11 Lemma 17. L et x ∈ X σ b e a minimal p oint. Then, lim inf n →∞ γ ( x 0 . . . x n ) n log( θ 2 ) log( θ 1 ) > 0 and lim inf n →∞ − γ ( x − n . . . x − 1 ) n log( θ 2 ) log( θ 1 ) > 0 Pr o of. W e o nly prov e the first inequality , the other o ne can b e shown analogo us ly . Assume the re s ult do es no t ho ld. Then, for a subse q uence ( n i ) i ∈ N , lim i →∞ γ ( x 0 . . . x n i ) n log( θ 2 ) log( θ 1 ) i = 0 Let ( p i , c i , s i ) i ∈ N be the pr efix-suffix deco mpo sition of x and let a ∈ ˆ A such that ( p ( a ) , a, s ( a ) ) is the p erio dic part of it. (1) First we assume s ( a ) is different fro m the e mpt y word. Let N i be the minimal int eger such that x 1 . . . x n i is the prefix of σ N i ( a ). Consider the prefix-suffix decomp osition ( p ( n i ) j , c ( n i ) j , s ( n i ) j ) j ∈ N of T n i +1 ( x ). Clearly , σ N i − 1 ( p ( n i ) N i − 1 ) . . . σ ( p ( n i ) 1 ) p ( n i ) 0 = σ N i − 1 ( p N i − 1 ) . . . σ ( p 1 ) p 0 x 0 . . . x n i Then, 0 X j = N i − 1 θ j 2 γ ( p ( n i ) j ) = 0 X j = N i − 1 θ j 2 γ ( p j ) + γ ( x 0 . . . x n i ) Dividing by θ N i 2 one gets, N i X j =1 θ − j 2 γ ( p ( n i ) N i − j ) = N i X j =1 θ − j 2 γ ( p N i − j ) + θ − N i 2 γ ( x 0 . . . x n i ) T aking the limit when i → ∞ a nd using the fact that x is minimal one gets lim i →∞ N i X j =1 θ − j 2 γ ( p ( n i ) N i − j ) = v ( a ) since b y assumption lim i →∞ θ − N i 2 γ ( x 0 . . . x n i ) = 0 . Observe that n i behaves like θ N i 1 . This prop erty allows to sho w, following the same idea s used to pr ov e Lemma 10, that there is y = ( p y i , c y i , s y i ) i ∈ N ∈ S such that v ( a ; y ) = v ( a ). By Le mma 16, y 1 = ( p ( a ) , a, s ( a ) ). This implies n i + 1 = 0 for so me la rge i , which is a contradiction. (2) Now suppose s ( a ) is the empty w ord. Then, (considering a p ow er o f σ if neces- sary) ( x n ) n ≥ N = lim m →∞ σ m ( b ) for s o me N ∈ N and b ∈ A . If we write σ ( b ) = bs one obtains x m . . . = bsσ ( s ) σ 2 ( s ) . . . . W e claim v ( b ) = 0. Supp o se this is no t true. Then for k ∈ N large enough one has P k i =1 θ K − i 2 γ ( p x ( b ) i ) ≤ K θ k 2 with K < 0. That is, γ applied to a prefix o f σ k ( b ) can b e a s negative as w e wan t if k incr eases. This implies that γ n ( x ) < 0 for some n ∈ N , whic h is imp os ible since x is a minimal p oint. Then v ( b ) = 0. F urthermore, we hav e prov ed that γ ( x N . . . x N + i ) > 0 for all i ≥ 1. One also deduces, by the algebraic condition, that x ( b ) = ( ε, b, s ) i ∈ N , where ε is the empty w ord. T o conclude one uses part (1) with b ins tead of a .  The following prop osition is pla in. 12 Xavier Bressaud , Pascal Hub e rt, Alejand ro Maass Prop ositi o n 18. L et x ∈ X σ b e a minimal p oint. Then, X n ≥ 1 e − γ ( x 0 ...x n − 1 ) < ∞ and X n ≥ 1 e γ ( x − n ...x − 1 ) < ∞ 4. Proof of the Main Theorem The arguments of this section follows the strateg y developed in the works of [CG] and [C]. Let T ( λ,π ) be a self-similar interv al exchange transformatio n a nd R its asso ciated matrix. As s ume R verifies h yp o thes es of Theo rem 1. Let X σ be the substitutiv e system a sso ciated to T ( λ,π ) and let M = t R be the asso ciated matrix. Consider a minimal p oint x ∈ X σ . By Prop osition 18, K = X n ≥ 1 e γ ( x − n ...x − 1 ) + 1 + X n ≥ 1 e − γ ( x 0 ...x n − 1 ) < ∞ Let t = ϕ ( x ). That is, x is the co ding o f t or x is the co ding of (lim s → t − T i ( s )) i ∈ Z in the case t is in the or bit o f one of the a i ’s. T o simplify nota tions w e assume the first case ho lds, the other one is analog ous. Define the pr obability mea sure µ t on [0 , 1) by µ t = 1 K   X n ≥ 1 e γ ( x − n ...x − 1 ) δ T − n ( λ,π ) t + δ t + X n ≥ 1 e − γ ( x 0 ...x n − 1 ) δ T n ( λ,π ) t   Lemma 19. F or every Bor el set I ⊆ [0 , 1) µ t ( T ( λ,π ) ( I )) = r X i =1 e − γ i µ t ( I ∩ [ a i − 1 , a i )) Pr o of. It is enough to consider I = [ a i − 1 , a i ) for i ∈ A . One has, µ t ( T ( λ,π ) ( I )) = 1 K   X n ≥ 1 e γ ( x − n ...x − 1 ) δ T − n ( λ,π ) t + δ t + X n ≥ 1 e − γ ( x 0 ...x n − 1 ) δ T n ( λ,π ) t   ( T ( λ,π ) ( I )) = 1 K   X n ≥ 1 e γ ( x − n ...x − 1 ) δ T − n − 1 ( λ,π ) t + δ T − 1 ( λ,π ) t + X n ≥ 1 e − γ ( x 0 ...x n − 1 ) δ T n − 1 ( λ,π ) t   ( I ) = 1 K   X n ≥ 1 e − γ ( x − n ) e γ ( x − n ...x − 1 ) δ T − n ( λ,π ) t + e − γ ( x 0 ) δ t + X n ≥ 1 e − γ ( x n ) e − γ ( x 0 ...x n − 1 ) δ T n ( λ,π ) t   ( I ) = e − γ i µ t ( I ) where in the la st equality we use the fa c t tha t T n ( λ,π ) ( t ) ∈ I if a nd only if γ ( x n ) = γ i .  Define g : [0 , 1 ) → [0 , 1) by g ( s ) = µ t ([0 , s ]). This function is nondecr easing, right contin uous and has left limits. Let i ∈ A . Denote a ′ i = T ( a i ) a nd define b i = lim a → a − i g ( a ) and b ′ i = lim a ′ → ( a ′ i ) − g ( a ′ ). Then at interv al [ b i − 1 , b i ) define linearly Persistence of wandering interv als in self -similar AIET 13 the AIET f with image [ b ′ i − 1 , b ′ i ). The slop e vector of f is w = ( e − γ 1 , . . . , e − γ r ). Indeed, b ′ i − b ′ i − 1 b i − b i − 1 = µ t ([ a ′ i − 1 , a ′ i )) µ t ([ a i − 1 , a i )) = e − γ i where the la st equality follows fro m Lemma 1 9. Let h : [0 , 1) → [0 , 1) b e the map defined by: h ( v ) = u if g ( u ) = v a nd h ( v ) = u if lim w → u − g ( w ) ≤ v ≤ g ( u ). Clea rly h is surjective, co n tin uous a nd non decreasing . Since µ t has atoms, then h is not injective The following lemma allows to conclude Theor e m 1. Lemma 20. The map h defines a semi-c onjugacy b etwe en t he AIET f and T ( λ,π ) . Mor e over, f has wandering intervals. Pr o of. The semi-conjugacy follows from construction. The interv a l I = ( lim s → t − g ( s ) , g ( t )] is a wandering in terv al for h .  5. Pseudo-Anosov diffeomorphisms and eigenv al ues of ma trices obt ained by Rauzy induction In this section, w e discuss the hypothes is of Theor em 1 in a geometric langua ge. Our hypothesis is that the Perro n-F rob enius eig env alue θ 1 of the matrix R has a real conjugate θ 2 > 1. W e recall that every in ter v al exchange transformation T ( λ,π ) is realized a s the first return map of a flow on a trans la tion surface S which gen us g ( π ) only dep ends o n the permutation π (a nd not on λ ). This transla tion surface is not unique. If T ( λ,π ) is a p erio dic p oint of the Rauzy induction, one can c ho ose S fixed by a pseudo- Anosov diffeomorphism φ (see [Th] for an enlightening discussion on pseudo- Anosov diffeomorphisms). The eigenv alue θ 1 is the dominant eigenv alue o f the action of φ on the a bsolute ho mology of S . Therefore θ 1 is an algebra ic num b er of deg r ee a t most 2 g ( π ) ov er Q . Heuristically , a fter the w ork of Avila and Viana [A V], it is rea sonable to believe that a “generic” pseudo-Anosov satisfies our hypothesis. Nevertheless, it seems extremely difficult to understand the eigenv alue s of al l pseudo-Anosov diffeomo r- phisms. In this section, we w ant to explain that our hyp o thesis are often satisfied. They are no t alwa ys satisfied: for instance, the conjugates of the Arnoux-Y o ccoz pseudo-Anosov are not real. Situations muc h worse do exist. 5.1. Existence of a conjugate θ 2 with | θ 2 | ≥ 1 . A pseudo-Anosov diffeomor- phism preser ves the symplectic form induced by the in tersection form. Thus if z is an eig e n v alue of the automor phism φ ∗ of H 1 ( S , Z ), its in v erse z − 1 is also an eigenv alue of φ ∗ . Co nsequently , 1 θ 1 is an eige nv alue of φ ∗ . If it is the only Galois conjugate of θ 1 , it means that θ 1 is an alge braic num b er o f degree 2. It is cla ssical (see [K S] for instance) that the surface S is then a cov er ing of a torus (a square tiled surface up to no rmalization). Ther efore hypothesis (1) is satisfied if and only if the surface S is not a square tiled surface. Thus, this hypothesis is very natural and simple to chec k. 14 Xavier Bressaud , Pascal Hub e rt, Alejand ro Maass 5.2. Real conjugates. The second h ypo thesis is more subtle to analyz e . A ps e udo -Anosov diffeomorphism is obtained by Thu rston’s construction if it is the pro duct of tw o affine Dehn t wists T h and T v along tw o m ulti-curves filling a surface (see [Th]). After no rmalization, the deriv atives of the Dehn twists in the natural parameters of the tra nslation surfac e ar e T h =  1 a 0 1  , T v =  1 0 b 1  where a and b ar e p ositive real n um ber s a nd a b is a n algebra ic n um ber . An ele ment f of the gro up generated by T h and T v is a ps eudo-Anosov diffeomor- phism if the abso lute v alue o f the tr ace t ( f ) of the corres p o nding matrix is lar ger than 2. F or every pse udo -Anosov diffeomorphism obtained by Thurston’s construc- tion, the conjugates of t ( f ) ar e real n um ber s (see [HL]). The dominant root of the action o f f on the homology is the real num b er θ 1 > 1 with θ 1 + θ − 1 1 = t ( f ). The nu mber θ 1 (or one o f its p ow er) is the P erron-F rob enius eigenv alue of the matr ix obtained by Rauzy induction considered in the pr esent pap er (see [V e]). Let θ ′ be a conjugate of θ 1 and t ′ ( f ) = θ ′ + θ ′− 1 a ( r e al ) conjugate of t ( f ). θ ′ is a real num b er with θ ′ > 1 if | t ′ ( f ) | > 2. It is a complex n umber of modulus one if | t ′ ( f ) | < 2. This directly comes from the fact that θ ′ + θ ′− 1 = t ′ ( f ). F or instance, the diffeomorphisms f n,m = T n h T m v are pse udo-Anosov diffeo mo r- phisms if n, m are p os itive in teg ers. In fact the abs o lute v alue of the tra ce o f  1 a 0 1  n  1 0 b 1  m is lar ger than 2 b ecaus e nmab > 0. Thu s θ ′ > 1 if | t ′ ( f ) | = | 2 + nm ( ab ) ′ | > 2 (wher e ( ab ) ′ is a re a l num ber). This is satisfied for all co uples ( n, m ) except for finite num b er of exceptions. Using mor e sophisticated ar gument, | t ′ ( f ) | > 2 if n and m are p o sitive in teg ers. Ac knowledgmen ts. The second author is supp o rted b y pro ject bla nc ANR: ANR- 06-BLAN-00 38. The third author is suppo rted b y Nucleus Millennium Information and Randomness P 04-06 9-F. References [Ad] B. Adamcz ewski, Symbolic discrepancy and self -simil ar dynamics, Ann. Inst. F ourier (Grenoble) 54 (2004), 22012234 (2 005). [A V] A. Avila, M. Viana, Simplicity of Lyapunov sp e ctr a: pr o of of the Zorich-Kontsevich c onje ctur e , Acta Mathematica 198 (2007 ), 1- 56. 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[Zo] A. Zorich, Finite Gauss me asur e on the sp ac e of interval exc hange tr ansformations, Lyapunov exp onents , Annales de l’Institut F ourier 46:2, ( 1996), 325–370. [Zo2] A. Zorich , Flat surfac es , in “F ront iers in Number Theory , Physics and Geometry , volume I. On Random matrices, Zet a F unctions and Dynamical Syste ms”, P . Cartier, B. Julia, P . M oussa, P . V anhov e (Editors), Springer V erlag, Berlin 2006, 439–585. Institut de M a th ´ ema tiques de Lum iny, 16 3 a venue de Luminy, Case 90 7, 1328 8 Ma rseille Cedex 9, France. E-mail ad dr ess : bress aud@iml.u niv-mrs. fr Labora toire Anal yse, Topologie et Pr obabilit ´ es, Case cour A, F acul t ´ e des Sciences de Saint-Jer ˆ ome, A venu e Escadrille Norm a ndie-Niemen, 13397 Marseille Cedex 2 0, France. E-mail ad dr ess : huber t@cmi.uni v-mrs.fr Centro de Modelamiento Ma tem ´ atico and Dep ar t amento de Ingen ier ´ ıa Ma tem ´ atica, Uni- versidad de Chile, A v. Blanco Encalada 2 1 20, Sa ntiago, Chile. E-mail ad dr ess : amaas s@dim.uch ile.cl