Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transfomations
EExactness of the Euclidean algorithm and of the Rauzyinduction on the space of interval exchange transformations
Tomasz Miernowski Arnaldo NogueiraOctober 30, 2018
Abstract
The two-dimensional homogeneous Euclidean algorithm is the central motivation for the defi-nition of the classical multidimensional continued fraction algorithms, as Jacobi-Perron, Poincar´e,Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, isa key tool in the study of interval exchange transformations. Both maps are known to be dissipativeand ergodic with respect to Lebesgue measure. Here we prove that they are exact.
Here we study the dynamics of a class of piecewise linear maps defined in the n -dimensionalEuclidean space, which are dissipative and ergodic with respect to Lebesgue measure. Theaim of this paper is to prove that they bare a stronger property: they are exact, thatis, they satisfy a kind of Kolmogorov 0 − T : X → X is exact with respect to a measure µ if, andonly if, for every positive measure set Ω ⊂ X , there exists a positive integer k whichdepends on the set Ω, such that the measure of the intersection T k +1 (Ω) ∩ T k (Ω) ispositive. The exactness property of an n -dimensional homogeneous algorithm implies thatsuitable projections of the map also bare this property. In particular, the radial projectionof the algorithm on the ( n − G , admits a finite invariantmeasure. Later Bufetov [4] showed that the map G = G ◦G is exact and used this propertyto prove a result on the decay of correlations for the map G . On the other hand, the map G is not exact. The domain ∆ on which G acts splits into two disjoint non trivial sets,namely ∆ + and ∆ − , such that G (∆ + ) = ∆ − and G (∆ − ) = ∆ + .In order to illustrate our approach, we begin our analysis with the Euclidean algorithmdefined as follows. Let R = { ( λ , λ ) ∈ R : λ ≥ , λ ≥ } and consider the map E : R → R given by E ( λ , λ ) = (cid:26) ( λ − λ , λ ) , if λ ≥ λ , ( λ , λ − λ ) , if λ < λ . (1.1)1 a r X i v : . [ m a t h . D S ] M a y ther classical versions of the Euclidean algorithm are defined at the end of Section 4.Since E is piecewise linear, we may describe its dynamics in matrix notation. Let B = (cid:18) (cid:19) and B = (cid:18) (cid:19) , (1.2)be the two elementary matrices which generate the group SL (2 , Z ). The definition (1.1)may be rewritten as E : λ = (cid:18) λ λ (cid:19) (cid:55)→ (cid:26) B − λ , if λ ≥ λ ,B − λ , if λ < λ . (1.3)The above expression relates the dynamics of E to the linear action of SL (2 , Z ). Theorem ([10], Proposition 8.1) . Let λ ∈ R , then ∪ ∞ m =0 ∪ ∞ k =0 E − m ( {E k ( λ ) } ) = R ∩ SL (2 , Z ) { λ } . In particular, the ergodicity of the action of SL (2 , Z ) on R implies the ergodicity of E with respect to Lebesgue measure. Here we prove the following. Theorem 1.1.
The Euclidean algorithm is exact with respect to Lebesgue measure.
Next we consider the Rauzy induction algorithm which acts on the space of intervalexchange transformations. To an n -interval exchange one associates a first return mapinduced on a suitable subinterval, which is itself an exchange of n intervals. The aim ofthe process is to relate the dynamics of an interval exchange to the dynamics of a classof interval exchange maps. Later it was noticed that the Rauzy induction allows thesuspension of an interval exchange and one may define a flow on the resulting surface,which is related to the so-called T eichmuller f low . Due to the work of Veech [16], andother, the Rauzy induction became a central tool in the dynamical study of the intervalexchange transformations.It is known that the Rauzy induction is dissipative and ergodic [17]. Here we provethe following result.
Theorem 1.2.
The Rauzy induction algorithm acting on the space of interval exchangetransformations is exact with respect to Lebesgue measure.
Since the Rauzy induction on the space of 2-interval exchanges coincides with theEuclidean algorithm, the statement of Theorem 1.1 may be seen as a particular case ofthe last theorem. However, we decided to treat it independently, since the presentation ofthe proof becomes more transparent and the techniques involved extend rather easily tothe multidimensional case.There is a generalization of the Rauzy induction acting on the space of linear invo-lutions (see Danthony and Nogueira [5]). Therefore it is natural to conjecture that thistransformation is exact as well. This places the exactness property in the context of mea-sured foliations on orientable surfaces.The article is organized in the following way. Section 2 concernes the exactness prop-erty. We prove a technical lemma which contains an exactness criterium for ergodic maps.The criterium is general and suits the class of multidimensional continued fraction algo-rithms. In Section 3, we present the main dynamical features of the Euclidean algorithmwhich are exploited in Section 4 to prove that E is an exact map - Theorem 1.1. It is known2hat iterations of E generate natural Markov partitions of the cone R into subcones. Ba-sically, we prove that we can extract new partitions whose subcones have distortions aslarge as we want. At the end of Section 4 we introduce two alternative versions of theEuclidean algorithm which are shown to be exact as well. In Section 5 we apply Theorem1.1 to show exactness of two maps which are conjugate to the Euclidean algorithm. InSection 6 we define interval exchange transformations and the inducing process for intervalexchanges, called Rauzy induction. It may be seen as an algorithm acting on copies of thepositive cone of R n . Section 7 is devoted to the study of Rauzy classes of permutations, inparticular we prove that every Rauzy graph has a loop, a central fact exploited in the proofof our main theorem. In Section 8, we prove that the Rauzy induction defines suitablepartitions of the positive cone R n + which bare similar properties as those defined by theEuclidean algorithm. In Section 9 we adapt the argument developed in Section 4 to themultidimensional case in order to prove our main result, Theorem 1.2. In the last sectionwe give examples of other exact multidimensional continued fraction algorithms which areadapted to the approach developed in our work. Let ( X, B , µ ) be a measure space and let T : X → X be a measurable transformation. Themap T is said to be ergodic with respect to µ , if for every Ω ∈ B such that T − (Ω) = Ω, µ (Ω) = 0 or µ ( X \ Ω) = 0. The map T is said to be nonsingular , if for Ω ∈ B , µ ( T − (Ω)) =0 if, and only if, µ (Ω) = 0. The map T is said to be exact , if Ω ∈ ∩ m ≥ T − m ( B ) implies µ (Ω) = 0 or µ ( X \ Ω) = 0. For more information about exact maps one may refer to [1,Section 1.2]. The measurable map T is said to be bi-measurable , if, for every Ω ∈ B , wehave T (Ω) ∈ B .In order to study exactness we introduce an additional dynamical property: the bi-measurable map T satisfies the intersection property , with respect to the measure µ if, forevery Ω ∈ B with µ (Ω) >
0, there exists k = k (Ω) ≥ µ ( T k +1 (Ω) ∩ T k (Ω)) > Lemma 2.1.
Let ( X, B , µ ) be a measure space and let T : X → X be bi-measurable,nonsingular and ergodic. Then T is exact if, and only if, it satisfies the intersectionproperty.Proof. Assume that T is exact. Let Ω ∈ B be of positive measure and consider thefollowing nested sequence of subsets of X : T (Ω) ⊂ T − ( T ( T (Ω))) ⊂ . . . ⊂ T − k ( T k ( T (Ω))) ⊂ . . . . Let S = ∪ k ≥ T − k ( T k ( T (Ω))). We have S = ∪ k ≥ m T − k ( T k ( T (Ω))) = T − m ( T m ( S )) , for every m ≥ . Therefore S ∈ ∩ m ≥ T − m ( B ). Moreover, since T (Ω) ⊂ S , by the nonsingularity of T weget µ ( S ) >
0. The exactness of T implies that S is of full measure in X . Since µ (Ω) > k ≥ µ ( T − k ( T k ( T (Ω)) ∩ Ω) >
0. Again, by the nonsingularity of T , we get µ ( T k +1 (Ω) ∩ T k (Ω)) > . T satisfies the intersection property. Let Ω ∈ ∩ m ≥ T − m ( B ),which is equivalent to Ω = T − m ( T m (Ω)) for all m ≥
1. Assume that µ (Ω) >
0. In orderto show that T is exact we have to show that Ω is of full measure.Let Λ = Ω \ T (Ω). We have Λ ∩ T (Λ) = ∅ . Moreover, for every n ≥ T − n ( T n (Ω)) \ T − n ( T n ( T (Ω))) = T − n ( T n (Ω) \ T n +1 (Ω)) , we obtain T n (Λ) ∩ T n +1 (Λ) = ∅ which implies µ (Λ) = 0 by the intersection property. Wehave Ω ⊂ T (Ω) up to a null measure set. The same argument applied to Λ (cid:48) = T (Ω) \ Ωgives T (Ω) ⊂ Ω which implies Ω = T (Ω) up to a null measure set. We may write T − (Ω) = T − ( T (Ω)) = Ω . Since T is ergodic and Ω a positive measure set, we get that it is a full measure set. Themap T is thus exact.In what follows the measure space ( X, B , µ ) is essentially a positive cone of an Eu-clidean space R n with Borel σ -algebra and Lebesgue measure µ . The maps considered(the Euclidean algorithm and Rauzy induction) are bi-measurable and nonsingular. Al-though they do not preserve Lebesgue measure, they admit invariant measures absolutelycontinuous with respect to µ . In this section we recall some results about the dynamics of the Euclidean algorithm thatwill be used to prove Theorem 1.1. For more details one may refer to [11] and referencestherein.Let E : R → R be the Euclidean algorithm (1.3). To a point λ ∈ R we mayassociate a { B , B } -valued (1.2) sequence of matrices ( B m k ) k ≥ such that E k : (cid:18) λ λ (cid:19) (cid:55)→ B − m k · . . . · B − m (cid:18) λ λ (cid:19) for every k ≥ . (3.1)One may easily verify that (3.1) holds if and only if (cid:18) λ λ (cid:19) ∈ B m · . . . · B m k ( R ) for every k ≥ . (3.2)The sequence ( B m k ) is called the expansion of the point λ . This comes from the factthat it is closely related to the continued fraction expansion of the ratio λ /λ . Assumethis ratio irrational. In such case, the sequence ( B m k ) contains infinitely many of bothmatrices B and B . We may define a sequence of integers ( a i ) i ≥ as follows. Let a = 0if the sequence begins with B , otherwise let a be the number of consecutive matrices B at the beginning of the sequence. Next, let a be the number of consecutive matrices B that follow. Then, define a to be the number of consecutive matrices B that come next,and so on: ( B m k ) = B . . . B (cid:124) (cid:123)(cid:122) (cid:125) a B . . . B (cid:124) (cid:123)(cid:122) (cid:125) a B . . . B (cid:124) (cid:123)(cid:122) (cid:125) a B . . . B (cid:124) (cid:123)(cid:122) (cid:125) a . . . It may be shown that λ λ = [ a ; a , a , . . . ] = a + 1 a + 1 a + 1. . . . .1 Partitions of R From (3.2) we deduce that for every k ≥
1, the positive cone R is decomposed into2 k subcones, of disjoint nonempty interiors, which correspond to different sequences ofelementary matrices involved in the iterations of E . Let P ( k ) be the k th partition P ( k ) = { B m · . . . · B m k ( R ) : m , . . . , m k ∈ { , }} . (3.3)For every k ≥
1, the partition P ( k +1) is a refinement of P ( k ) . To be more precise, if C ( k ) ∈ P ( k ) is defined by a couple of vectors ( l , l ), then it generates two elements of P ( k +1) defined by the couples ( l , l + l ) and ( l + l , l ).Fix λ ∈ R with irrational ratio λ /λ . For every k ≥
1, let C ( k ) λ ∈ P ( k ) be thecone defined by (3.2). We obtain a nested sequence of subcones of R and one may showthat the intersection ∩ k ≥ C ( k ) λ equals the radial line { αλ : α ≥ } . For every k ≥
1, let l ( k )1 and l ( k )2 stand for the two column-vectors of the matrix B m · . . . · B m k , generatingthe corresponding cone C ( k ) λ . From (3.1) we deduce that E k ( C ( k ) λ ) = R and the vectors l ( k )1 , l ( k )2 are sent onto the canonical basis of R : E k ( l ( k ) i ) = e i , i = 1 , . (3.4)The next lemma shows a central property of the family of partitions ( P ( k ) ) k ≥ of R .Namely, the angles formed by the column-vectors l ( k )1 , l ( k )2 which define the subcones C ( k ) go to zero uniformly as k → ∞ . Lemma 3.1.
Let K be a compact measurable subset of R . Then lim k →∞ max C ∈P ( k ) µ ( K ∩ C ) = 0 . Proof.
Let K be a compact subset of R . Let α > K ⊂ ∆( α ) = { λ ∈ R : λ + λ ≤ α } . Let C ∈ P ( k ) and l , l be the column-vectors of the matrix which defines C . A trivial computation gives µ (∆( α ) ∩ C ) = 12 α (cid:107) l (cid:107) (cid:107) l (cid:107) which implies that µ ( K ∩ C ) ≤ α (cid:107) l (cid:107) (cid:107) l (cid:107) ≤ α k + 1) . We conclude that lim k →∞ max C ∈P ( k ) µ ( K ∩ C ) = 0 . Next, let (cid:107) · (cid:107) stand for the Euclidean norm. We have the following result which comesfrom the continued fraction expansion interpretation of E . Theorem 3.2 ([11], Section 4) . Let λ ∈ R with irrational ratio λ /λ and l ( k )1 , l ( k )2 bethe column-vectors which define C ( k ) λ , k ≥ . Then the following properties hold: . There exist infinitely many integers k ≥ such that θ ≤ (cid:107) l ( k )1 (cid:107)(cid:107) l ( k )2 (cid:107) ≤ θ, where θ > is a constant independent of λ .2. For every N > , for almost every λ there exist infinitely many integers s ≥ suchthat (cid:107) l ( s )1 (cid:107)(cid:107) l ( s )2 (cid:107) > N or (cid:107) l ( s )2 (cid:107)(cid:107) l ( s )1 (cid:107) > N. Next we introduce the notion of distortion of a subcone which will be needed in thenext section. Let C ∈ P ( k ) and l , l be the column-vectors of the matrix which defines C .We call the distortion of C the numbermax (cid:26) (cid:107) l (cid:107)(cid:107) l (cid:107) , (cid:107) l (cid:107)(cid:107) l (cid:107) (cid:27) . The second part of the last theorem implies that there exists a partition of R formedonly by subcones of ∪ k ≥ P ( k ) whose distortions are as large as we want. Corollary 3.3.
Let
N > . Then there exists a partition P N of R formed by subconesof ∪ k ≥ P ( k ) such that, for every C ∈ P N , its distortion is greater than N .Proof. From Theorem 3.2 we know that for almost every λ ∈ R there exist infinitelymany integers s ≥ C ( s ) λ is a distorted cone. Let s ( λ ) be the smallest of thoseintegers and define P N to be the collection of all cones C ( s ( λ )) λ obtained this way. We claimthat P N is a partition of R .Clearly, the union of all the members of P N covers R up to a null measure set.Moreover, let C and C be subcones belonging to the collection P N . We claim that C = C , otherwise C ∩ C is a null measure set.Assume µ ( C ∩ C ) >
0. There exist λ and λ such that C i = C ( s ( λ i )) λ i , i = 1 , s = s ( λ ) ≥ s = s ( λ ). Since the partition P ( s ) is a refinement of the partition P ( s ) and C and C have a non trivial intersection, the cone C must be contained in C .This implies λ ∈ C and thus s = s by the definition of s ( λ ). This implies C = C and proves that P N is a partition of R with the desired properties. To prove that the Euclidean algorithm is exact with respect to Lebesgue measure we useLemma 2.1. The main steps of the proof are the following.Let Ω ⊂ R be a positive Lebesgue measure set and λ a density point of Ω. First,we construct a sequence ( Q n ) n ≥ of quadrilateral domains that shrink to { λ } as n → ∞ .Using a version of Lebesgue density theorem, we show that given ε >
0, for every n sufficiently large we have µ (Ω ∩ Q n ) ≥ (1 − ε ) µ ( Q n ) , (4.1)where µ stands for Lebesgue measure.Fix Q n satisfying (4.1). We will show that a similar density property holds for a smallerquadrilateral Q which is the intersection of Q n with a subcone coming from a “distorted”partition P N given by Corollary 3.3. 6 λ p n q n r n s n φ n ρ n Figure 1. Quadrilateral Q n .This new quadrilateral Q is the intersection of Q n with a cone C ( s ) defined by (3.2).We know that E s ( C ( s ) ) = R , which implies that the vertices of the quadrilateral E s ( Q )are fixed by E . Moreover, it still satisfies a density condition close to (4.1). Finally we usethe distortion property of the cone C ( s ) to show that the intersection E s ( Q ) ∩ E s +1 ( Q ) islarge enough to imply µ ( E s (Ω) ∩ E s +1 (Ω)) > N can be chosen inde-pendently of the values of ε and n . Let Ω ⊂ R be a set of positive Lebesgue measure and fix λ = ( λ , λ ) a density pointof Ω, which is an interior point of R . For every n ≥
1, let Q n be the quadrilateral(trapezoid) p n q n r n s n whose vertices are defined as follows (see Figure 1): • p n = λ − n ( λ , − λ ) = ( λ − n λ , λ + n λ ) , • q n = λ + n ( λ , − λ ) = ( λ + n λ , λ − n λ ) , • r n = (1 + n ) q n and s n = (1 + n ) p n .The point λ is the middle point of the segment p n q n and the Euclidean distance betweenthe parallel segments p n q n and s n r n is equal to the length of p n q n , that is n (cid:107) λ (cid:107) . Thenested sequence of quadrilaterals ( Q n ) n ≥ satisfies ∩ n ≥ Q n = { λ } . Moreover, for n largeenough the quadrilateral Q n is contained in R . Lemma 4.1.
For every n ≥ there exists a ball B ( λ , ρ n ) centered at λ of radius ρ n ,such that Q n ⊂ B ( λ , ρ n ) , ρ n → as n → ∞ and lim n →∞ µ ( Q n ) µ ( B ( λ , ρ n )) = 45 π . (4.2) Proof.
We have µ ( Q n ) = 2 n + 12 n (cid:107) λ (cid:107) . Q n situated the farthest from λ are the vertices r n and s n . This impliesthat Q n is contained in the ball centered at λ of radius ρ n = √ n + 2 n + 1 (cid:107) λ (cid:107) n . Wehave ρ n →
0, as n → ∞ , and µ ( Q n ) µ ( B ( λ , ρ n )) = 2 n (2 n + 1) π (5 n + 2 n + 1) → π . Corollary 4.2.
For every ε > there exists n ≥ such that µ (Ω ∩ Q n ) ≥ (1 − ε ) µ ( Q n ) .Proof. Let B ( λ , ρ n ) be the sequence of balls defined in the previous lemma. Since λ isa density point of Ω, we know thatlim n →∞ µ (Ω ∩ B ( λ , ρ n )) µ ( B ( λ , ρ n )) = 1 . Fix δ ∈ (0 , n large enough we have µ (Ω ∩ B ( λ , ρ n )) ≥ (1 − δ ) µ ( B ( λ , ρ n )). Since Q n ⊂ B ( λ , ρ n ), we get µ (Ω ∩ Q n ) = µ (Ω ∩ B ( λ , ρ n )) − µ (Ω ∩ ( B ( λ , ρ n ) \ Q n ))which implies µ (Ω ∩ Q n ) ≥ (1 − δ ) µ ( B ( λ , ρ n )) − µ ( B ( λ , ρ n ) \ Q n ) = µ ( Q n ) − δµ ( B ( λ , ρ n )) . The relation (4.2) implies µ (Ω ∩ Q n ) µ ( Q n ) ≥ − δ µ ( B ( λ , ρ n )) µ ( Q n ) ≥ − δ π, for n large enough. Since δ may be chosen as small as we wish, the claim follows. Let Q n be a quadrilateral defined above and satisfying (4.1). In the next lemma weconsider the intersection of Q n with cones of the partition P N given by Corollary 3.3. Inparticular, for N large enough, one of the new smaller quadrilaterals satisfies a densitycondition close to (4.1). Lemma 4.3.
For N ≥ large enough there exists C ∈ P N such that the quadrilateral Q = Q n ∩ C satisfies the density condition µ (Ω ∩ Q ) ≥ (1 − ε ) µ ( Q ) . Proof.
Let C ( Q n ) be the subcone of R generated by the couple of vectors corresponding tothe vertices p n and q n of Q n . Thus C ( Q n ) is the smallest cone containing the quadrilateral Q n . For every N ≥
1, the family, by Corollary 3.3, the family P N is a partition of R andlim N →∞ max C ∈P N µ ( Q n ∩ C ) = 0 , by Lemma 3.1. This means that for N large enough we get (cid:88) C ∈P N C ⊂ C ( Q n ) µ ( Q n ∩ C ) ≥ (1 − ε ) µ ( Q n ) . Q n and Q . q n r n p n s n l l QQ n α l β l α l β l µ (Ω ∩ Q n ∩ C ) < (1 − ε ) µ ( Q n ∩ C ) for every cone C in the above sum, we get µ (Ω ∩ Q n ) < (1 − ε ) µ ( Q n ) + εµ ( Q n ) = (1 − ε ) µ ( Q n ) , which contradicts the choice of Q n .Let N ≥ Q be a quadrilateral given by the previous lemma. It isdefined as Q = Q n ∩ C ( s ) , where C ( s ) ∈ P N ∩ P ( s ) for some s ≥
1. Let also l ( s )1 and l ( s )2 be the column-vectors of the matrix B m · · · B m s generating C ( s ) . Since C ( s ) ∈ P N , thosevectors satisfy the corresponding distortion condition. Without loss of generality, assume (cid:107) l (cid:107) ≥ N (cid:107) l (cid:107) .The vertices of Q may be written as α l , β l , β l and α l , where 0 < α < β and0 < α < β (see Figure 2). Let φ n be the angle between the vectors corresponding to thepoints q n and λ of the quadrilateral Q n (see Figure 1). We have the following estimates. Lemma 4.4.
The vertices of Q satisfy α β = α β = nn + 1 and α α = β β ≥ N cos φ n . Proof.
The first equality above comes from the fact that Q is a trapezoid and from thedefinition of Q n . To prove the second one, recall that Q is the intersection of the trapezoid Q n with a subcone of the initial cone C ( Q n ). This means, according to Figures 1 and 2,that α (cid:107) l (cid:107) ≥ (cid:107) λ (cid:107) and α (cid:107) l (cid:107) ≤ (cid:107) q n (cid:107) . We get α α ≥ (cid:107) l (cid:107)(cid:107) λ (cid:107)(cid:107) l (cid:107)(cid:107) q n (cid:107) ≥ N cos φ n . The same argument holds for β /β . 9igure 3. Trapezoids T , T + and E ( T + ). α β β β β + β α α α + α α β β β β + β α α α + α T + ∩ E ( T + )0 We know that E s ( C ( s ) ) = R and E s ( l i ) = e i , i = 1 , Q are α l , β l , β l and α l , the image E s ( Q ) is the trapezoid T of vertices( α , , ( β , , (0 , β ) and (0 , α ) (see Figure 3). Moreover, the map E s restricted to C ( s ) is bijective and preserves Lebesgue measure, which implies µ ( E s (Ω) ∩ T ) ≥ (1 − ε ) µ ( T ) . Consider the smaller trapezoid T + = { λ ∈ T : λ ≥ λ } whose vertices are (0 , α ),(0 , β ), ( β β β + β , β β β + β ) and ( α α α + α , α α α + α ). A short calculation and Lemma 4.4 give µ ( T + ) µ ( T ) = (cid:18) β ( β ) β + β − α ( α ) α + α (cid:19) β β − α α = β β + β ≥ N cos φ n N cos φ n + 1 . (4.3)The choice of N is independent of the choice of ε and of the first quadrilateral Q n . Choosing N large enough, we may assume µ ( T + ) ≥ (1 − ε ) µ ( T ) which implies µ ( E s (Ω) ∩ T + ) ≥ (1 − ε ) µ ( T + ) . (4.4) Lemma 4.5.
For N large enough µ ( E ( T + ) ∩ T + ) ≥ µ ( T + ) = 12 µ ( E ( T + )) . (4.5)10 roof. The image E ( T + ) is the trapezoid of vertices (0 , α ), (0 , β ), ( β β β + β , α α α + α , N large enough we get β β β + β ≥ β + α β β ≥ ( β ) + α β + α β ,β β ≥ α β + α β β β ,β β ≥ n + 1 , by Lemma 4.4. Since the choice of N is independent of the choice of the first quadrilateral Q n and since β /β ≥ N cos φ n , the claim follows.Now we conclude the proof of Theorem 1.1. Let ε < /
12 and N be large enough for(4.4) and (4.5) to hold. We get µ ( E s (Ω) ∩ T + ∩ E ( T + )) > µ ( T + ∩ E ( T + )) and µ ( E s +1 (Ω) ∩ T + ∩ E ( T + )) > µ ( T + ∩ E ( T + ))which imply µ ( E s (Ω) ∩ E s +1 (Ω)) >
0. Therefore E is exact with respect to Lebesguemeasure and the proof of Theorem 1.1 is complete. (cid:3) E In the literature one may find other definitions of the Euclidean algorithm. We would liketo mention two of them. The first one, denoted by E σ , is defined on R by E σ ( λ , λ ) = ( λ σ (1) , λ σ (2) − λ σ (1) ) = (cid:26) ( λ , λ − λ ) if λ ≤ λ , ( λ , λ − λ ) if λ < λ . Here σ stands for the permutation (depending on λ ) of the set { , } such that λ σ (1) ≤ λ σ (2) . It differs from E only by the permutation of coordinates in the upper sub-cone { λ ≥ λ } .The other one is a projection of E onto the subset Λ = { λ ∈ R : λ ≤ λ } . We defineit as follows. E π ( λ , λ ) = π ( λ , λ − λ ) = (cid:26) ( λ , λ − λ ) if λ ≤ λ − λ ( λ − λ , λ ) if λ − λ < λ , where π stands for the permutation of coordinates (depending on λ ) arranging them innondecreasing order.From the ergodicity of E one may deduce the ergodicity of E σ and E π . The proof ofTheorem 1.1 presented above applies with minor changes to those two transformations. Corollary 4.6.
The maps E σ and E π are exact with respect to Lebesgue measure. E σ and E π In this section we present two examples of transformations which are conjugate to theEuclidean algorithm and inherit therefore the exactness property. The first one is a nor-malization of the three-dimensional Poincar´e algorithm. The other one is an example of aso-called fully subtractive algorithm introduced by Schweiger [15, Chapter 9].11 .1 The Poincar´e algorithm
For every point λ ∈ R let σ be a permutation of the set { , , } such that λ σ (1) ≤ λ σ (2) ≤ λ σ (3) . The three-dimensional Poincar´e algorithm is the map P : R → R defined by P ( λ , λ , λ ) = ( λ σ (1) , λ σ (2) − λ σ (1) , λ σ (3) − λ σ (2) ) . In [10, Theorem 2.1] it is shown that P is not ergodic with respect to Lebesgue measure. Infact P admits a nontrivial absorbing set Γ ⊂ R and its restriction to this set is conjugateto an extension of the Euclidean algorithm E σ . To be more precise, the map P : Γ → Γ isconjugate to E σ × id : R → R .The nonergodic transformation P cannot be exact. However, being conjugate to E σ × id ,it satisfies the intersection property. Now consider the projection of the algorithm P ontothe two-dimensional simplex ∆ = { λ ∈ R : λ + λ + λ = 1 } given by˜ P : ∆ (cid:51) λ (cid:55)→ P ( λ ) (cid:107)P ( λ ) (cid:107) ∈ ∆ . This normalization of P , called Daniels-Parry map [15, p.185], is ergodic with respectto Lebesgue measure on ∆ [10, Theorem 2.3]. From the intersection property of P wededuce the analogous property for ˜ P . Corollary 5.1.
The normalized Poincar´e algorithm ˜ P : ∆ → ∆ is exact with respect toLebesgue measure. Let the map S : { λ ∈ R : λ ≤ λ ≤ λ } → { λ ∈ R : λ ≤ λ ≤ λ } be defined by S ( λ , λ , λ ) = π ( λ , λ − λ , λ − λ ) , where π is a permutation arranging the coordinates in the nondecreasing order. In anearlier paper [9] we showed that S is not ergodic with respect to Lebesgue measure.However, one may normalize S by imposing the last coordinate to be equal to one. Weobtain a new transformation˜ S ( λ , λ ) = − λ ( λ , λ − λ ) if λ ≤ λ − λ , − λ ( λ − λ , λ ) if λ − λ < λ ≤ − λ , λ ( λ − λ , − λ ) if 1 − λ < λ , defined on the set D = { λ ∈ R : 0 ≤ λ ≤ λ ≤ } . The map ˜ S is ergodic with respect toLebesgue measure [9, Theorem 1.3]. This comes from the fact that ˜ S admits a nontrivialabsorbing set on which its dynamics is conjugate to E π . Now we may improve our earlierresult. Corollary 5.2.
The normalized algorithm ˜ S : D → D is exact with respect to Lebesguemeasure. Throughout the remaining part of the paper let n ≥ C = { λ = ( λ , . . . , λ n ) ∈ R n : λ i > , ≤ i ≤ n, } and let S be the group of permutations of the set { , , . . . , n } .12 .1 Interval exchanges Here our main reference is Veech [V1]. An exchange of n intervals is a map which permutes n given intervals. It is defined by a couple of parameters ( λ, π ) ∈ C × S in the followingway. Let I λ = [0 , (cid:107) λ (cid:107) ), where (cid:107) λ (cid:107) = λ + . . . + λ n . We set α ( λ ) = 0 and α i ( λ ) = λ + . . . + λ i , for 1 ≤ i ≤ n . The points α i ( λ ) partition the interval I λ into n subintervals I λi = [ α i − ( λ ) , α i ( λ )) of length λ i . Finally we use π to permute those subintervals. We set λ π = ( λ π , . . . , λ πn ), where λ πi = λ π − ( i ) , for 1 ≤ i ≤ n .The ( λ, π )- interval exchange is the one-one onto map T = T ( λ,π ) : I λ → I λ , defined by T ( x ) = x − α i − ( λ ) + α π ( i ) − ( λ π ) , for x ∈ I λi , for 1 ≤ i ≤ n. The map T acts as a translation on each subinterval I λi and thus T preserves Lebesguemeasure. Moreover we have T ( I λi ) = I λ π π ( i ) .We say that a permutation π ∈ S is irreducible , if 1 ≤ k ≤ n and π { , . . . , k } = { , . . . , k } imply k = n . In other words, for an irreducible permutation π , if x > T ( λ,π ) [0 , x ) = [0 , x ), then x = α n ( λ ). We denote by S the set of irreducible permutationsof S .If π ∈ S is not irreducible, for every λ ∈ C the corresponding ( λ, π )-interval exchangemay be seen as two independent exchanges of k and n − k intervals. In particular it is notergodic with respect to Lebesgue measure. In what follows only irreducible permutationswill be considered. Here we follow the approach given in [12, Section 2]. A vector λ ∈ C is called irrational ifits coordinates λ , . . . , λ n are rationally independent. Let T ( λ,π ) be an interval exchangegiven by an irrational vector λ and an irreducible permutation π . The so-called Rauzyinduction assigns to T ( λ,π ) a first return map induced on a suitable subinterval of I λ . Wesplit C into two subcones C (cid:48) = { λ : λ n ≥ λ πn } and C (cid:48)(cid:48) = { λ : λ πn ≥ λ n } and define the induction for each of them separately.If λ ∈ C (cid:48) , we define T (cid:48) : [0 , α n − ( λ π )) → [0 , α n − ( λ π ))to be the first return map induced by T ( λ,π ) on the interval [0 , α n − ( λ π )). A computationshows that T (cid:48) is still an n -interval exchange. The couple of parameters ( λ (cid:48) , π (cid:48) ) ∈ C × S corresponding to T (cid:48) is described as follows. Consider the n × n -matrix A (cid:48) π = − , (6.1)where ( A (cid:48) π ) n,π − n = −
1. Then λ (cid:48) = A (cid:48) π λ and the permutation π (cid:48) is given by π (cid:48) ( j ) = π ( j ) , if π ( j ) ≤ π ( n ), π ( j ) + 1 , if π ( n ) < π ( j ) < n , π ( n ) + 1 , if π ( j ) = n . (6.2)13f λ ∈ C (cid:48)(cid:48) , we define T (cid:48)(cid:48) : [0 , α n − ( λ )) → [0 , α n − ( λ ))by inducing T ( λ,π ) on the interval [0 , α n − ( λ )). Then T (cid:48)(cid:48) is also an n -interval exchange.We consider the n × n -matrix A (cid:48)(cid:48) π = −
10 0 . . . , (6.3)where ( A (cid:48)(cid:48) π ) π − n,n = −
1, and set λ (cid:48)(cid:48) = A (cid:48)(cid:48) π λ . Let the permutation π (cid:48)(cid:48) be given by π (cid:48)(cid:48) ( j ) = π ( j ) , if j ≤ π − ( n ) π ( n ) , if j = π − ( n ) + 1 π ( j − , otherwise. (6.4)We have T (cid:48)(cid:48) = T ( λ (cid:48)(cid:48) ,π (cid:48)(cid:48) ) .The following lemma lets us iterate the inductive process described above. Lemma 6.1 ([12], Lemma 2.4) . Let λ be irrational and π irreducible. Then both λ (cid:48) , λ (cid:48)(cid:48) are irrational and both π (cid:48) , π (cid:48)(cid:48) irreducible. Let π ∈ S be a fixed permutation and define R ( π ) to be the set of all permutations π ∈ S which can be reached by the successive iterations of the Rauzy induction startingat some T ( λ,π ) , λ ∈ C . The set R ( π ) is called the Rauzy class of permutations of π , orthe Rauzy class of π for short.In order to study the possible sequences of permutations arising from this process, weconstruct a directed graph G ( π ) whose nodes are the permutations π ∈ R ( π ). For every π ∈ R ( π ) an arrow goes from π to each of π (cid:48) and π (cid:48)(cid:48) given by (6.2) and (6.4) respectively.For n = 2 we have only one Rauzy class whose graph consists of one node with two loopsattached. The following lemma concerns the structure of the Rauzy graph for n ≥ Lemma 6.2 ([12], Lemma 2.2 and 2.4) . Let π ∈ S . For every π , π ∈ G ( π ) there is apath in G ( π ) starting at π and reaching π . Moreover, every π ∈ G ( π ) has exactly twofollowers and two predecessors in G ( π ) . Let R be a Rauzy class in S . The inductive process described in the previous subsectiondefines an algorithm I acting on the parameter space C × R by I : ( λ, π ) ∈ C × R (cid:55)−→ (cid:26) ( λ (cid:48) , π (cid:48) ) if λ n > λ πn , ( λ (cid:48)(cid:48) , π (cid:48)(cid:48) ) if λ n < λ πn . (6.5)It is called the Rauzy induction of interval exchange transformations.The space
C × R is endowed with Lebesgue measure denoted by µ . Theorem 6.3 ([17], Theorem 1.6) . For every Rauzy class R , the map I is ergodic on C × R with respect to Lebesgue measure.
14n order to illustrate the definition of I , we will now describe explicitly its action inthe easiest cases n = 2 ,
3. In what follows, we represent the permutations in the form π = ( π − (1) , . . . , π − ( n )). In the case of n = 2, we have only one irreducible permutation on two letters, thetransposition (2 , I on the secondcoordinate is thus trivial. On the first coordinate I acts as the Euclidean algorithm E defined by (1.1): I ( λ, (2 , (cid:26) (( λ − λ , λ ) , (2 , , if λ > λ , (( λ , λ − λ ) , (2 , , if λ > λ . We have the corresponding Rauzy graph: 21 For n = 3 we also have an unique Rauzy class that contains all irreducible permutationson three letters, namely (231), (321), (312). The Rauzy induction is decribed as follows: I ( λ, (2 , , (cid:26) (( λ , λ , λ − λ ) , (2 , , , if λ > λ , (( λ − λ , λ , λ ) , (3 , , , if λ > λ , I ( λ, (3 , , (cid:26) (( λ , λ , λ − λ ) , (3 , , , if λ > λ , (( λ − λ , λ , λ ) , (2 , , , if λ > λ , I ( λ, (3 , , (cid:26) (( λ , λ , λ − λ ) , (3 , , , if λ > λ , (( λ , λ − λ , λ ) , (3 , , , if λ > λ . From this expression the Rauzy graph can be deduced.231 321 312 One may check that for n = 4 we get two distinct Rauzy classes of irreducible permu-tations, one generated by (4321) and the other by (3412). We will need more information about permutations within a given Rauzy class. A permu-tation π ∈ S is said to be standard , if π (1) = n and π ( n ) = 1. Lemma 7.1 ([13]) . Every Rauzy class contains a standard permutation.
The notion of standard permutation was rediscovered in [12], where it was noticed thatthe existence of standard permutations in every Rauzy class was a central tool to provethe weak-mixing property of interval exchanges (see also [2]). Here standard permutationswill also be used. First we prove a technical lemma which concerns permutations whichare fixed by Rauzy induction.
Lemma 7.2.
Let π ∈ S be such that π ( n −
1) = n and π (cid:48)(cid:48) be the permutation defined by(6.4). Then π (cid:48)(cid:48) = π . roof. We have π − ( n ) = n −
1. By (6.4), π (cid:48)(cid:48) reduces to π (cid:48)(cid:48) ( j ) = (cid:26) π ( j ) , if j ≤ n − , , if j = n − n, which implies π (cid:48)(cid:48) = π .The above lemma proves that, at such node, the Rauzy graph has a loop. We call looppermutation an irreducible permutation π with π ( n −
1) = n . Lemma 7.3.
Every Rauzy class contains a loop permutation.Proof.
Let R be a Rauzy class. By Lemma 7.1, R contains a standard permutation σ .We will show that there is a loop permutation π in the orbit of σ under I . The idea ofthe proof is depicted in the figures in [12, p.1192] and corresponds to the case i = n − λ ∈ C satisfy λ n > λ + . . . + λ n − . For k ≥ λ ( k ) , σ ( k ) ) = I k ( λ, σ ). Since σ is standard, we have σ ( n ) = 1 < σ ( n − < n = σ (1). For 1 ≤ j ≤ n − σ ( n −
1) one gets λ ( j ) = ( λ , . . . , λ n − , λ n − λ σ − ( n ) − . . . − λ σ − ( n +1 − j ) ) ,σ ( j ) ( n ) = 1 and σ ( j ) ( n −
1) = σ ( n −
1) + j . This implies that π = σ ( n − σ ( n − is a looppermutation. Moreover, since π is obtained as an image of σ by the Rauzy induction I , itbelongs to the same Rauzy class R . Fix a permutation π ∈ S . For every irrational λ ∈ C , one may consider the iterations I k ( λ, π ) = ( λ ( k ) , π kλ ) of the Rauzy induction algorithm (6.5). This generates an infinitesequence of permutations ( π λ ) = π , π λ , . . . , π kλ , . . . (8.1)which is an infinite path in the Rauzy graph G ( π ). Together with ( π λ ) we get an infinitesequence of matrices A λ , A λ , . . . , A kλ , . . . , such that λ ( k ) = A kλ · · · A λ λ , for k ≥
1. We set A ( k ) λ = A kλ · · · A λ A λ and B ( k ) λ = ( A ( k ) λ ) − , where B ( k ) λ is a non-negative matrix.Conversely, to any infinite path π , π , π , . . . in G ( π ), there corresponds a nonemptyclosed subset of vectors λ ∈ C that generate that path via Rauzy induction. To be moreprecise, let π , π , . . . , π k be a finite path in G ( π ) and define C π ,...,π k π = { λ ∈ C : π iλ = π i , ≤ i ≤ k } . Every λ ∈ C π ,...,π k π generates the same beginning of the sequence of matrices A λ , A λ , . . . , A kλ .The set C π ,...,π k π is an Euclidean cone satisfying C π ,...,π k π = B ( k ) λ ( C ) = { α l ( k )1 ( λ ) + . . . + α n l ( k ) n ( λ ) : α i ≥ , ≤ i ≤ n } , l ( k )1 ( λ ) , . . . , l ( k ) n ( λ ) stand for the column-vectors of the matrix B ( k ) λ .Since every permutation in G ( π ) has exactly two followers, the finite path π , π , . . . , π k may be continued in two different ways, choosing π k +1 = ( π k ) (cid:48) or π k +1 = ( π k ) (cid:48)(cid:48) . Thischoice splits the cone C π ,...,π k π into two nontrivial subcones C π ,...,π k , ( π k ) (cid:48) π and C π ,...,π k , ( π k ) (cid:48)(cid:48) π .For k ≥ P ( k ) ( π ) be the family of cones C π ,...,π k π , where π , . . . , π k runsthrough all possible k -paths in G ( π ) starting at π . The family P ( k ) ( π ) forms a parti-tion of C ×{ π } into 2 k subcones and P ( k +1) ( π ) is a refinement of P ( k ) ( π ) for every k ≥ λ ∈ C be irrational. For each k ≥
1, we denote by C ( k ) ( λ, π ) the unique subconeof the partition P ( k ) ( π ) which contains ( λ, π ). We need the following result. Lemma 8.1 ([6], Corollary 1.9) . Let π ∈ S . There is a positive constant c = c ( π ) suchthat for almost every λ ∈ C there are infinitely many integers k ≥ with1. π kλ = π ,2. max i (cid:107) l ( k ) i ( λ ) (cid:107) min i (cid:107) l ( k ) i ( λ ) (cid:107) ≤ c ( π ) , where l ( k )1 ( λ ) , . . . , l ( k ) n ( λ ) are the column-vectors of the matrix B ( k ) λ which generate the subcone C ( k ) ( λ, π ) . Corollary 8.2.
Let π ∈ S n . There exists a partition of C × { π } whose elements aresubcones C ( k ) ( λ, π ) which satisfy Lemma 8.1.Proof. The argument is easily adapted from the one used in the proof of Corollary 3.3.The next lemma is equivalent to the unique ergodicity of almost every interval ex-change.
Lemma 8.3 ([6], Theorem 1.10, [7], [16]) . Let π ∈ S . For almost every λ ∈ C , ∩ k ≥ C ( k ) ( λ, π ) = { αλ : α ≥ } . For π ∈ S let P ( π ) be a partition of C × { π } given by Corollary 8.2. The followinglemma is a generalization of Corollary 3.3 to the case of Rauzy induction. Lemma 8.4.
Let π ∈ S be a loop permutation. For every N ≥ there exists a partition P N of C × { π } which is a refinement of the partition P ( π ) and satisfies the followingproperties:1. its elements are subcones of type C ( k ) ( λ, π ) ,2. (cid:107) l ( k ) n ( λ ) (cid:107) (cid:107) l ( k ) n − ( λ ) (cid:107) > N , where l ( k )1 ( λ ) , . . . , l ( k ) n ( λ ) are the column-vectors of the matrix B ( k ) λ which generate the subcone C ( k ) ( λ, π ) .Proof. Let C (cid:48) × { π } = C ( k ) ( λ, π ) × { π } be an element of the partition P ( π ) and l , . . . , l n be the column-vectors of the matrix B = B ( k ) λ which defines the subcone C (cid:48) = B ( C ).We recall that π kλ = π (see Lemma 8.1). Since π is a loop permutation, we may continuethe path π, π λ , . . . , π kλ choosing π j = π for k + 1 ≤ j ≤ k + N . Let C (cid:48) N be the subcone of C (cid:48) corresponding to this path. It is generated by a matrix B N whose column-vectors are l , . . . , l n − , l n + N l n − . We have (cid:107) l n + N l n − (cid:107) (cid:107) l n − (cid:107) = N + (cid:107) l n (cid:107) (cid:107) l n − (cid:107) ≥ N + 1 c ( π ) .
17o show that almost every λ ∈ C (cid:48) belongs to such a cone, we will show that the cone C (cid:48) N occupies a large proportion of the volume of the cone C (cid:48) . To this end, let∆ n − = { x ∈ C : x + . . . + x n = 1 } (8.2)be the ( n − C (cid:48) and C (cid:48) N on ∆ n − .By [8, Lemma 3.2], the ratio of the volumes of the projections of C (cid:48) and C (cid:48) N equals (cid:107) l n (cid:107) · · · (cid:107) l n (cid:107) (cid:107) l (cid:107) · · · (cid:107) l n − (cid:107) (cid:107) l n + Kl n − (cid:107) . It is bounded from below by Nc ( π ) , which is a constant independent of the initial cone C (cid:48) . We may thus deduce that iterating the same construction on C (cid:48) \ C (cid:48) N will result in adesired partition of C (cid:48) . We will adapt the proof of Theorem 1.1 to the multidimensional case of Rauzy induction.Let R be a Rauzy class and π ∈ R be a loop permutation. Let Ω ⊂ C × R be a set ofpositive measure which, without loss of generality, is assumed to be contained in C × { π } .As the map I is non-singular and ergodic, in order to prove that it is exact, according toLemma 2.1, it suffices to show that there exists k ≥ µ ( I k +1 (Ω) ∩ I k (Ω)) > . (9.1)Let λ ∈ Ω be a Lebesgue density point of Ω satisfying Lemma 8.3. Let ρ > λ and radius ρ is entirely contained in C . Let D ρ = { λ + x : x ∈C , x λ + . . . + x n λ n = 0 , (cid:107) x (cid:107) ≤ ρ } which is an ( n − λ ofradius ρ . Next we define a section of a cylindrical coneΣ( λ , ρ ) = { tλ : λ ∈ D ρ , δ ≤ t ≤ } , where δ = 1 − ρ (cid:107) λ (cid:107) . Given ε >
0, for ρ > µ (Ω ∩ Σ( λ , ρ )) > (1 − ε ) µ (Σ( λ , ρ )) , by Lebesgue density theorem.Let N ≥ P N ( π ) be a partition of C × { π } given by Lemma 8.4. The setsΣ( λ , ρ ) ∩ C N , C N ∈ P N ( π ) , partition Σ( λ , ρ ) into a family of polyhedral slices. In virtue of Lemma 8.3, taking arefinement of P N ( π ) if necessary, one may assume that there exists a subcone C N ∈ P N ( π )such that µ (Ω ∩ Σ( λ , ρ ) ∩ C N ) ≥ (1 − ε ) µ (Σ( λ , ρ ) ∩ C N )(see the proof of Lemma 4.3).Recall that C N = C ( k ) ( λ, π ) for some k ≥ λ ∈ C . In particular, this implies I k ( C N × { π } ) = C × { π } . l , . . . , l n be the column-vectors of the matrix B ( k ) λ generating the polyhedral cone C N .As in the case of the Euclidean algorithm, the slice Σ( λ , ρ ) ∩ C N may be described asΣ( λ , ρ ) ∩ C N = { x t l + . . . + x n t n l n : x i ≥ , x + . . . + x n = 1 , α i ≤ t i ≤ β i } , where α i β i = δ for 1 ≤ i ≤ n and α n − α n = β n − β n ≥ N , (9.2)if ρ is small enough (see Lemma 4.4).The matrix A ( k ) λ associated to C N sends the vectors l , . . . , l n onto the canonical basisof R n . Let P stand for the polyhedral slice A ( k ) λ (Σ( λ , ρ ) ∩ C N ). We have I k ((Σ( λ , ρ ) ∩ C N ) × { π } ) = P × { π } and P = { ( x t , . . . , x n t n ) ∈ C : x i ≥ , x + . . . + x n = 1 , α i ≤ t i ≤ β i } . A calculation shows that µ ( P ) = 1 n ! β β · · · β n − n ! α α · · · α n = 1 n ! (1 − δ n ) β β · · · β n . As in the case of the Euclidean algorithm, we are interested in the subset P + of P defined by P + = { λ ∈ P : λ n − > λ n } . In virtue of (9.2), a calculation analogous to (4.3)gives µ ( P + ) ≥ NN + 2 µ ( P ) . For N large enough we may thus assume µ (Ω ∩ P + ) ≥ (1 − ε ) µ ( P + ) . We want to show that I ( P + × { π } ) intersects P + × { π } and that the volume of thisintersection is large enough to imply (9.1). First, since P + is contained in the set { λ ∈ C : λ n − > λ n } and π is a loop permutation, we have I ( P + ×{ π } ) ⊂ C ×{ π } . It is thus enoughto show the intersection property on the first coordinate. To this end, we remark that theonly coordinate that changes when applying I on P + is λ n − . Moreover, the action on thecouple of coordinates ( λ n − , λ n ) corresponds to that of the Euclidean algorithm E . Theargument of Lemma 4.5 is then valid also in this case. We get µ ( P + × { π } ∩ I ( P + × { π } )) µ ( P + × { π } ) → N → ∞ . Choosing N large enough we get the intersection property (9.1). The Rauzyinduction is exact with respect to Lebesgue measure. Theorem 1.2 is proved. (cid:50)
10 Remarks
In many cases, in particular in [16], instead of the homogenous algorithm I defined by(6.5), a normalized version is considered, for example its radial projection on the simplex∆ n − (8.2), ˜ I ( λ, π ) ∈ ∆ n − × R (cid:55)−→ ( λ (cid:48) (cid:107) λ (cid:48) (cid:107) , π (cid:48) ) ∈ ∆ n − × R . The map ˜ I is conservative and ergodic with respect to Lebesgue measure on the simplex∆ n − . The following result may be deduced from Theorem 1.2.19 orollary 10.1. The map ˜ I is exact with respect to Lebesgue measure. For completeness, we give examples of multidimensional continued fraction algorithmswhich are adapted to our approach and should satisfy the intersection property.First we define the map σ : λ ∈ C (cid:55)→ σ ( λ ) = ( λ σ λ (1) , . . . , λ σ λ ( n ) ) ∈ C , where σ λ arranges the coordinates λ , . . . , λ n in non decreasing order. We recall that, if λ is irrational, the permutation σ λ is unique.Let 1 ≤ i ≤ n − T i be the homogeneous algorithm given by λ ∈ C (cid:55)→ σ − λ ( λ σ λ (1) , . . . , λ σ λ ( n − , λ σ λ ( n ) − λ σ λ ( i ) ) ∈ C . The map T i is nonsingular and dissipative. The subcones C j = { λ ∈ C : σ λ ( j ) = i } , for1 ≤ j ≤ n , define a partition of the cone C . The map T i satisfies a Markov partitionproperty: T i ( C j ) = C , for j = 1 , . . . , n .The map T n − is known as homogeneous Brun algorithm (see [15, p.45]) and the map T is the called homogeneous Selmer algorithm (see [15, p.45]). Our definitions coincidewith those of the reference up to a permutation, however, as far as ergodicity and exactnessare concerned, they bare the same properties.Another example of multidimensional algorithm that could be studied this way is the Jacobi-Perron algorithm . Following [15, p.24], it is convenient to define it on a subcone of C . Let ˜ C = { λ ∈ C : λ > , λ ≥ λ i , ∀ ≤ i ≤ n } and [ x ] be the integer part of x ∈ R .The Jacobi-Perron algorithm, denoted by J , is the map defined by J : λ ∈ ˜ C (cid:55)→ J ( λ ) = ( λ , λ − a λ , . . . , λ n − a n − λ , λ − a n λ ) ∈ C , where a j = [ λ j +1 /λ ], for 2 ≤ j ≤ n − a n = [ λ /λ ].The Jacobi-Perron algorithm is ergodic and has a finite invariant measure absolutelycontinuous with respect to Lebesgue measure. Although it is not homogeneous, it maybe seen as a suitable first return time of a homogeneous algorithm defined in [11, Section3.1]. This underlying algorithm is adapted to our approach and we conjecture that J isexact. References [1] Jon Aaronson.
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Ann. Inst. Fourier (Grenoble) , 46(2):325–370, 1996.Tomasz Miernowski and Arnaldo NogueiraInstitut de Math´ematiques de Luminy163, avenue de Luminy, Case 90713288 Marseille Cedex 9, FranceE-mail: [email protected] and [email protected]@iml.univ-mrs.fr