Metrical theory for α-rosen fractions

Transcription

1 J Eur Math Soc, c European Mathematical Society 2009 Karma Dajani Cor Kraaikamp Wolfgang Steiner Metrical theory for α-rosen fractions Received February 5, 2007 and in revised form December 20, 2007 Abstract The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions In this paper we introduce a new class of continued fractions related to the Rosen fractions, the α-rosen fractions The metrical properties of these α-rosen fractions are studied We find planar natural extensions for the associated interval maps, and show that their domains of definition are closely related to the domains of the classical Rosen fractions This unifies and generalizes results of diophantine approximation from the literature Keywords Rosen fractions, natural extension, Diophantine approximation Introduction Although David Rosen Ros] introduced as early as 954 an infinite family of continued fractions which generalize the nearest-integer continued fraction, it is only very recently that the metrical properties of these so-called Rosen fractions have been investigated; see eg Schm], N2], GH] and BKS] In this paper we will introduce α-rosen fractions, and study their metrical properties for special choices of α These choices resemble Nakada s α-expansions, in fact for q = 3 these are Nakada s α-expansions; see also N] To be more precise, let q Z, q 3, and λ = λ q = 2 cos π/q Then we define for α /2, /λ] the map T α : λ(α ), λα] λ(α ), λα) by T α (x) := x λ xλ, + α x = 0, () and T α (0) := 0 Here, ξ denotes the floor (or entier) of ξ, ie, the greatest integer smaller than or equal to ξ In order to have positive digits, we demand that α /λ Setting d(x) = xλ + α (with d(0) = ), ε(x) = sgn(x), and more generally ε n (x) = ε n = ε(t n α (x)) and d n (x) = d n = d(t n α (x)) (2) K Dajani: Department of Mathematics, Utrecht University, Postbus 80000, 3508 TA Utrecht, the Netherlands; kdajani@uunl C Kraaikamp: TU Delft, EWI (DIAM), Mekelweg 4, 2628 CD Delft, the Netherlands; ckraaikamp@tudelftnl W Steiner: LIAFA, CNRS UMR 7089, Université Paris Diderot Paris 7, Case 704, Paris Cedex 3, France; steiner@liafajussieufr Mathematics Subject Classification (2000): Primary 28D05; Secondary K55

2 260 Karma Dajani et al for n, one obtains for x I q,α := λ(α ), αλ] an expression of the form x = d λ + ε d 2 λ + + where ε i {±, 0} and d i N { } Setting R n S n = d λ + ε we will show in Section 23 that and for convenience we will write ε 2 d 2 λ + + ε n d n λ ε x = ε 2 d λ + d 2 λ + ε 2 ε n d n λ + T n α (x), =: ε : d, ε 2 : d 2,, ε n : d n ], (3) R n lim = x, n S n =: ε : d, ε 2 : d 2, ] (4) We call R n /S n the nth α-rosen convergent of x, and (4) the α-rosen fraction of x The case α = /2 yields the Rosen fractions, while the case α = /λ is the Rosen fraction equivalent of the classical regular continued fraction expansion (RCF) In case q = 3 (and /2 α /λ), the above defined α-rosen fractions are in fact Nakada s α- expansions (and the case α = /λ = is the RCF) Already from BKS] it is clear that in order to construct the underlying ergodic system for any α-rosen fraction and the planar natural extension for the associated interval map T α, it is fundamental to understand the orbit under T α of the two endpoints λ(α ) and λα of X = X α := λ(α ), λα] Although the situation is in general more complicated than the classical case from BKS], the natural extension together with the invariant measure can be given, and it is shown that this dynamical system is weakly Bernoulli Using the natural extension, metrical properties of the α-rosen fractions will be given in Section 4 2 Natural extensions In this section we find the smallest domain α R 2 on which the map ( T α (x, y) = T α (x), d(x)λ + ε(x)y ), (x, y) α, (5) is bijective ae We will deal with the general case, resembling Nakada s α-expansions, ie, /2 α /λ and λ = λ q = 2 cos π/q for some fixed q Z, q 4 (the case q = 3 is in fact the case of Nakada s α-expansions; see also N]) As in BKS], we need to discern between odd and even q s, but some properties are shared by both cases, and these are collected here first

3 Metrical theory for α-rosen fractions 26 For x λ(α ), λα], setting ( ) 0 εi A i = and M d i λ n = A A n = ( ) Kn R n, L n R n it immediately follows from M n = M n A n that K n = R n, L n = S n, and R :=, R 0 := 0, R n = d n λr n + ε n R n 2 for n =, 2,, S := 0, S 0 :=, S n = d n λs n + ε n S n 2 for n =, 2,, (6) if d n < For a matrix A = ( ) a b c d with det(a) = 0, we define the corresponding Möbius (or fractional linear) transformation by A(x) = ax + b cx + d Consequently, considering M n as a Möbius transformation, we find that M n (0) = R n S n, and M n (0) = A A n (0) = = ε : d,, ε n : d n ] It follows that the numerators and denominators of the α-rosen convergents of x from (3) satisfy the usual recurrence relations (6); see also BKS, p 279] Furthermore, since ( ) 0 εn x = M n d n λ + Tα n(x) (0), we have x = R n + T n α (x)r n S n + T n α (x)s n and T n α (x) = R n S n x S n x R n (7) Let l 0 = (α )λ be the left endpoint of the interval on which the continued fraction map T α was defined in (), r 0 = αλ its right endpoint, and let (ε : d) = {x (α )λ, αλ] ε (x) = ε, d (x) = d} be the cylinders of order of numbers with same first digits given by (4) If we set δ d = (α + d)λ for all d, then the cylinders are given by the following table: ( : ) ( : d), d 2 (0 : ) (+ : d), d 2 (+ : ) l 0, δ ) δ d, δ d ) {0} (δ d, δ d ] (δ, r 0 ] where we have used the fact that r 0 > δ since λ 2 for q 4 Note that by definition for all x (ε, d), x = 0 T α (x) = ε/x λd

4 262 Karma Dajani et al Setting l n = T n α (l 0), r n = T n α (r 0), n 0, we have r = αλ λ = αλ2 < 0 αλ In case α = /2, we write φ n instead of l n, for n 0 In BKS], it was shown that { ( : ) p ] if q = 2p, λ/2 = ( : ) h, : 2, ( : ) h ] if q = 2h + 3, from which it immediately follows that φ 0 = λ 2 < φ < < φ p 2 = λ < φ p = 0 if q = 2p, (8) and that for q = 2h + 3, φ 0 = λ 2 < φ < < φ h 2 < φ h < 2 3λ < φ h < 2 5λ, φ 0 < φ h+ < φ, φ h+ = λ, and see also Figure φ h+ < φ h+2 < < φ 2h = λ < 2 3λ < φ 2h+ = 0; λ/2 δ δ 2 δ 2 δ λ/2 λ/2 δ δ 2 δ 2 δ λ/2 Fig The map T /2 and the orbit of λ/2 (dashed broken line) for q = 8 (left) and q = 7 (right) Thus we see that the behavior of the orbit of λ/2 is very different in the even case compared to the odd case; see also Figure 2, where the relevant terms of (φ n ) n 0, (l n ) n 0, and (r n ) n 0 are displayed for even q Direct verification yields the following lemma

5 Metrical theory for α-rosen fractions 263 Lemma 2 For q 4 and /2 α /λ, we have φ 0 = λ 2 l 0 r < φ = λ2 2, λ with φ 0 = l 0 if and only if α = /2, and l 0 = r if and only if α = /λ In BKS], the sequence (φ n ) n 0 plays a crucial role in the construction of the natural extension of the Rosen fractions Due to the fact that the orbits of both λ/2 and λ/2 would become constant 0 after a finite number of steps (depending on q), the natural extension of the Rosen fraction could be easily constructed In this paper, the (l n ) n 0 and (r n ) n 0 play a role comparable to that of the sequence (φ n ) n 0 (even though the φ n s are frequently used as well) Let x λ(α ), αλ] be such that (ε n (x) : d n (x)) = ( : ) for n =,, m Then it follows from (6) that the α-rosen convergents of x satisfy R =, R 0 = 0, R n = λr n R n 2 for n =,, m, S = 0, S 0 =, S n = λs n S n 2 for n =,, m As in BKS], we define the auxiliary sequence (B n ) n 0 by B 0 = 0, B =, B n = λb n B n 2 for n = 2, 3, (9) This shows, for n =,, m, that R n = B n, S n = B n+, and T n α (x) = B n+b n+ x B n +B n x by (7) It follows that l n = B n + B n+ (α )λ B n + B n (α )λ = B n+αλ B n+2 B n αλ B n+ if l 0 = ( : ) n, ] (0) For x = + :, ( : ) n, ] = ( : ) n, ], we obtain similarly R n = B n, S n = B n+, thus T n α (x) = B n B n+ x B n x B n and r n = B n+αλ B n B n αλ B n if r 0 = + :, ( : ) n, ] () It is easy to see that B n = sin nπ q / sin π q (see also W, Equation 5 in Section 44]) Clearly B n, n 0, is a periodic sequence, with period length q 2 Even indices Let q = 2p, p N, p 2 Essential in the construction of the natural extension is the following theorem

6 264 Karma Dajani et al Theorem 22 Let q = 2p, p N, p 2, and let the sequences (l n ) n 0 and (r n ) n 0 be defined as before If /2 < α < /λ, then l 0 < r < l < < r p 2 < l p 2 < δ < r p < 0 < l p < r 0, (2) d p (r 0 ) = d p (l 0 ) + and l p = r p If α = /2, then If α = /λ, then l 0 < r = l < < r p 2 = l p 2 < δ < r p = l p = 0 < r 0 (3) l 0 = r < l = r 2 < < l p 2 = r p = δ < 0 < r 0 (4) Proof If α = /2, then l 0 = φ 0 and r 0 = φ 0, hence (3) is an immediate consequence of (8) In general, in view of Lemma 2 and the fact that φ 0 = ( : ) p ], we have the following situation: T α (l 0, φ )) = l, φ 2 ) and T α (φ j, φ j )) = φ j, φ j+ ) for j = 2,, p 2 (cf Figure 2) This yields l 0 = ( : ) p 2, ] φ 0 l 0 r φ l r 2 φ 2 φ p 3 l p 3 r p 2 φ p 2 δ 0 r 0 Fig 2 The relevant terms of (φ n ) n 0, (l n ) n 0, and (r n ) n 0 for even q Since sin (p )π 2p By (0), we have therefore = sin (p+)π 2p, we obtain B p = B p+, B p = λ ( λ 2 ) 2 B p, B p 2 = 2 B p l p 2 = B p αλ B p 2 αλ 2 = B p 2 αλ B p λ( αλ 2 + 2α) (α + )λ = δ, with l p 2 = δ if and only if α = /λ For α = /λ, we clearly have r 0 =, thus r = λ = l 0 and (4) is proved If /2 < α < /λ, then l p 2 < δ, hence l 0 = ( : ) p,, ( : d p ), ] with d p, and again due to (0) we obtain l p = B pαλ B p (2α )λ = B p αλ B p 2 αλ 2 > 0 Similarly, r 0 = + :, ( : ) p 2, ] and, by (), r p = B pαλ B p (2α )λ = B p αλ B p 2 2 ( α)λ 2 ( δ, 0),

7 Metrical theory for α-rosen fractions 265 thus (2) is proved Since r p l p = λ, it follows from the definition of T α that l p = r p and d (r p ) = d (l p ) + With d p (x) = d (T p α (x)), the theorem is proved Remark The structure of the l n s and r n s allows us to determine all possible sequences of digits For example, the longest consecutive sequence of digits ( : ) contains p terms if α < /λ, since l p 2 < δ and l p δ In case α = /λ, we only have ( : ) p 2 In particular in case q = 4, α = /λ, the cylinder ( : ) is empty On the other hand, (+ : ) is always followed by ( : ) p 2 since r p 2 < δ, with (+ : ) = {r 0 } in case α = /λ Now we construct the domain α upon which T α is bijective Theorem 23 Let q = 2p with p 2 Then the system of relations (R ) : H = /(λ + H 2p ), (R 2 ) : H 2 = /λ, (R n ) : H n = /(λ H n 2 ) for n = 3,, 2p, (R 2p ) : H 2p 2 = λ/2, (R 2p+ ) : H 2p 3 + H 2p = λ, admits the (unique) solution H 2n = φ p n = B n B n+ = sin H 2n = B p n B p+ n = cos B p n B p n nπ 2p sin (n+)π 2p nπ 2p cos (n )π 2p cos (n+)π 2p cos nπ 2p for n =,, p, for n =,, p, in particular H 2p = Let /2 < α < /λ and α = 2p n= J n 0, H n ] with J 2n = l n, r n ), J 2n = r n, l n ) for n =,, p, and J 2p = l p, r 0 ) Then the map T α : α α given by (5) is bijective off a set of Lebesgue measure zero Proof It is easily seen that the solution of this system of relations is unique and valid, and that T α is injective We thus concern ourselves with the surjectivity of T α ; see also Figure 3 By (2), we have J n 2 ( : ) for n = 3,, 2p 2, thus ] T α (J n 2 0, H n 2 ]) = J n λ, = J n H 2, H n ], λ H n 2 where we have used (R 2 ) and (R n ) Furthermore, (R 2p ) gives ] T α (l p 2, δ ) 0, H 2p 3 ]) = l p, r 0 ) λ, = J 2p H 2, H 2p ] λ H 2p 3

8 266 Karma Dajani et al H 5 H 5 H 4 H 4 H 3 H 2 H 3 H 2 H H /2λ /3λ l 0 r l δ δ 2 r 2 δ3 0 δ 2 l 2 δ r 0 l 0 r l r 2 0 l 2 l 3 =r 3 r 0 Fig 3 The natural extension domain α (left) and its image under T α (right) of the α-rosen continued fraction (δ n = δ n ); here q = 6, α = 053, d p (l 0 ) = 2, d p (r 0 ) = 3 For n = 2,, d (r p ) = d p (r 0 ), we have ] T α ( δ n, δ n ) 0, H 2p 3 ]) = l 0, r 0 ) nλ, nλ H 2p 3 The remaining part of the rectangle J 2p 3 0, H 2p 3 ] is mapped to T α ( δ dp (r 0 ), r p ) 0, H 2p 3 ]) = l 0, r p ) d p (r 0 )λ, d p (r 0 )λ H 2p 3 Now consider the image of J 2p 0, H 2p ] If d p (l 0 ) 2, then it is split into ] T α ((l p, δ dp (l 0 ) ] 0, H 2p ]) = l 0, r p ),, d p (l 0 )λ + H 2p d p (l 0 )λ T α (( δ n, δ n ] 0, H 2p ]) = l 0, r 0 ), ] nλ + H 2p nλ for n = 2,, d p (l 0 ), T α ((δ, r 0 ) 0, H 2p ]) = (r, r 0 ), ] = (r, r 0 ) H, H 2 ], λ + H 2p λ where we have used (R ) Since H 2p 3 + H 2p = λ and d p (r 0 ) = d p (l 0 ) +, the different parts of T α ( δ, r p ) 0, H 2p 3 ]) and T α ((l p, δ ] 0, H 2p ]) lie one under the other and fill up like a jig-saw puzzle the set ( l 0, r p ) d p (r 0 )λ, d p (l 0 )λ In case d p (l 0 ) =, we simply have ]) ( l 0, r 0 ) ]) d p (r 0 )λ, H T α ( δ, r p ) 0, H 2p 3 ]) = l 0, r p ) /(2λ), H ], T α ((l p, r 0 ) 0, H 2p ]) = (r, r p ) H, H 2 ] ]

9 Metrical theory for α-rosen fractions 267 Finally, the image of the central rectangle J 2p 2 0, H 2p 2 ] is split into T α (r p, δ dp (r 0 )) 0, H 2p 2 ]) = r p, r 0 ) d p (r 0 )λ, ] T α ( δ n, δ n ) 0, H 2p 2 ]) = l 0, r 0 ) nλ, nλ H 2p 2 T α ((δ n, δ n ] 0, H 2p 2 ]) = l 0, r 0 ), ] nλ + H 2p 2 nλ T α ((δ dp (l 0 ), l p ] 0, H 2p 2 ]) = r p, r 0 ) d p (r 0 )λ H 2p 2 d p (l 0 )λ + H 2p 2, Since H 2p 2 = λ/2 and d p (r 0 ) = d p (l 0 ) +, the union of these images is ( ( l 0, r 0 ) 0, ]) ( r p, r 0 ) d p (r 0 )λ d p (r 0 )λ, d p (l 0 )λ ], for n > d p (r 0 ), for n > d p (l 0 ), ] d p (l 0 )λ ]) Therefore T α ( α ) and α differ only by a set of Lebesgue measure zero Remark If α = /2, then the intervals J 2n are empty and r p = l p = 0 The proof of Theorem 23 remains valid, with d (r p ) = d (l p ) = ; see also BKS] Since l n = φ n for n = 0,,, p, we have p ( /2 = n= B n B n+, B n B n ) 0, B ]) ( n B n+ 0, λ ) ) 0, ] B n B n 2 For α = /λ, we just have the intervals J 2n, n =,, p 2 and add J 2p 2 = r p, r 0 ) (= δ, )) Furthermore, r n = B n B n+ B n B n for n =,, p and r 0 = B p B p+ B p B p = This provides the following theorem Theorem 24 Let q = 2p with p 2 and p Bn B n+ /λ = n= B n B n, B n+ B n+2 B n+ B n ) 0, B n B n+ Then T /λ : /λ /λ is bijective off a set of Lebesgue measure zero Proof By (4), we have l 0 = r < l = r 2 < < l p 2 = r p = δ, thus ) ( Bn B n+ T /λ, B n+ B n+2 B n B n B n+ B n Bn+ B n+2 =, B n+2 B n+3 B n+ B n B n+2 B n+ 0, ) B n B n+ ]) λ, B n+ B n+2 ] ] for n =,, p 2

10 268 Karma Dajani et al The different parts of r p, r 0 ) are mapped to T /λ ( δ n, δ n ) 0, λ ]) ] = λ, ) 2 nλ, 2 (2n )λ T /λ ((δ n, δ n ] 0, λ ]) 2 = λ, ) 2 (2n + )λ, ] nλ T /λ ((δ, ) 0, λ ]) 2 = λ, ) 2 3λ, ] λ and the union of these images is λ, ) (0, /λ] for n = 2, 3,, for n = 2, 3,, Remark Note that there is a simple relation between /2 and /λ, which will be useful in Section 3; reflect /2 in the line y = x in case x 0, and reflect /2 in the line y = x in case x 0, to find /λ ; see also Figure 4 λ λ 2 φ H λ 2 φ 2 3λ 0 2 3λ λ 2 λ H = +λ 0 +λ Fig 4 /2 (left) and /λ (right); here q = 6 As in BKS], a Jacobian calculation shows that T α preserves the probability measure ν α with density C q,α /( + xy) 2, where C q,α is a normalizing constant For the calculation of this constant, we need the following lemma Lemma 25 If m m 2 = m 3 m 4, then B n+m B n+m2 B n+m3 B n+m4 = B m m 3 B m2 +m 3 for all n Z Proof With ζ = exp(πi/q), we have B n+m B n+m2 B n+m3 B n+m4 = (ζ n+m ζ n m )(ζ n+m 2 ζ n m 2) (ζ n+m 3 ζ n m 3)(ζ n+m 4 ζ n m 4) (ζ ζ ) 2 = ζ m +m 2 ζ m m 2 ζ m 3+m 4 ζ m 3 m 4 (ζ ζ ) 2 = B m m 3 B m2 +m 3 Proposition 26 For /2 α /λ, the normalizing constant is C q,α = / log + cos π q sin π q

11 Metrical theory for α-rosen fractions 269 Proof Similarly to BKS], integration of the density over α gives C q,α = / ( p + r0 ) + r n H 2n + l n H 2n log + l p + l n= n H 2n + r n H 2n for /2 < α < /λ, by Theorem 23 Using (0), () and Lemma 25, we find + r n H 2n = B n B n αλ, + l n H 2n B n αλ B n + l n H 2n + r n H 2n = B nαλ B n B n+ B n αλ for n =,, p, and Putting everything together, we obtain + r 0 = B p B p αλ + l p B p B p C q,α = / log B p B p = / log sin π q cos π q = / log + cos π q sin π q For α = /2, we have the same constant by the remark following Theorem 23 and by BKS] Finally, the remark following Theorem 24 shows that C q,/λ is the same constant as well Let µ α be the projection of ν α on the first coordinate, B the restriction of the twodimensional σ -algebra on α, and B the Lebesgue σ -algebra on I q,α = λ(α ), αλ] In Roh], Rohlin introduced and studied the concept of natural extension of a dynamical system In our setting, a natural extension of (I q,α, B, µ α, T α ) is an invertible dynamical system (X α, B Xα, ρ α, S α ), which contains (I q,α, B, µ α, T α ) as a factor, such that B Xα = n 0 Sn α π B, where π is the factor map A natural extension is unique up to isomorphism We have the following theorem Theorem 27 Let q 4, q = 2p, and let /2 α /λ Then the dynamical system ( α, B, ν α, T α ) is the natural extension of the dynamical system (I q,α, B, µ α, T α ) Proof Let π : α I q,α be the projection onto the first coordinate An easy calculation shows that π T α = T α π, µ α = ν α π, and π B B so that π is a factor map It remains to show that B = T n B (5) n 0 α π For each admissible block (ε, d ),, (ε n, d n ), define n ((ε, d ),, (ε n, d n )) = (ε, d ) Tα (ε 2, d 2 ) Tα (n ) (ε n, d n )

12 270 Karma Dajani et al dλ+ T (l p, δ d ] 0, H 2p ]) T ( δ d, r p ] 0, H 2p 3 ]) T (δ d, l p ] 0, H 2p 2 ]) T (r p, δ d+ ] 0, H 2p 2 ]) dλ dλ+h 2p 2 (d+)λ l 0 r 0 lp = r p Fig 5 Blow-up of the relevant part of α for even q, /2 < α < /λ The intervals n defined above are called fundamental intervals of order n Since T α is expanding, the Lebesgue measure of n ((ε, d ),, (ε n, d n )) tends to 0 as n for any admissible sequence (ε, d ), (ε 2, d 2 ), Thus, the collection P = { n ((ε, d ),, (ε n, d n )) : n, with (ε, d ),, (ε n, d n ) admissible} generates B, ie σ ( n 0 T α np ) = B Let P α = π P To prove (5) it is enough to show that n 0 T α np α generates B, which is equivalent to showing that n 0 T α np α separates points of α To do this, we first study the action of Tα on α From Theorem 23, one sees that Tα must take horizontal stripes to vertical stripes, so we need a partition in the vertical direction Unfortunately, it is not always possible to find a uniform partition on the y-axis that works for all x; see Figure 5 Instead, we partition per fiber To be more specific, for each x I q,α, let D(x) = {y (x, y) α }, so D(x) is the fiber over x Consider the following partition of D(x): H 2, H 2p 2 ] if x l p, H 2, H 2p 3 ] if r p x < l p, H # 2, H 2p 4 ] if l p 2 x < r p 2, (,, x) = H 2, H 3 ] if l x < r 2, if x < l, { # H, H (,, x) = 2 ] if x r, if x < r For (ε, d) {(, d p (r 0 )), (, d p (l 0 ))}, set ] # (, d, x) = dλ,, # (, d, x) = dλ H 2p 2 Finally, ] # d p (r 0 )λ, (d p (r 0 ) )λ + (, d p (r 0 ), x) = ] d p (r 0 )λ, (d p (r 0 ) )λ + H 2p 2, ] dλ + H 2p dλ if x < l p, if x l p,

13 Metrical theory for α-rosen fractions 27 ] # d p (l 0 )λ +, d p (l 0 )λ (, d p (l 0 ), x) = ], d p (l 0 )λ + H 2p 2 d p (l 0 )λ if x < l p, if x l p One can give T α explicitly: T α (x, y) = ( ε dλ + x, ε ( )) y dλ if y # (ε, d, x) From the definitions of T α and Tα one sees that T α is expanding in the x-direction, while Tα is expanding in the y-direction Now, let (x, y) and (x, y ) be two distinct elements of α If we have x = x, then there exist two distinct fundamental intervals n ((ε, d ),, (ε n, d n )) and n ((ε, d ),, (ε n, d n )) such that (x, y) π n((ε, d ),, (ε n, d n )) and (x, y ) π n((ε, d ),, (ε n, d n )), ie they belong to different elements of P α Suppose now that x = x but y = y Since Tα is expanding in the y-coordinate, there exist n 0 and (ε, d ),, (ε n+, d n+ ), (ε n+, d n+ ) such that (i) (ε n, d n ) = (ε n, d n ), (ii) Tα j (x, y), Tα j (x, y ) T α π (ε j, d j ) for j = 0,, n (this is void if n = 0), (x, y) T α π (ε n, d n ) and Tα n (x, y ) T α π (ε n, d n ) (iii) T n α Then (x, y) Tα n+ π (ε n, d n ) Tα n π (ε n+, d n+ ) T α π (ε 0, d 0 ), (x, y ) Tα n+ π (ε n, d n ) T α n π (ε n+, d n+ ) T α π (ε 0, d 0 ) Thus, (x, y) and (x, y ) belong to different elements of n Z T α np α In all cases, we see that n Z T α np α separates points of α Therefore, ( α, B, ν α, T α ) is the natural extension of (I q,α, B, µ α, T α ) Remark In case α = /2 and α = /λ the proof of Theorem 27 is a straightforward application of Theorem 222 from Schw]; see also Examples 23 (the case of the RCF) and 232 (the NICF) in Schw] However, for /2 < α < /λ, a lot of extra work is needed, due to the problem mentioned in the above proof, and illustrated in Figure 5 22 Odd indices Let q = 2h + 3, h N The l n s and r n s are ordered in the following way Theorem 28 Let q = 2h + 3, h N, let the sequences (l n ) n 0 and (r n ) n 0 be defined as above, and let ρ = (λ 2 + λ 2 4λ + 8)/2 Then we have the following cases:

14 272 Karma Dajani et al α = /2 : l 0 < r h+ = l h+ < r = l < < r 2h = l 2h < r h = l h < r 2h = l 2h < δ < r h = l h < δ 2 < r 2h+ = l 2h+ = 0 < r 0 /2 < α < ρ/λ : l 0 < r h+ < l h+ < r < < l h 2 < r 2h < l 2h < r h < l h < r 2h < l 2h < δ < r h < l h < δ 2 < r 2h+ < 0 < l 2h+ < r 0 Furthermore, l 2h+2 = r 2h+2 and d 2h+2 (r 0 ) = d 2h+2 (l 0 ) + α = ρ/λ : l 0 = r h+ < l h+ = r < < l h < r h = δ < l h < δ 2 < 0 < r 0 ρ/λ < α < /λ : l 0 < r < < l h < r h < δ < l h < 0 < r h+ < r 0 Furthermore, l h+ = r h+2 and d h+ (l 0 ) = d h+2 (r 0 ) + α = /λ : l 0 = r < < l h = r h < δ < l h = r h+ = 0 < r 0 Proof In BKS], Section 32 (see also the introduction of this section), it was shown that φ 0 < φ h+ < φ, and φ 0 = λ 2 < φ < < φ h 2 < φ h < 2 3λ < φ h < 2 5λ, φ h+ < φ h+2 < < φ 2h = λ < 2 3λ < φ 2h+ = 0 In view of this and Lemma 2, we have φ h l h r h < φ h An important question is to know where δ is located Since 3/2 +α, we have φ h < δ For q = 2h+3, we have sin (h + )π/q = sin (h + 2)π/q, thus B h+ = B h+2, B h = (λ )B h+, B h = (λ 2 λ )B h+ Hence we obtain, by (0), l h = αλb h B h+ αλ(λ ) = αλb h B h λ αλ(λ 2 λ ) < δ The position of r h with respect to δ leads us to distinguish between the possible cases We have r h = B h+αλ B h ( α)λ = B h αλ B h ( α)λ(λ ) < δ if and only if α 2 λ 2 + αλ(2 λ) > 0, ie, αλ > (λ 2 + λ 2 4λ + 8)/2 = ρ Note that 2 < λ 2 + λ 2 4λ + 8 < for 0 < λ < 2 2λ λ Assume first that α > ρ/λ Then r h < δ, which immediately yields r h+ = B h+ B h+2 αλ αλ = B h+ αλ B h ( α)λ 0, l h = B h+αλ B h+2 B h+ B h αλ = αλ αλ(λ ) 0

15 Metrical theory for α-rosen fractions 273 If α < /λ, then /l h /r h+ = λ, hence l h+ = r h+2 and d (r h+ ) = d (l h ) + In case α = /λ, we have r h+ = l h = 0 Hence the last two cases are proved It remains to consider /2 < α ρ/λ Now δ r h (< φ h ), with δ = r h if and only if α = ρ/λ Consequently, we immediately find that r 0 = + :, ( : ) h, : 2, ( : ) h, ] To see that l h < δ 2, note that this is equivalent to α 2 λ 2 αλ 2 + 2αλ < 2(λ )( αλ), which holds because of the assumption α 2 λ 2 + αλ(2 λ) 0 This assumption also implies that l h+ r, where again equality holds if and only if α = ρ/λ This proves the case α = ρ/λ For α < ρ/λ, we have l 0 = ( : ) h, : 2, ( : ) h, ], hence the convergents of l 0 satisfy R h = B h = (λ 2 λ )B h+, R h = (λ )B h+ and R h+ = 2λR h R h = (λ 2 λ + )B h+ The recurrence R h+n+ = λr h+n R h+n for n =,, h yields R h+n = (B n+2 B n+ + 2B n )B h+ for n = 0,,, h + For the S n s, we have similarly S h = (λ )B h+, S h = B h+, thus S h+ = 2λS h S h = (λ + )B h+ and S h+n = (B n+ + B n )B h+ for n = 0,,, h + By (7), we obtain, for n = 0,,, h +, l h+n = (B n+ + B n )(α )λ + B n+2 B n+ + 2B n (B n + B n )(α )λ + B n+ B n + 2B n (6) For the convergents of r 0, only the sign of the R n is different and we get This yields r h+n = (B n+ + B n )αλ (B n+2 B n+ + 2B n ) (B n + B n )αλ (B n+ B n + 2B n ) (7) (2α )λ r 2h+ = αλ 2 2λ + 2 (< 0), l (2α )λ 2h+ = ( α)λ 2 (> 0), 2λ + 2 hence /r 2h+ /l 2h+ = λ, d (r 2h+ ) = d (l 2h+ ) +, and the theorem is proved

16 274 Karma Dajani et al For the construction of the natural extension, we have to distinguish between the different cases of the previous theorem Consider first α > ρ/λ Theorem 29 Let q = 2h + 3 with h Then the system of relations (R ) : H = /(λ + H 2h+2 ), (R 2 ) : H 2 = /λ, (R n ) : H n = /(λ H n 2 ) for n = 3,, 2h + 2, (R 2h+3 ) : H 2h+ = λ/2, (R 2h+4 ) : H 2h + H 2h+2 = λ, admits the (unique) solution H 2n = φ 2h+ n = B n B n+ = H 2n = B n + B n = sin B n + B n+ sin nπ q (n )π q sin nπ q sin (n+)π q + sin nπ q (n+)π + sin q for n =,, h +, for n =,, h +, in particular H 2h+2 = Let ρ/λ < α /λ and α = 2h+2 n= J n 0, H n ] with J 2n = l n, r n ), J 2n = r n, l n ) for n =,, h, J 2h+ = l h, r h+ ) and J 2h+2 = r h+, r 0 ) Then the map T α : α α given by (5) is bijective off a set of Lebesgue measure zero Remark The case q = 3, ρ/λ α /λ, which is the case of Nakada s α-expansions for ( 5 )/2 α, has been dealt with in N]; see also NIT], TI], and K, K2] The proof of Theorem 29 is very similar to that of Theorem 23 and therefore omitted; see also Figure 6 In case α = /λ, the intervals J 2n are empty H 4 H 4 H 3 H 3 H 2 H 2 H H /2λ /3λ l 0 r δ δ 2 l δ 3 0 δ 2 r 2 δ r 0 l0 r l l 2=r 3 0 r 2 r 0 Fig 6 The natural extension domain α (left) and its image under T α (right) of the α-rosen continued fraction (δ n = δ n ); here q = 5, α = 056, d h+ (l 0 ) = 3, d h+2 (r 0 ) = 2 Once more, a Jacobian calculation shows that T α preserves the probability measure ν α with density C q,α /( + xy) 2, where C q,α is a normalizing constant given by the following proposition

17 Metrical theory for α-rosen fractions 275 Proposition 20 If q = 2h + 3 and ρ/λ < α /λ, then the normalizing constant is Proof Integration gives C q,α = / ( + r0 log + r h+ Using (0) and (), we find and C q,α = / log + αλ = / log + 2α cos π q 2 λ 2 sin 2q π h+ n= + r n H 2n = B n B n αλ, + l n H 2n B n αλ B n + r n H 2n + l n H 2n h n= ) + l n H 2n + r n H 2n + l n H 2n + r n H 2n = B nαλ B n B n+ B n αλ, + r 0 + r h+ = ( + αλ)(b h+αλ B h ) B h+ B h = ( + αλ)(b h+ αλ B h )B h+, where we have used (2 λ)bh+ 2 = 2( cos π (h+)π q ) sin2 q sin 2 π q Putting everything together, we obtain = 4 sin2 π 2q cos2 π 2q 4 sin 2 π 2q cos2 π 2q = C q,α = / log(( + αλ)b h+ ) = / log + αλ = / log + αλ 2 λ 2 sin 2q π Remark Note that for q = 3 this result confirms Nakada s result from N] for α between ( 5 )/2 and ; in this case, the normalizing constant is indeed / log( + α) Now consider α < ρ/λ Theorem 2 Let q = 2h + 3 with h Then the system of relations (R ) : H = /(2λ H 4h ), (R 2 ) : H 2 = /(2λ H 4h ), (R 3 ) : H 3 = /(λ + H 4h+3 ), (R 4 ) : H 4 = /λ, (R n ) : H n = /(λ H n 4 ) for n = 5,, 4h + 3, (R 4h+4 ) : H 4h+2 = λ/2, (R 4h+5 ) : H 4h+ + H 4h+3 = λ, admits the (unique) solution H 4n = in particular H 4h+3 = ρ B n B n+, H 4n 2 = B n + B n B n + B n+, H 4h+3 4n = B n+ρ B n B n ρ B n, H 4h+ 4n = B n+ρ B n+2 B n ρ B n+,

18 276 Karma Dajani et al Let /2 α < ρ/λ, and α = 4h+3 n= J n 0, H n ] with J 4n 3 = l n, r h+n ), J 4n 2 = r h+n, l h+n ), J 4n = l h+n, r n ), J 4n = r n, l n ) for n =,, h, J 4h+ = l h, r 2h+ ), J 4h+2 = r 2h+, l 2h+ ) and J 4h+3 = l 2h+, r 0 ) Then the map T α : α α given by (5) is bijective off a set of Lebesgue measure zero Proof The proof of the bijectivity runs along the same lines as the proof of Theorem 23 and is therefore omitted; see also Figure 7 The H 4n s are determined by (R 4 ), (R 8 ),, (R 4h ) The H 4n 2 s are determined by (R 2 ), (R 6 ),, (R 4h+2 ) and (R 4h+4 ) By (R 3 ), (R 7 ),, (R 4h+3 ), we obtain H 4h+3 4n = B n+h 4h+3 B n B n H 4h+3 B n and = H 3 = B h+h 4h+3 B h H 4h+3 (λ ) = λ + H 4h+3 B h H 4h+3 B h (λ )H 4h+3 (λ 2 λ ), thus H4h (2 λ)h 4h+3 = 0, ie H 4h+3 = ρ Finally, the H 4h+ 4n s are determined by (R ), (R 5 ),, (R 4h+5 ) For α = /2, the intervals J 4n and J 4n 2 are empty H 6 ρ H 5 H 6 ρ H 5 H 4 H 4 H 3 H 2 H H 3 H 2 /2λ /3λ /4λ l 0 r 2 l2 δ r l δ 2δ 3 r 3δ4 0 δ 3l 3δ 2 δ r 0 l 0 r 2 l2 r l r 3 0 l3 l 4 =r 4 r 0 Fig 7 The natural extension domain α (left) and its image under T α (right) of the α-rosen continued fraction (δ n = δ n ); here q = 5, α = 05038, d 2h+2 (l 0 ) = 2, d 2h+2 (r 0 ) = 3 Again, T α preserves the probability measure ν α with density C q,α /( + xy) 2, where C q,α is a normalizing constant given by the following proposition Proposition 22 If q = 2h+3 and /2 α < ρ/λ, then the normalizing constant is C q,α = / log + ρ = / log + ρ 2 λ 2 sin 2q π Proof Integration shows that C q,α is equal to / ( + h+ r0 ρ + r h+j H 4j 3 + l h+j H 4j 2 log + l 2h+ ρ + l j= j H 4j 3 + r h+j H 4j 2 Using (0), (), (6), (7) and Lemma 25, we find h j= ) + r j H 4j + l j H 4j + l h+j H 4j + r j H 4j + r h+j H 4j 3 (λ( 2α)ρ + αλ 2 λ 2 + 2λ 2)(B j B j αλ) = + l j H 4j 3 ((αλ )ρ αλ + λ )((B j + B j )αλ B j+ + B j 2B j ),

19 Metrical theory for α-rosen fractions l h+j H 4j 2 = (B j + B j )αλ B j+ + B j 2B j + r h+j H 4j 2 2B j B j + B j 2 (B j + B j )αλ, + r j H 4j = ((αλ )ρ αλ + λ )(2B j B j + B j 2 (B j + B j )αλ) + l h+j H 4j (λ( 2α)ρ + αλ 2 λ 2, + 2λ )(B j αλ B j ) + l j H 4j = B j αλ B j + r j H 4j B j+ B j αλ, + r 0 ρ + l 2h+ ρ = ( + αλρ)((2b h+ B h + B h (B h+ + B h )αλ) ((2α )λρ αλ 2 + λ 2 + 2λ 2)B h+ Putting everything together, we obtain C q,α = / log ( + αλρ) 2 λ + αλ λ + ( αλ)ρ = / log + ρ 2 λ This confirms again Nakada s result for q = 3, ie, C 3,α = / log 5+ 2 for 2 α < 5 2 The case α = ρ/λ is slightly different from both other cases (similarly to α = /λ for even q) Theorem 23 Let q = 2h + 3 with h, α = ρ/λ and ρ = h ( l j, r j ) j= 0, B j + B j B j + B j+ ]) ( r j, l j ) 0, B j B j+ Then T ρ : ρ ρ is bijective off a set of Lebesgue measure zero ]) ( l h, ρ) 0, λ ]) 2 The normalizing constant in this case is C q,ρ/λ = / log +ρ 2 λ as above As in the even case, we let µ α be the projection of ν α on the first coordinate, B be the restriction of the two-dimensional σ -algebra on α, and B be the Lebesgue σ -algebra on I q,α = λ(α ), αλ] We have the following theorem, whose proof is similar to the proof of Theorem 27 in the even case Theorem 24 Let q 3, q = 2h +, and let /2 α /λ Then the dynamical system ( α, B, ν α, T α ) is the natural extension of the dynamical system (I q,α, B, µ α, T α ) ρ H 5 λ/2 /λ /λ H 3 H 2 H λ 2 φ 2 2 3λ φ 0 2 3λ λ ρ λ 2 λ+ρ l 0 λ+ρ ρ λ λ+ 0 λ+ Fig 8 /2 (left), ρ/λ (middle) and /λ (right); here q = 5

20 278 Karma Dajani et al 23 Convergence of the continued fractions Now we can prove easily that the α-rosen continued fractions converge If Tα n(x) = 0 for some n 0, then this is clear Therefore assume that Tα n (x) = 0 for all n 0 Setting (t n, v n ) = Tα n(x, 0), it follows directly from the definition (5) of T α that v n = : d n, ε n : d n,, ε 2 : d ] Furthermore, an immediate consequence of (6) is that S n /S n = : d n, ε n : d n,, ε 2 : d ], ie, v n = S n /S n Theorems 23, 29 and 2 (see also Figures 3, 6 and 7) show that v n, ie, S n S n, and that S n = S n if and only if q = 2h + 3, n = h +, d =, (ε i : d i ) = ( : ) for i = 2,, h+, which is possible only if α > ρ/λ Furthermore, v n v n /c for some constant c >, ie, S n cs n 2 It follows from (7) that x R n S = Tα n(x)(r n S n R n S n ) n S n (S n + Tα n(x)s n ) = t n Sn 2( + t nv n ) αλ ( + αλ λ)sn 2, (8) hence the α-rosen convergents R n /S n converge to x as n 3 Mixing properties of α-rosen fractions In case q is even, and α = /λ, we saw in the previous section that there is a simple relation between /2 and /λ ; see also Figure 4 Define in this case the map M : /λ /2 by { ( y, x) if (x, y) /λ, x < 0, M(x, y) = (y, x) if (x, y) /λ, x 0 Clearly, M : /λ /2 is bijective and bi-measurable, and ν /λ (M (A)) = ν /2 (A) for every Borel set A /2 By comparing the partitions of T /λ (on /λ ) and that of T /2 (on /2), we find that T /λ (x, y) = M (T /2 (M(x, y))), (x, y) /λ, x = 0 This implies that the dynamical systems ( /2, B, ν /2, T /2 ) and ( /λ, B, ν /λ, T /λ ) are isomorphic In BKS] it was shown that the dynamical system ( /2, B, ν /2, T /2 ) is weakly Bernoulli with respect to the natural partition, hence ( /2, B, ν /2, T /2 ) and ( /2, B, ν /2, T /2 ) are isomorphic As a consequence we find that the dynamical systems ( /2, B, ν /2, T /2 ) and ( /λ, B, ν /λ, T /λ ) are isomorphic In this section, we will show that this result also holds for all q and all α strictly between /2 and /λ, using a result by M Rychlik Ry] For completeness, we state explicitly the hypothesis needed for Rychlik s result (the reader is referred to Ry] for more details) Let X be a totally ordered order-complete set Open intervals constitute a base of a complete topology in X, making X into a topological space If X is separable, then X is homeomorphic to a closed subset of an interval Let B be the Borel σ -algebra on X, and

21 Metrical theory for α-rosen fractions 279 m a fixed regular, Borel probability measure on X (in our case m will be the normalized Lebesgue measure restricted to X) Let U X be an open dense subset of X such that m(u) = Let S = X \ U; clearly m(s) = 0 Let T : U X be a continuous map, and β a countable family of closed intervals with disjoint interiors such that U β Furthermore, suppose that for any B β the set B S consists only of endpoints of B, and T restricted to B U admits an extension to a homeomorphism of B with some interval in X Suppose that T (x) = 0 for x U, and let g(x) = / T (x) for x U, and g S = 0 Let P : L (X, m) L (X, m) be the Perron Frobenius operator of T, Pf (x) = g(y)f (y) y T x In Ry], it was proved (among many other things) that if g < and Varg <, then there exist functions ϕ,, ϕ s of bounded variation such that (i) P ϕ i = ϕ i ; (ii) ϕ i dm = ; (iii) there exists a measurable partition C,, C s of X with T C i = C i for i =,, s; (iv) the dynamical systems (C i, T i, ν i ), where T i = T Ci and ν i (B) = B ϕ i dm, are exact, and ν i is the unique invariant measure for T i, absolutely continuous with respect to m Ci Rychlik also showed that if s =, ie, if is the only eigenvalue of P on the unit circle, and if there exists only one ϕ L (X, M) with P ϕ = ϕ and m(ϕ) = (ϕ 0), then the natural extension of (X, T, ν) is isomorphic to a Bernoulli shift Returning to our map T α, defined on X = I q,α = λ(α ), αλ], and using the same notation as above, we let m be normalized Lebesgue measure on X, { } S = {λ(α )} ± λ(α + d) d =, 2, and U = X \ S Note that T α : U X is continuous, and that the restriction of T α to each open interval is homeomorphic to an interval (in fact to X itself, except for the first and last interval) We have g(x) = / T α (x) = x2 on U, hence g < (since α = /λ), and Varg < It is easy, but tedious (cf DK] for a proof of the regular case), to see that T is exact, hence s =, and we can apply Rychlik s result to obtain the following theorem Theorem 3 The natural extension ( α, ν α, T α ) of (X α, µ α, T α ) is weakly Bernoulli Hence, the natural extension is isomorphic to any Bernoulli shift with the same entropy 4 Metrical properties of regular Rosen fractions An important reason to introduce and study the natural extension of the ergodic system underlying any continued fraction expansion is that such a natural extension facilitates

22 280 Karma Dajani et al the study of the continued fraction expansion at hand; see eg DK] and IK, Chapter 4] The following theorem is a consequence of this; see BJW], DK], or IK, Chapter 4] Theorem 4 Let q 3, and let /2 α /λ For almost all G q -irrational numbers x, the two-dimensional sequence T α (x, 0) = (T n α (x), S n /S n ), n, is distributed over α according to the density function g α given by g α (t, v) = C q,α ( + tv) 2 for (t, v) α, and g α (t, v) = 0 otherwise Here C q,α is the normalizing constant of the T α -invariant measure ν α Due to Proposition 4, it is possible to study the distribution of various sequences related to the α-rosen expansion of almost every x X α Classical examples of these are the frequency of digits, or the analogs of various classical results by Lévy and Khinchin However, these results can already be obtained from the projection (X α, B α, µ α, T α ) of ( α, B α, ν α, T α ) which is also ergodic and the Ergodic Theorem For the distribution of the so-called approximation coefficients, the natural extension ( α, B α, ν α, T α ) is necessary These approximation coefficients n = n (x) are defined by n = n (x) := Sn 2 x R n, n 0, (9) where R n /S n is the nth α-rosen convergent, which is obtained by truncating the α-rosen expansion With (t n, v n ) = Tα n (x, 0), it follows from (8) that n = S n ε n+t n + t n v n for n (20) Similarly, since t n = ε n /t n d n λ and v n = S n /S n, it follows from (6) that n = v n + t n v n for n (2) In view of (20) and (2), we define the map ( ) v F (t, v) = + tv, t =: (ξ, η) for tv = + tv It is now an easy calculation (see eg BKS, p 293]) that due to Proposition 4, for almost all x X α, the sequence ( n (x), ε n+ n (x)) n 0 is distributed on F ( α ) according to the density function C q,α / 4ξη Setting Ɣ + α = F ({(t, v) α t 0}) and Ɣ α = F ({(t, v) α t 0}), we obtain the following theorem

23 Metrical theory for α-rosen fractions 28 Theorem 42 Let q 3, /2 α /λ, and define the functions d + α and d α by d ± α (ξ, η) = C q,α 4ξη for (ξ, η) Ɣ ± α, (22) and d ± α (ξ, η) = 0 otherwise Then the sequence ( n (x), n (x)) n lies in the interior of Ɣ = Ɣ + α Ɣ α for all G q-irrational numbers x, and for almost all x this sequence is distributed according to the density function d α, where d α (ξ, η) = d α + (ξ, η) + d α (ξ, η) By this last statement we mean that, for almost all x and for all a, b 0, the limit exists, and equals lim N N #{j j N, j (x) < a, j (x) < b} a b 0 0 d α (ξ, η) dξ dη Several corollaries can be drawn from Theorem 42; see eg K], where (for q = 3) for almost all x the distributions of the sequences ( n ) n, ( n + n ) n, ( n n ) n were determined Here we only mention the following result for even values of q (it was previously obtained in BKS] for both even and odd values of q, and α = /2) Proposition 43 Let q 4 be an even integer, /2 α /λ, and let { λ L α := min λ + 2, λ(2 } αλ2 ) 4 λ 2 Then for almost all G q -irrational numbers x and all c /L α, we have { lim N N # n n N, n(x) < } = λc q,α c c Proof In view of the expression of n (x) in (20), we consider curves given by c = v + tv, where c > 0 is a constant and t l 0, r 0 ] Note that these curves are increasing on l 0, r 0 ], and that the curve given by v = c c t lies above the curve given by v = c 2 c 2 t if and only if c > c 2 Now let L α be defined as the largest positive c for which the curve c = +tv v lies in α for t l 0, r 0 ], ie, { ( ) } L α = max c > 0 c t, α for all t l 0, r 0 ] ct

24 282 Karma Dajani et al It follows from Theorem 4 that for all z L α, and for almost all G q -irrationals x, r0 ( z ) lim N N #{n n N, zt n(x) < z} = g α (t, v) dv dt = λc q,α z, where C q,α is the normalizing constant of the invariant measure (which has density g α ) So we are left to show that L α = min { λ λ+2, λ(2 αλ2 )} 4 λ 2 In the even case we can discern three cases: α = /2, /2 < α < /λ, and α = /λ Note that the first case has been dealt with in BKS]; in case α = /2 one has L α = λ+2 λ In case /2 < α < /λ, first note that the curve c = +tv v goes through (r, H ) = ( ) αλ λ, λ+ if and only if c = αλ+ αλ Since in this case v l 0 αλ αλ + < 2 < λ = H 2, and the curve c = +tv is increasing on l 0, r 0 ], we immediately find that this curve is in α for t l 0, 0], yielding L α αλ+ αλ From Theorem 22 we see that l p 2 < 0 < l p Set c 2 = H 2p 2 = λ(2 αλ2 ) + l p H 2p 2 4 λ 2, c 3 = H 2p = + r 0 H p + αλ It follows from Theorem 23 (see also Figure 3 for q = 6) that L α = min{c, c 2 }, since c < c 3 For q (and therefore λ) fixed, and for α /2, /λ], one easily shows that c = c (α) is an increasing function of α, with c (/2) = λ/(λ + 2) and c (/λ) = /2, while c 2 = c 2 (α) is a line with slope λ 3 /(4 λ 2 ) Since c 2 (/2) = λ/2 > /2 > λ/(λ + 2) and c 2 (/λ) = λ/(λ + 2) = c (/2) < c (/λ) = /2, we find for /2 < α < /λ that { λ L α = min λ + 2, λ(2 αλ2 ) 4 λ 2 0 } v +tv is In case α = /λ, the point (, λ/2) yields c = λ/(λ + 2) Since the curve c = increasing on l 0, r 0 ], and from the fact that for t = 0 we have v = c = λ/(λ+2) < /λ, where /λ is the smallest height of α, we find that This proves the theorem L /λ = λ λ + 2 Remark We only deal with the even case in Proposition 43; a result for the odd case is obtained similarly, but has a more involved expression Acknowledgments We thank the referee for many valuable remarks, which improved considerably the quality of the presentation of this paper The third author was supported by NWO Bezoekersbeurs NB 6-572, the Austrian Science Fund FWF, grant S8302-MAT, and by Agence Nationale de la Recherche, grant ANR-06-JCJC- 0073

25 Metrical theory for α-rosen fractions 283 References BJW] Bosma, W, Jager, H, Wiedijk, F: Some metrical observations on the approximation by continued fractions Nederl Akad Wetensch Indag Math 45, (983) Zbl MR BKS] Burton, R M, Kraaikamp, C, Schmidt, T A: Natural extensions for the Rosen fractions Trans Amer Math Soc 352, (2000) Zbl MR DK] Dajani, K, Kraaikamp, C: Ergodic Theory of Numbers Carus Math Monogr 29, Math Assoc Amer, Washington, DC (2002) Zbl MR GH] Gröchenig, K, Haas, A: Backward continued fractions and their invariant measures Canad Math Bull 39, (996) Zbl MR IK] Iosifescu, M, Kraaikamp, C: Metrical Theory of Continued Fractions Math Appl 547, Kluwer, Dordrecht (2002) Zbl MR K] Kraaikamp, C: The distribution of some sequences connected with the nearest integer continued fraction Indag Math 49, 77 9 (987) Zbl MR K2] Kraaikamp, C: A new class of continued fraction expansions Acta Arith 57, 39 (99) Zbl MR N] Nakada, H: Metrical theory for a class of continued fraction transformations Tokyo J Math 4, (98) Zbl MR N2] Nakada, H: Continued fractions, geodesic flows and Ford circles In: Algorithms, Fractals and Dynamics, Y Takahashi (ed), Plenum Press, 79 9 (995) Zbl MR NIT] Nakada, H, Ito, Sh, Tanaka, S: On the invariant measure for the transformations associated with some real continued-fractions Keio Engrg Rep 30, no 3, (977) Zbl MR Roh] Rohlin, V A: Exact endomorphisms of Lebesgue spaces Izv Akad Nauk SSSR Ser Mat 25, (96) (in Russian); English transl: Amer Math Soc Transl Ser 2, 39, 36 (964) Zbl MR Ros] Rosen, D: A class of continued fractions associated with certain properly discontinuous groups Duke Math J 2, (954) Zbl MR Ry] Rychlik, M: Bounded variation and invariant measures Studia Math 76, (983) Zbl MR Schm] Schmidt, T A: Remarks on the Rosen λ-continued fractions In: Number Theory with an Emphasis on the Markoff Spectrum, A Pollington and W Moran (eds), Dekker, New York, (993) Zbl MR Schw] Schweiger, F: Ergodic Theory of Fibred Systems and Metric Number Theory Clarendon Press, Oxford (995) Zbl MR TI] Tanaka, S, Ito, Sh: On a family of continued-fraction transformations and their ergodic properties Tokyo J Math 4, (98) Zbl MR W] Weber, H: Lehrbuch der Algebra, I 2nd ed, Vieweg, Braunschweig (898) Zbl