Applications of (a,b)-continued fraction transformations
AAPPLICATIONS OF ( a, b ) -CONTINUED FRACTIONTRANSFORMATIONS SVETLANA KATOK AND ILIE UGARCOVICI
Dedicated to the memory of Dan Rudolph
Abstract.
We describe a general method of arithmetic coding of geodesicson the modular surface based on the study of one-dimensional Gauss-like mapsassociated to a two parameter family of continued fractions introduced in [16].The finite rectangular structure of the attractors of the natural extension mapsand the corresponding “reduction theory” play an essential role. In specialcases, when an ( a, b )-expansion admits a so-called “dual”, the coding sequencesare obtained by juxtaposition of the boundary expansions of the fixed points,and the set of coding sequences is a countable sofic shift. We also prove thatthe natural extension maps are Bernoulli shifts and compute the density of theabsolutely continuous invariant measure and the measure-theoretic entropy ofthe one-dimensional map. Introduction and background
In [16], the authors studied a new two-parameter family of continued fractiontransformations. These transformations can be defined using the standard gener-ators T ( x ) = x + 1, S ( x ) = − /x of the modular group SL (2 , Z ) and considering f a,b : ¯ R → ¯ R given by(1.1) f a,b ( x ) = x + 1 if x < a − x if a ≤ x < bx − x ≥ b . Under the assumption that the parameters ( a, b ) belong to the set P = { ( a, b ) | a ≤ ≤ b, b − a ≥ , − ab ≤ } , one can introduce corresponding continued fraction algorithms by using the firstreturn map of f a,b to the interval [ a, b ). Equivalently, these so called ( a, b ) -continuedfractions can be defined using a generalized integral part function:(1.2) (cid:98) x (cid:101) a,b = (cid:98) x − a (cid:99) if x < a a ≤ x < b (cid:100) x − b (cid:101) if x ≥ b , where (cid:98) x (cid:99) denotes the integer part of x and (cid:100) x (cid:101) = (cid:98) x (cid:99) + 1. Mathematics Subject Classification.
Primary 37D40, 37B40; Secondary 11A55, 20H05.
Key words and phrases.
Continued fractions, modular surface, geodesic flow, invariantmeasure.The second author is partially supported by the NSF grant DMS-0703421. a r X i v : . [ m a t h . D S ] M a y SVETLANA KATOK AND ILIE UGARCOVICI
A starting point of the theory is the following result [16, Theorem 2.1]: if ( a, b ) ∈P , then any irrational number x can be expressed uniquely as an infinite continuedfraction of the form x = n − n − n −
1. . . = (cid:98) n , n , · · · (cid:101) a,b , ( n k (cid:54) = 0 for k ≥ , where n = (cid:98) x (cid:101) a,b , x = − x − n and n k +1 = (cid:98) x k +1 (cid:101) a,b , x k +1 = − x k − n k , i.e. thesequence of partial fractions r k = (cid:98) n , n , . . . , n k (cid:101) a,b converges to x .It is possible to construct ( a, b )-continued fraction expansions for rational num-bers, too. However, such expansions will terminate after finitely many steps if b (cid:54) = 0. If b = 0, the expansions of rational numbers will end with a tail of 2’s, since0 = (cid:98) , , , . . . (cid:101) a, .The above family of continued fraction transformations contains three classicalexamples: the case a = − b = 0 described in [22, 12] gives the “minus” (back-ward) continued fractions, the case a = − / b = 1 / a = − b = 1was presented in [19, 14] in connection with a method of coding symbolically thegeodesic flow on the modular surface following Artin’s pioneering work [6] and cor-responds to the regular “plus” continued fractions with alternating signs of thedigits.The main object of study in [16] is a two-dimensional realization of the naturalextension map of f a,b , F a,b : ¯ R \ ∆ → ¯ R \ ∆, ∆ = { ( x, y ) ∈ ¯ R | x = y } , defined by(1.3) F a,b ( x, y ) = ( x + 1 , y + 1) if y < a (cid:16) − x , − y (cid:17) if a ≤ y < b ( x − , y −
1) if y ≥ b . Here is the main result of that paper:
Theorem 1.1 ([16]) . There exists an explicit one-dimensional Lebesgue measurezero, uncountable set E that lies on the diagonal boundary b = a + 1 of P such that: (1) for all ( a, b ) ∈ P \ E the map F a,b has an attractor D a,b = ∩ ∞ n =0 F na,b ( ¯ R \ ∆) on which F a,b is essentially bijective. (2) The set D a,b consists of two (or one, in degenerate cases) connected com-ponents each having finite rectangular structure , i.e. bounded by non-decreasing step-functions with a finite number of steps. (3) Almost every point ( x, y ) of the plane ( x (cid:54) = y ) is mapped to D a,b afterfinitely many iterations of F a,b . An essential role in the argument is played by the forward orbits associated to a and b : to a , the upper orbit O u ( a ) (i.e. the orbit of Sa ) and the lower orbit O (cid:96) ( a )(i.e. the orbit of T a ), and to b , the upper orbit O u ( b ) (i.e. the orbit of T − b ) and the lower orbit O (cid:96) ( b ) (i.e. the orbit of Sb ). It was proved in [16] that if ( a, b ) ∈ P \ E ,then f a,b satisfies the finiteness condition , i.e. for both a and b , their upper andlower orbits are either eventually periodic, or they satisfy the cycle property , i.e.they meet forming a cycle; more precisely, there exist k , m , k , m ≥ f m a,b ( Sa ) = f k a,b ( T a ) = c a , (resp., f m a,b ( T − b ) = f k a,b ( Sb ) = c b ) , PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 3
APPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 3 - - - - - - ab Figure 1.
Attracting domain D a,b for a = − , b = where c a and c b are the ends of the cycles . If the products of transformationsover the upper and lower sides of the cycle are equal, the cycle property is strong ,otherwise, it is weak . In both cases the set L a,b of the corresponding values is finite;ends of the cycles belong to the set L a,b if and only if they are equal to 0, i.e. if thecycle is weak. The structure of the attractor D a,b is explicitly “computed” fromthe finite set L a,b .The paper is organized as follows. In Section 2 we give some background infor-mation about geodesic flows and their representations as special flows over symbolicdynamical systems, and define the coding map. In Section 3 we describe the reduc-tion procedure for coding geodesics via ( a, b )-continued fractions based on the studyof the attractor of the associated natural extension map, define the correspondingcross-section set, and introduce the notion of reduced geodesic . In Section 4 we provethat the first return map to the cross-section corresponds to a shift of the codingsequence (Theorem 4.1) and, as a consequence, show that ( a, b )-continued fractionssatisfy the Tail Property , i.e. two SL (2 , Z )-equivalent real numbers have the sametails in their ( a, b )-continued fraction expansions. In Section 5 we introduce a notionof a dual code and show that if an ( a, b )-expansion has a dual ( a ! , b ! )-expansion, thenthe coding sequence of a reduced geodesic is obtained by juxtaposition of the ( a, b )-expansion of its attracting endpoint w and the ( a ! , b ! )-expansion of 1 /u , where u isits repelling endpoint. We also prove that if the ( a, b )-expansion admits a dual, thenthe set of admissible coding sequences is a sofic shift (Theorem 5.8). In Section 6 wederive formulas for the density of the absolutely continuous invariant measure andthe measure-theoretic entropy of the one-dimensional Gauss-type maps and theirnatural extensions. We also prove that the natural extension maps are Bernoullishifts. And finally, in Section 7 we apply results of [16] to obtain explicit for-mulas for invariant measure for the one-dimensional maps for some regions of theparameter set P . Figure 1.
Attracting domain D a,b for a = − , b = where c a and c b are the ends of the cycles . If the products of transformationsover the upper and lower sides of the cycle are equal, the cycle property is strong ,otherwise, it is weak . In both cases the set L a,b of the corresponding values is finite;ends of the cycles belong to the set L a,b if and only if they are equal to 0, i.e. if thecycle is weak. The structure of the attractor D a,b is explicitly “computed” fromthe finite set L a,b .The paper is organized as follows. In Section 2 we give some background infor-mation about geodesic flows and their representations as special flows over symbolicdynamical systems, and define the coding map. In Section 3 we describe the reduc-tion procedure for coding geodesics via ( a, b )-continued fractions based on the studyof the attractor of the associated natural extension map, define the correspondingcross-section set, and introduce the notion of reduced geodesic . In Section 4 we provethat the first return map to the cross-section corresponds to a shift of the codingsequence (Theorem 4.1) and, as a consequence, show that ( a, b )-continued fractionssatisfy the Tail Property , i.e. two SL (2 , Z )-equivalent real numbers have the sametails in their ( a, b )-continued fraction expansions. In Section 5 we introduce a notionof a dual code and show that if an ( a, b )-expansion has a dual ( a (cid:48) , b (cid:48) )-expansion, thenthe coding sequence of a reduced geodesic is obtained by juxtaposition of the ( a, b )-expansion of its attracting endpoint w and the ( a (cid:48) , b (cid:48) )-expansion of 1 /u , where u isits repelling endpoint. We also prove that if the ( a, b )-expansion admits a dual, thenthe set of admissible coding sequences is a sofic shift (Theorem 5.8). In Section 6 wederive formulas for the density of the absolutely continuous invariant measure andthe measure-theoretic entropy of the one-dimensional Gauss-type maps and theirnatural extensions. We also prove that the natural extension maps are Bernoullishifts. And finally, in Section 7 we apply results of [16] to obtain explicit for-mulas for invariant measure for the one-dimensional maps for some regions of theparameter set P . SVETLANA KATOK AND ILIE UGARCOVICI Geodesic flow on the modular surface and its representation as aspecial flow over a symbolic dynamical system
Let H = { z = x + iy : y > } be the upper half-plane endowed with thehyperbolic metric, F = { z ∈ H : | z | ≥ , | Re z | ≤ } be the standard fundamentalregion of the modular group P SL (2 , Z ) = SL (2 , Z ) / {± I } , and M = P SL (2 , Z ) \H be the modular surface. Let S H denote the unit tangent bundle of H . We will usethe coordinates v = ( z, ζ ) on S H , where z ∈ H , ζ ∈ C , | ζ | = Im (z). The quotientspace P SL (2 , Z ) \ S H can be identified with the unit tangent bundle of M , SM ,although the structure of the fibered bundle has singularities at the elliptic fixedpoints (see [11, § { ˜ ϕ t } on H is defined as an R -action on theunit tangent bundle S H which moves a tangent vector along the geodesic defined bythis vector with unit speed. The geodesic flow { ˜ ϕ t } on H descents to the geodesicflow { ϕ t } on the factor M via the canonical projection(2.1) π : S H → SM of the unit tangent bundles. Geodesics on M are orbits of the geodesic flow { ϕ t } .A cross-section C for the geodesic flow is a subset of the unit tangent bundle SM visited by (almost) every geodesic infinitely often both in the future and in thepast. In other words, every v ∈ C defines an oriented geodesic γ ( v ) on M whichwill return to C infinitely often. The “ceiling” function g : C → R giving the timeof the first return to C is defined as follows: if v ∈ C and t is the time of the firstreturn of γ ( v ) to C , then g ( v ) = t . The map R : C → C defined by R ( v ) = ϕ g ( v ) ( v )is called the first return map . Thus { ϕ t } can be represented as a special flow onthe space C g = { ( v, s ) : v ∈ C, ≤ s ≤ g ( v ) } , given by the formula ϕ t ( v, s ) = ( v, s + t ) with the identification ( v, g ( v )) = ( R ( v ) , Geodesic flow on the modular surface and its representation as aspecial flow over a symbolic dynamical system
Let H = { z = x + iy : y > } be the upper half-plane endowed with thehyperbolic metric, F = { z ∈ H : | z | ≥ , | Re z | ≤ } be the standard fundamentalregion of the modular group P SL (2 , Z ) = SL (2 , Z ) / {± I } , and M = P SL (2 , Z ) \H be the modular surface. Let S H denote the unit tangent bundle of H . We will usethe coordinates v = ( z, ζ ) on S H , where z ∈ H , ζ ∈ C , | ζ | = Im (z). The quotientspace P SL (2 , Z ) \ S H can be identified with the unit tangent bundle of M , SM ,although the structure of the fibered bundle has singularities at the elliptic fixedpoints (see [11, § { ˜ ϕ t } on H is defined as an R -action on theunit tangent bundle S H which moves a tangent vector along the geodesic defined bythis vector with unit speed. The geodesic flow { ˜ ϕ t } on H descents to the geodesicflow { ϕ t } on the factor M via the canonical projection(2.1) π : S H → SM of the unit tangent bundles. Geodesics on M are orbits of the geodesic flow { ϕ t } .A cross-section C for the geodesic flow is a subset of the unit tangent bundle SM visited by (almost) every geodesic infinitely often both in the future and in thepast. In other words, every v ∈ C defines an oriented geodesic γ ( v ) on M whichwill return to C infinitely often. The “ceiling” function g : C → R giving the timeof the first return to C is defined as follows: if v ∈ C and t is the time of the firstreturn of γ ( v ) to C , then g ( v ) = t . The map R : C → C defined by R ( v ) = ϕ g ( v ) ( v )is called the first return map . Thus { ϕ t } can be represented as a special flow onthe space C g = { ( v, s ) : v ∈ C, ≤ s ≤ g ( v ) } , given by the formula ϕ t ( v, s ) = ( v, s + t ) with the identification ( v, g ( v )) = ( R ( v ) , v R ( v ) gCC g Figure 2.
Geodesic flow is a special flowLet N be a finite or countable alphabet, N Z = { x = { n i } i ∈ Z | n i ∈ N } be thespace of all bi-infinite sequences endowed with the Tikhonov (product) topology, σ : N Z → N Z defined by ( σx ) i = n i +1 be the left shift map, and Λ ⊂ N Z be a closed σ -invariant subset. Then (Λ , σ )is called a symbolic dynamical system . There are some important classes of suchdynamical systems. The space ( N Z , σ ) is called the full shift (or the topologicalBernoulli shift ). If the space Λ is given by a set of simple transition rules whichcan be described with the help of a matrix consisting of zeros and ones, we say Figure 2.
Geodesic flow is a special flowLet N be a finite or countable alphabet, N Z = { x = { n i } i ∈ Z | n i ∈ N } be thespace of all bi-infinite sequences endowed with the Tikhonov (product) topology, σ : N Z → N Z defined by ( σx ) i = n i +1 be the left shift map, and Λ ⊂ N Z be a closed σ -invariant subset. Then (Λ , σ )is called a symbolic dynamical system . There are some important classes of suchdynamical systems. The space ( N Z , σ ) is called the full shift (or the topologicalBernoulli shift ). If the space Λ is given by a set of simple transition rules which PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 5 can be described with the help of a matrix consisting of zeros and ones, we saythat (Λ , σ ) is a one-step topological Markov chain or simply a topological Markovchain (also called a subshift of finite type ). A factor of a topological Markov chainis called a sofic shift . (See [10, § C and code it, i.e. to findan appropriate symbolic dynamical system (Λ , σ ) and a continuous surjective mapCod : Λ → C (in some cases the actual domain of Cod is Λ except a finite orcountable set of excluded sequences) defined such that the diagramΛ σ −−−−→ Λ Cod (cid:121) (cid:121)
Cod C R −−−−→ C is commutative. We can then talk about coding sequences for geodesics definedup to a shift which corresponds to a return of the geodesic to the cross-section C .Notice that usually the coding map is not injective but only finite-to-one (see e.g.[2, § § a, b )-continued fractions thatis based on study of the attractor of the associated natural extension map. Thisapproach, coupled with ideas of Bowen and Series [7], may be useful for coding ofgeodesics on quotients by general Fuchsian groups.3. The reduction procedure
In what follows we will denote the end points of geodesics on H by u and w ,and whenever we refer to such geodesics, we use ( u, w ) as their coordinates on ¯ R ( u (cid:54) = w ).The reduction procedure for coding symbolically the geodesic flow on the mod-ular surface via continued fraction expansions was presented for the three classicalcases in [14]; for a survey on symbolic dynamics of the geodesic flow see also [15].Here we describe the reduction procedure for ( a, b )-continued fractions and explainhow it can be used for coding purposes.Let γ be an arbitrary geodesic on H from u to w (irrational end points), and w = (cid:98) n , n , . . . (cid:101) a,b . We construct the sequence of real pairs { ( u k , w k ) } ( k ≥ u = u, w = w and w k +1 = ST − n k w k , u k +1 = ST − n k u k . Each geodesic γ k from u k to w k is P SL (2 , Z )-equivalent to γ by construction. Itis convenient to describe this procedure using the reduction map that combines theappropriate iterate of the map F a,b : R a,b : R \ ∆ → R \ ∆ SVETLANA KATOK AND ILIE UGARCOVICI given by the formula R a,b ( u, w ) = ( ST − n u, ST − n w ), where n is the first digit inthe ( a, b )-expansion of w . Notice that ( u k , w k ) = R ka,b ( u, w ). Definition 3.1.
A geodesic in H from u to w is called ( a, b ) -reduced if ( u, w ) ∈ Λ a,b ,where Λ a,b = F a,b ( D a,b ∩ { a ≤ w ≤ b } ) = S ( D a,b ∩ { a ≤ w ≤ b } ) . According to Theorem 1.1, for (almost) every geodesic γ from u to w in H ,the above algorithm produces in finitely many steps an ( a, b )-reduced geodesic P SL (2 , Z )-equivalent to γ , and an application of this algorithm to an ( a, b )-reducedgeodesic produces another ( a, b )-reduced geodesic. In other words, there exists apositive integer (cid:96) such that R (cid:96)a,b ( u, w ) ∈ Λ a,b and R a,b : Λ a,b → Λ a,b is bijective(with the exception of some segments of the boundary of Λ a,b and their images).Let γ be a reduced geodesic with the repelling point u (cid:54) = 0 and the attractingpoint(3.2) w = (cid:98) n , n , . . . (cid:101) a,b . Then, by successive applications of the map R a,b , we obtain a sequence of real pairs { ( u k , w k ) } ( k ≥
0) (see (3.1) above) such that each geodesic γ k from u k to w k is( a, b )-reduced. Using the bijectivity of the map R a,b , we extend the sequence (3.2)to the past to obtain a bi-infinite sequence of integers(3.3) (cid:98) γ (cid:101) = (cid:98) . . . , n − , n − , n , n , n , . . . (cid:101) , called the coding sequence of γ , as follows. There exists an integer n − (cid:54) = 0 and a realpair ( u − , w − ) ∈ Λ a,b such that ST − n − w − = w = w and ST − n − u − = u = u .Notice that (cid:98) w − (cid:101) a,b = n − . By uniqueness of the ( a, b )-expansion, we concludethat w − = (cid:98) n − , n , n , . . . (cid:101) a,b . Continuing inductively, we define the sequence ofintegers n − k and the real pairs ( u − k , w − k ) ∈ Λ a,b ( k ≥ w − k = (cid:98) n − k , n − k +1 , n − k +2 , . . . (cid:101) a,b by ST − n − k w − k = w − ( k − and ST − n − k u − k = u − ( k − . We also associate to γ abi-infinite sequence of ( a, b )-reduced geodesics(3.4) ( . . . , γ − , γ − , γ , γ , γ , . . . ) , where γ k is the geodesic from u k to w k . Remark . Notice that all “intermediate” geodesics T − s γ k (1 ≤ s ≤ n k ) obtainedfrom γ k using the map F a,b are not ( a, b )-reduced. Proposition 3.3.
A formal minus continued fraction comprised from the digits ofthe “past” of (3.3), n − − n − − n − − . . . = ( n − , n − , n − , . . . ) converges to /u .Proof. By [13, Lemma 1.1], it will be sufficient to check that | n − k | = 1 implies n − k · n − ( k +1) <
0, i.e. the digit 1 must be followed by a negative integer and thedigit − PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 7 the set Λ a,b that can be derived from our knowledge of the shape of the set D a,b determined in [16, Lemmas 5.6, 5.10, 5.11]. The upper part of Λ a,b is contained inthe region(3.5) [ − , × (cid:104) − a , + ∞ (cid:105) ∪ [0 , × (cid:104) − b − , + ∞ (cid:105) if b < − , × (cid:104) − a , + ∞ (cid:105) if b ≥ . The lower part of Λ a,b is contained in the region(3.6) [ − , × (cid:104) −∞ , − a + 1 (cid:105) ∪ [0 , × (cid:104) −∞ , − b (cid:105) if a > − , × (cid:104) −∞ , − b (cid:105) if a ≤ − . Recall that ( u − ( k +1) , w − ( k +1) ) = ( T n − ( k +1) Su − k , T n − ( k +1) Sw − k ) for an appropriateinteger n − ( k +1) (cid:54) = 0. Suppose n − k = 1. Then w − k >
0. If u − k <
0, then Su − k > Sw − k <
0, and it takes a negative power of T to bring it back to (the lowercomponent of) Λ a,b , i.e. n − ( k +1) <
0. The case u − k >
0, according to (3.5), canonly take place if b ≤
1. In this case, − / ( b − ≤ w − k < b + 1, which is equivalentto b >
1, a contradiction. Therefore n − k = 1 implies n − ( k +1) <
0. A similarargument shows that n − k = − n − ( k +1) >
0. We conclude that the formalminus continued fraction converges. In order to prove that the limit is equal to 1 /u we use the recursive definition of the digits n − , n − , . . . , to write1 u = n − − u − = n − − n − − u − = · · · = ( n − , n − , . . . , n − k − u − k ) = · · · , and the conclusion follows since the formal minus continued fraction converges. (cid:3) Let C = { z ∈ H | | z | = 1 , − ≤ Re z ≤ } be the upper-half of the unit circle, and C − = { z ∈ H | | z + 1 | = 1 , − ≤ Re z ≤ } and C + = { z ∈ H | | z − | = 1 , ≤ Re z ≤ } be the images of the two vertical boundary components of the fundamental region F under S (see Figure 3). Proposition 3.4.
Every ( a, b ) -reduced geodesic either intersects C or both curves C − and C + .Proof. If a, b are such that − ≤ a ≤ ≤ b ≤
1, then by properties (3.5) and(3.6) of the set Λ a,b , if ( u, w ) ∈ Λ a,b , then − ≤ u ≤ w ≥ − a or w ≤ − b ,and hence all ( a, b )-reduced geodesics intersect C . For the case b > − < u <
0, then either w > − a > b > w < − a +1 < −
1, i.e. the geodesicintersects C ; if 0 < u <
1, then (3.5) implies that w < − b < a <
0, thus thecorresponding geodesic intersects C if w < −
1, and it intersects first C + and then C − , if − < w <
0. Similarly, for the case a < − < u <
1, then either w < − b < a < − w > − b − >
1, i.e. the geodesic intersects C ; if − < u < SVETLANA KATOK AND ILIE UGARCOVICI then (3.6) implies that w > − a > b >
0, therefore the corresponding geodesicintersects C if w >
1, and it intersects first C − and then C + if 0 < w < (cid:3) Based on Proposition 3.4 we introduce the notion of the cross-section point . Itis either the intersection of a reduced geodesic γ with C , or, if γ does not intersect C , its first intersection with C − ∪ C + .Now we can define a map ϕ : Λ a,b → S H , ϕ ( u, w ) = ( z, ζ )where z ∈ H is the cross-section point on the geodesic γ from u to w , and ζ is theunit vector tangent to γ at z . The map is clearly injective. Composed with thecanonical projection π introduced in (2.1) we obtain a map π ◦ ϕ : Λ a,b → SM.
Let C a,b = π ◦ ϕ (Λ a,b ) ⊂ SM . This set can be described as follows: C a,b = P ∪ Q ∪ Q , where P consists of the unit vectors based on the circular boundaryof the fundamental region F pointing inward such that the corresponding geodesic γ on the upper half-plane H is ( a, b )-reduced, Q consists of the unit vectors basedon the right vertical boundary of F pointing inward such that either Sγ or T Sγ is ( a, b )-reduced (notice that they cannot both be reduced), and Q consists of theunit vectors based on the left vertical boundary of F pointing inward such thateither Sγ or T − Sγ is ( a, b )-reduced (see Figure 3). Then a.e. orbit of { ϕ t } returnsto C a,b , i.e. C a,b is a cross-section for { ϕ t } , and Λ a,b is a parametrization of C a,b .The map π ◦ ϕ is injective, as follows from Remark 3.2: only one of the geodesics γ , Sγ , T − Sγ , and T Sγ can be reduced.
Based on Proposition 3.4 we introduce the notion of the cross-section point . Itis either the intersection of a reduced geodesic γ with C , or, if γ does not intersect C , its first intersection with C − ∪ C + .Now we can define a map ϕ : Λ a,b → S H , ϕ ( u, w ) = ( z, ζ )where z ∈ H is the cross-section point on the geodesic γ from u to w , and ζ is theunit vector tangent to γ at z . The map is clearly injective. Composed with thecanonical projection π introduced in (2.1) we obtain a map π ◦ ϕ : Λ a,b → SM.
Let C a,b = π ◦ ϕ (Λ a,b ) ⊂ SM . This set can be described as follows: C a,b = P ∪ Q ∪ Q , where P consists of the unit vectors based on the circular boundaryof the fundamental region F pointing inward such that the corresponding geodesic γ on the upper half-plane H is ( a, b )-reduced, Q consists of the unit vectors basedon the right vertical boundary of F pointing inward such that either Sγ or T Sγ is ( a, b )-reduced (notice that they cannot both be reduced), and Q consists of theunit vectors based on the left vertical boundary of F pointing inward such thateither Sγ or T − Sγ is ( a, b )-reduced (see Figure 3). Then a.e. orbit of { ϕ t } returnsto C a,b , i.e. C a,b is a cross-section for { ϕ t } , and Λ a,b is a parametrization of C a,b .The map π ◦ ϕ is injective, as follows from Remark 3.2: only one of the geodesics γ , Sγ , T − Sγ , and T Sγ can be reduced.
P Q Q − C − C + F ab − − Figure 3.
The cross-section (left) and its Λ a,b parametrization (right)4.
Symbolic coding of the geodesic flow via ( a, b ) -continuedfractions. If γ is a geodesic on H , we denote by ¯ γ the canonical projection of γ on M . Fora given geodesic on M that can be reduced in finitely many steps, we can alwayschoose its lift γ to H to be ( a, b )-reduced.The following theorem provides the basis for coding geodesics on the modularsurface using ( a, b )-coding sequences. Theorem 4.1.
Let γ be an ( a, b ) -reduced geodesic on H and ¯ γ its projection to M .Then Based on Proposition 3.4 we introduce the notion of the cross-section point . Itis either the intersection of a reduced geodesic γ with C , or, if γ does not intersect C , its first intersection with C − ∪ C + .Now we can define a map ϕ : Λ a,b → S H , ϕ ( u, w ) = ( z, ζ )where z ∈ H is the cross-section point on the geodesic γ from u to w , and ζ is theunit vector tangent to γ at z . The map is clearly injective. Composed with thecanonical projection π introduced in (2.1) we obtain a map π ◦ ϕ : Λ a,b → SM.
Let C a,b = π ◦ ϕ (Λ a,b ) ⊂ SM . This set can be described as follows: C a,b = P ∪ Q ∪ Q , where P consists of the unit vectors based on the circular boundaryof the fundamental region F pointing inward such that the corresponding geodesic γ on the upper half-plane H is ( a, b )-reduced, Q consists of the unit vectors basedon the right vertical boundary of F pointing inward such that either Sγ or T Sγ is ( a, b )-reduced (notice that they cannot both be reduced), and Q consists of theunit vectors based on the left vertical boundary of F pointing inward such thateither Sγ or T − Sγ is ( a, b )-reduced (see Figure 3). Then a.e. orbit of { ϕ t } returnsto C a,b , i.e. C a,b is a cross-section for { ϕ t } , and Λ a,b is a parametrization of C a,b .The map π ◦ ϕ is injective, as follows from Remark 3.2: only one of the geodesics γ , Sγ , T − Sγ , and T Sγ can be reduced.
P Q Q − C − C + F ab − − Figure 3.
The cross-section (left) and its Λ a,b parametrization (right)4.
Symbolic coding of the geodesic flow via ( a, b ) -continuedfractions. If γ is a geodesic on H , we denote by ¯ γ the canonical projection of γ on M . Fora given geodesic on M that can be reduced in finitely many steps, we can alwayschoose its lift γ to H to be ( a, b )-reduced.The following theorem provides the basis for coding geodesics on the modularsurface using ( a, b )-coding sequences. Theorem 4.1.
Let γ be an ( a, b ) -reduced geodesic on H and ¯ γ its projection to M .Then Figure 3.
The cross-section (left) and its Λ a,b parametrization (right)4.
Symbolic coding of the geodesic flow via ( a, b ) -continuedfractions. If γ is a geodesic on H , we denote by ¯ γ the canonical projection of γ on M . Fora given geodesic on M that can be reduced in finitely many steps, we can alwayschoose its lift γ to H to be ( a, b )-reduced.The following theorem provides the basis for coding geodesics on the modularsurface using ( a, b )-coding sequences. PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 9
Theorem 4.1.
Let γ be an ( a, b ) -reduced geodesic on H and ¯ γ its projection to M .Then (1) each geodesic segment of ¯ γ between successive returns to the cross-section C a,b produces an ( a, b ) -reduced geodesic on H , and each reduced geodesic SL (2 , Z ) -equivalent to γ is obtained this way; (2) the first return of ¯ γ to the cross-section C a,b corresponds to a left shift ofthe coding sequence of γ .Proof. (1) By lifting a geodesic segment on M starting on C a,b to H , we obtain asegment of a geodesic γ on H that is reduced by the definition of the cross-section C a,b . A coding sequence of γ = γ that connects u to w = (cid:98) n , n , . . . (cid:101) a,b , (cid:98) γ (cid:101) = (cid:98) . . . , n − , n − , n , n , n , . . . (cid:101) , is obtained by extending the sequence of digits of w to the past as explained inthe previous section.Let us assume that w >
0, i.e. n ≥
1. The case w < ST − n γ = γ is reduced by Theorem 1.1. Let z and z be the cross-section points on γ and γ , respectively. Then z (cid:48) = T n Sz ∈ γ ;it is the intersection point of γ with the circle | z − n | = 1. We will show thatthe geodesic segment of γ , [ z , z (cid:48) ] projected to M is the segment between twosuccessive returns to the cross-section C a,b . Since ST − n ( z (cid:48) ) = z is the cross-section point on γ , the geodesic segment [ z , z (cid:48) ] projected to M is between tworeturns to C a,b . Recall that a geodesic in F consists of countably many orientedgeodesic segments between consecutive crossings of the boundary of F that areobtained by the canonical projection of γ to F .If z is the intersection of γ with C , there are two possibilities. First, when γ intersects F or γ does not intersect F and ST − γ exists F through it circularboundary, and, second, when γ does not intersect F and ST − γ exists F throughit left vertical boundary. In the first case the segments in F are represented by theintersection with F of the following geodesics in H : T − γ , T − γ , . . . , T − n +1 γ ,either ST − n +1 γ or T − n γ , and either γ , or ST − γ .Suppose that for some intermediate point z ∈ γ , z ∈ [ z , z (cid:48) ] the unit vectortangent to γ at z , ( z, ζ ) is projected to C a,b . By tracing the geodesic γ inside F ,we see that ( z, ζ ) must be projected to (¯ z, ¯ ζ ) with ¯ z on the boundary of F and ¯ ζ directed inward. Then the geodesic through (¯ z, ¯ ζ )(a) enters F through its vertical boundary and exits it also through the verticalboundary,(b) enters F through its vertical boundary and exits through its circular bound-ary, or(c) enters F through its circular boundary and exits through its vertical bound-ary.The following assertions are implied by the analysis of the attractor D a,b . In case(a), T − ST − s γ is not reduced for 1 ≤ s < n since s < n , T − s w > b , hence ST − s w > − b , i.e. ( ST − s u , ST − s w ) / ∈ D a,b , therefore( T − ST − s u , T − ST − s w ) / ∈ Λ a,b . In case (b), either the segment T − n γ exits through the circular boundary of F , ST − n γ = γ is reduced, and we reached the point z on the cross-section. If the segment T − n +1 γ intersects the circular boundary of F , ST − n +1 γ is notreduced. In case (c), ST − n +1 is not reduced.In the second case the first digit of w , n = 2. This is because n = 1 wouldimply b + 1 < w < − b − which is impossible. Thus ST − γ = γ is reduced. Inthis case the geodesic in F consists of the intersection with F of a single geodesic ST − γ that enters F through its right vertical and leave it through its left verticalboundary, since ( T S ) T ( ST − γ ) = ST − γ = γ is reduced. In all cases thegeodesic segment [ z , z (cid:48) ] projected to M is between two consecutive returns to C a,b .If z / ∈ C , by Proposition 3.4, since w > z ∈ C − . Notice that this impliesthat a < − n = 1, and γ = ST − γ is reduced. In this case the geodesicin F also consists of the intersection with F of a single geodesic Sγ that enters F through its right vertical and leave it through its left vertical boundary, since( T S ) T ( Sγ ) = ST − γ = γ is reduced, and hence the geodesic segment [ z , z (cid:48) ]projected to M is between two consecutive returns to C a,b . Continuing this argu-ment by induction in both positive and negative direction, we obtain a bi-infinitesequence of points ( . . . , z − , z − , z , z , z , . . . ) , where z k is the cross-section point of the reduced geodesic γ k in the sequence of γ ,that represents the sequence of all successive returns of the geodesic γ in M to thecross-section C a,b .If ˜ γ is a reduced geodesic in H , SL (2 , Z )-equivalent to γ , then both project tothe same geodesic on M . Therefore, the cross-section point ˜ z of ¯ γ projects on C a,b to a cross-section point z k of γ k for some k . This completes the proof of (1).(2) Since γ = ST − n γ , w = ST − n w = (cid:98) n , n , . . . (cid:101) a,b . The first digit ofthe past is evidently n , and the remaining digits are the same as for γ . Thus (2)follows. (cid:3) The following corollary is immediate.
Corollary 4.2. If γ (cid:48) is SL (2 , Z ) -equivalent to γ , and both geodesics can be reducedin finitely many steps, then the coding sequences of γ and γ (cid:48) differ by a shift. It implies a very important property of ( a, b )-continued fractions that escapes adirect proof.
Corollary 4.3. (The Tail Property)
For almost every pair of real numbersthat are SL (2 , Z ) -equivalent, the “tails” of their ( a, b ) -continued fraction expansionscoincide.Remark . The set of exceptions in Corollary 4.3 is the same as the one describedin Theorem 1.1(3).Thus we can talk about coding sequences of geodesics on M . To any geodesic γ that can be reduced in finitely many steps we associate the coding sequence (3.3)of a reduced geodesic SL (2 , Z )-equivalent to it. Corollary 4.2 implies that thisdefinition does not depend on the choice of a particular representative: sequencesfor equivalent reduced geodesics differ by a shift.Let X a,b be the closure of the set of admissible sequences and σ be the left shiftmap. The coding map Cod : X a,b → C a,b is defined by(4.1) Cod( (cid:98) . . . , n − , n − , n , n , . . . (cid:101) ) = (1 / ( n − , n − , . . . ) , (cid:98) n , n , . . . (cid:101) a,b ) . PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 11
This map is essentially bijective.The symbolic system ( X a,b , σ ) ⊂ ( N Z , σ ) is defined on the infinite alphabet N ⊂ Z \ { } . The product topology on N Z is induced by the distance function d ( x, x (cid:48) ) = 1 m , where x = ( n i ) , x (cid:48) = ( n (cid:48) i ) ∈ N Z , and m = max { k | n i = n (cid:48) i for | i | ≤ k } . Proposition 4.5.
The map
Cod is continuous.Proof. If d ( x, x (cid:48) ) < m , then the ( a, b )-expansions of the attracting end points w ( x )and w ( x (cid:48) ) of the corresponding geodesics given by (3.2) have the same first m digits. Hence the first m convergents of their ( a, b )-expansions are the same, andusing the properties of ( a, b ) continued fraction and the rate of convergence of[16,Theorem 2.1] we obtain | w ( x ) − w ( x (cid:48) ) | < m . Similarly, the first m digits in theconvergent formal minus continued fraction of u ( x ) and u ( x (cid:48) ) are the same, andhence | u ( x ) − u ( x (cid:48) ) | < | u ( x ) u (cid:48) ( x ) | m < m . Therefore the geodesics are uniformly m -close. But the tangent vectors v ( x ) , v ( x (cid:48) ) ∈ C a,b are determined by the intersectionof the corresponding geodesic with the unit circle or the curves C + and C − . Hence,by making m large enough we can make v ( x (cid:48) ) as close to v ( x ) as we wish. (cid:3) In conclusion, the geodesic flow becomes a special flow over a symbolic dynamicalsystem ( X a,b , σ ) on the infinite alphabet N ⊂ Z \ { } . The ceiling function g a,b ( x )on X a,b coincides with the time of the first return of the associated geodesic γ ( x )to the cross-section C a,b . One can establish an explicit formula for g a,b ( x ) as thefunction of the end points of the corresponding geodesic γ ( x ), u ( x ), w ( x ), followingthe ideas explained in [8]. If − ≤ a ≤ ≤ b ≤
1, then g a,b ( x ) is cohomologousto 2 log | w ( x ) | ; more precisely, g a,b ( x ) = 2 log | w ( x ) | +log h ( x ) − log h ( σx ) where h ( x ) = | w ( x ) − u ( x ) | (cid:112) w ( x ) − w ( x ) (cid:112) − u ( x ) . Dual codes
We have seen that a coding sequence for a reduced geodesic from u to w (3.3) iscomprised from the sequence of digits in ( a, b )-expansion of w and the “past”, aninfinite sequence of non-zero integers, each digit of which depends on w and u . Insome special cases the “past” only depends on u , and, in fact, it will coincide withthe sequence of digits of 1 /u by using a so-called dual expansion to ( a, b ).Let ψ ( x, y ) = ( − y, − x ) be the reflection of the plane about the line y = − x . Definition 5.1. If ψ ( D a,b ) coincides with the attractor set D a (cid:48) ,b (cid:48) for some ( a (cid:48) , b (cid:48) ) ∈P , then the ( a (cid:48) , b (cid:48) )-expansion is called the dual expansion to ( a, b ). If ( a (cid:48) , b (cid:48) ) = ( a, b ),then the ( a, b )-expansion is called self-dual . Example 5.2.
The classical situations of ( − , − , −√ , −√ ) and ( − , ), respectively,are shown in Figure 4. Example 5.3.
The expansions ( − n , − n ), n ≥
1, satisfy a weak cycle propertyand have dual expansions that are periodic. A classical example in this series is theHurwitz case ( − , ) whose dual is ( −√ , − √ ) (see [9, 14]). Their domains areshown in Figure 5. The following result gives equivalent characterizations for an expansion to admita dual.
Proposition 5.4.
The following are equivalent: (i) the ( a, b ) -expansion has a dual; (ii) the boundary of the lower part of the set D a,b does not have y -levels with a < y < , and the boundary of the upper part of the set D a,b does not have y -levels with < y < b ; (iii) a and b do not have the strong cycle property.Proof. If the ( a, b )-expansion has a dual ( a ! , b ! )-expansion, then the parameters a ! , b ! are obtained from the boundary of D a,b as follows: the right vertical boundary of -3 -2 -1 0 1 2 3-4-2024 ( a, b ) = ( −√ , −√ ) -4 -3 -2 -1 0 1 2 3-2024-4 ( a, b ) = ( − , ) Figure 4.
Domains of self-dual expansions -3 -2 -1 0 1 2 3-3-2-10123 ( a, b ) = ( − , ) -3 -2 -1 0 1 2 3-4-2024 ( a ! , b ! ) = ( −√ , − √ ) Figure 5.
Domains of dual expansions( a, b ) = ( −√ , −√ )
12 SVETLANA KATOK AND ILIE UGARCOVICI
The following result gives equivalent characterizations for an expansion to admita dual.
Proposition 5.4.
The following are equivalent: (i) the ( a, b ) -expansion has a dual; (ii) the boundary of the lower part of the set D a,b does not have y -levels with a < y < , and the boundary of the upper part of the set D a,b does not have y -levels with < y < b ; (iii) a and b do not have the strong cycle property.Proof. If the ( a, b )-expansion has a dual ( a ! , b ! )-expansion, then the parameters a ! , b ! are obtained from the boundary of D a,b as follows: the right vertical boundary of -3 -2 -1 0 1 2 3-4-2024 ( a, b ) = ( −√ , −√ ) -4 -3 -2 -1 0 1 2 3-2024-4 ( a, b ) = ( − , ) Figure 4.
Domains of self-dual expansions -3 -2 -1 0 1 2 3-3-2-10123 ( a, b ) = ( − , ) -3 -2 -1 0 1 2 3-4-2024 ( a ! , b ! ) = ( −√ , − √ ) Figure 5.
Domains of dual expansions( a, b ) = ( − , ) Figure 4.
Domains of self-dual expansions
12 SVETLANA KATOK AND ILIE UGARCOVICI -3 -2 -1 0 1 2 3-4-2024 ( a, b ) = ( −√ , −√ ) -4 -3 -2 -1 0 1 2 3-2024-4 ( a, b ) = ( − , ) Figure 4.
Domains of self-dual expansions -3 -2 -1 0 1 2 3-3-2-10123 ( a, b ) = ( − , ) -3 -2 -1 0 1 2 3-4-2024 ( a , b ) = ( −√ , − √ ) Figure 5.
Domains of dual expansionsThe following result gives equivalent characterizations for an expansion to admita dual.
Proposition 5.4.
The following are equivalent: (i) the ( a, b ) -expansion has a dual; (ii) the boundary of the lower part of the set D a,b does not have y -levels with a < y < , and the boundary of the upper part of the set D a,b does not have y -levels with < y < b ; (iii) a and b do not have the strong cycle property.Proof. If the ( a, b )-expansion has a dual ( a , b )-expansion, then the parameters a , b are obtained from the boundary of D a,b as follows: the right vertical boundary of( a, b ) = ( − , )
12 SVETLANA KATOK AND ILIE UGARCOVICI -3 -2 -1 0 1 2 3-4-2024 ( a, b ) = ( −√ , −√ ) -4 -3 -2 -1 0 1 2 3-2024-4 ( a, b ) = ( − , ) Figure 4.
Domains of self-dual expansions -3 -2 -1 0 1 2 3-3-2-10123 ( a, b ) = ( − , ) -3 -2 -1 0 1 2 3-4-2024 ( a , b ) = ( −√ , − √ ) Figure 5.
Domains of dual expansionsThe following result gives equivalent characterizations for an expansion to admita dual.
Proposition 5.4.
The following are equivalent: (i) the ( a, b ) -expansion has a dual; (ii) the boundary of the lower part of the set D a,b does not have y -levels with a < y < , and the boundary of the upper part of the set D a,b does not have y -levels with < y < b ; (iii) a and b do not have the strong cycle property.Proof. If the ( a, b )-expansion has a dual ( a , b )-expansion, then the parameters a , b are obtained from the boundary of D a,b as follows: the right vertical boundary of( a (cid:48) , b (cid:48) ) = ( −√ , − √ ) Figure 5.
Dual expansionsThe following result gives equivalent characterizations for an expansion to admita dual.
Proposition 5.4.
The following are equivalent: (i) the ( a, b ) -expansion has a dual; (ii) the boundary of the lower part of the set D a,b does not have y -levels with a < y < , and the boundary of the upper part of the set D a,b does not have y -levels with < y < b ; (iii) a and b do not have the strong cycle property.Proof. If the ( a, b )-expansion has a dual ( a (cid:48) , b (cid:48) )-expansion, then the parameters a (cid:48) , b (cid:48) are obtained from the boundary of D a,b as follows: the right vertical boundary of PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 13 the upper part of D a,b is the ray x = 1 − b (cid:48) , and the left vertical boundary of thelower part of D a,b is the ray x = − − a (cid:48) . Now assume that (ii) does not hold.Then at least one of the parameters a, b has the strong cycle property, and eitherthe left boundary of the upper part of Λ a,b or the right boundary of the lower partof Λ a,b is not a straight line. Assume the former. Then the reflection of D a,b withrespect to the line y = − x is not D a (cid:48) ,b (cid:48) since the map F a (cid:48) ,b (cid:48) is not bijective on it:the black rectangle in Figure 6 belongs to it, but its image under T − , colored ingrey, does not. Thus (i) ⇒ (ii). APPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 13
Then at least one of the parameters a, b has the strong cycle property, and eitherthe left boundary of the upper part of Λ a,b or the right boundary of the lower partof Λ a,b is not a straight line. Assume the former. Then the reflection of D a,b withrespect to the line y = − x is not D a ! ,b ! since the map F a ! ,b ! is not bijective on it:the black rectangle in Figure 6 belongs to it, but its image under T − , colored ingrey, does not. Thus (i) ⇒ (ii). b ! b ! − y = − x Figure 6.
Dual expansions and D a,b Conversely, let the vertical line x = 1 − b " be the right boundary of the upperpart of D a,b and the vertical line x = − − a " be the left boundary of the lowerpart of D a,b . Let [ x a , ∞ ] × { a } be the intersection of D a,b with the horizontal lineat the level a , and [ −∞ , x b ] × { b } be the intersection of D a,b with the horizontalline at the level b . Then a " = x b and b " = x a . We also see that 1 − b " = − t ,where t = x b or t < x b if [ t, x b ] × { } is a segment of the boundary of D a,b . Then − b " + 1 = − t ≤ a " , which implies b " − a " ≥
1. By Lemma 5.6 of [16] x b ≤ − x a ≥
1, therefore(5.1) − ≤ a " ≤ ≤ b " ≤ , and(5.2) Λ a,b = D a,b ∩ { ( u, w ) ∈ ¯ R : − b " ≤ u ≤ − a " } . We now show that ψ ( D a,b ) = D a ! ,b ! is the attractor for F a ! ,b ! , where(5.3) F a ! ,b ! = ψ ◦ F − a,b ◦ ψ − . For ( u, w ) ∈ D a ! ,b ! with a " < w < b " we have ψ − ( u, w ) = ( − w, − u ) with − b " b " we have ψ − ( u, w ) =( − w, − u ) with u < − b " , so F − a,b ( − w, − u ) = ( − w +1 , − u +1), and F a ! ,b ! ( u, w ) = ( u − , w − u, w ) ∈ D a ! ,b ! with w < a " we have ψ − ( u, w ) = ( − w, − u )with u > − a " , so F − a,b ( − w, − u ) = ( − w − , − u − F a ! ,b ! ( u, w ) = ( u +1 , w +1).This proves that (ii) ⇒ (i).Notice that (ii) and (iii) are equivalent by Theorems 4.2 and 4.5 of [16]. ! Figure 6.
Dual expansions and D a,b Conversely, let the vertical line x = 1 − b (cid:48) be the right boundary of the upperpart of D a,b and the vertical line x = − − a (cid:48) be the left boundary of the lowerpart of D a,b . Let [ x a , ∞ ] × { a } be the intersection of D a,b with the horizontal lineat the level a , and [ −∞ , x b ] × { b } be the intersection of D a,b with the horizontalline at the level b . Then a (cid:48) = x b and b (cid:48) = x a . We also see that 1 − b (cid:48) = − t ,where t = x b or t < x b if [ t, x b ] × { } is a segment of the boundary of D a,b . Then − b (cid:48) + 1 = − t ≤ a (cid:48) , which implies b (cid:48) − a (cid:48) ≥
1. By Lemma 5.6 of [16] x b ≤ − x a ≥
1, therefore(5.1) − ≤ a (cid:48) ≤ ≤ b (cid:48) ≤ , and(5.2) Λ a,b = D a,b ∩ { ( u, w ) ∈ ¯ R : − b (cid:48) ≤ u ≤ − a (cid:48) } . We now show that ψ ( D a,b ) = D a (cid:48) ,b (cid:48) is the attractor for F a (cid:48) ,b (cid:48) , where(5.3) F a (cid:48) ,b (cid:48) = ψ ◦ F − a,b ◦ ψ − . For ( u, w ) ∈ D a (cid:48) ,b (cid:48) with a (cid:48) < w < b (cid:48) we have ψ − ( u, w ) = ( − w, − u ) with − b (cid:48) b (cid:48) we have ψ − ( u, w ) =( − w, − u ) with u < − b (cid:48) , so F − a,b ( − w, − u ) = ( − w +1 , − u +1), and F a (cid:48) ,b (cid:48) ( u, w ) = ( u − , w − u, w ) ∈ D a (cid:48) ,b (cid:48) with w < a (cid:48) we have ψ − ( u, w ) = ( − w, − u )with u > − a (cid:48) , so F − a,b ( − w, − u ) = ( − w − , − u − F a (cid:48) ,b (cid:48) ( u, w ) = ( u +1 , w +1).This proves that (ii) ⇒ (i). Notice that (ii) and (iii) are equivalent by Theorems 4.2 and 4.5 of [16]. (cid:3)
Remark . Notice that if an ( a, b )-expansion has a dual, then − ≤ a ≤ ≤ b ≤ Theorem 5.6.
If an ( a, b ) -expansion admits a dual expansion ( a (cid:48) , b (cid:48) ) , and γ is an ( a, b ) -reduced geodesic, then its coding sequence (5.4) (cid:98) γ (cid:101) = (cid:98) . . . , n − , n − , n , n , n , . . . (cid:101) , is obtained by juxtaposing the ( a, b ) -expansion of w = (cid:98) n , n , n , . . . (cid:101) a,b and the ( a (cid:48) , b (cid:48) ) -expansion of /u = (cid:98) n − , n − , . . . (cid:101) a (cid:48) ,b (cid:48) . This property is preserved underthe left shift of the sequence.Proof. We will show that the digits in the ( a (cid:48) , b (cid:48) )-expansion of 1 /u coincide withthe digits of the “past” of (5.4). By (5.3), the following diagramΛ a,b S ψ −−−−→ Λ a (cid:48) ,b (cid:48) R -1a , b (cid:121) (cid:121) R a’ , b’ Λ a,b S ψ −−−−→ Λ a (cid:48) ,b (cid:48) is commutative. The pair ( u , w ) ∈ Λ a,b , therefore ( Su , Sw ) ∈ S Λ a,b ⊂ D a,b ,and (1 /w , /u ) ∈ Λ a (cid:48) ,b (cid:48) . The first digit of the ( a (cid:48) , b (cid:48) )-expansion of 1 /u is n − , so R a (cid:48) ,b (cid:48) (1 /w , /u ) = ( ST − n − (1 /w ) , ST − n − (1 /u ))maps Λ a (cid:48) ,b (cid:48) to itself. Then( u − , w − ) := R − a,b ( u , w ) = ( T n − Su , T n − Sw ) ∈ Λ a,b and ( ST − n − u − , ST − n − w − ) = ( u , w ). Also w − = (cid:98) n − , n , n , . . . (cid:101) a,b , and ST − n − (1 /u ) = 1 /u − = (cid:98) n − , . . . (cid:101) a (cid:48) ,b (cid:48) .Continuing by induction, one proves that all digits of the “past” of the sequence(5.4) are the digits of the ( a (cid:48) , b (cid:48) )-expansion of 1 /u .In order to see what happens under a left shift, we reverse the diagram to obtain:Λ a,b S ψ −−−−→ Λ a (cid:48) ,b (cid:48) R a , b (cid:121) (cid:121) R -1a’ , b’ Λ a,b S ψ −−−−→ Λ a (cid:48) ,b (cid:48) Since the first digit of ( a, b )-expansion of w is n , R a,b ( u , w ) = ( ST − n u , ST − n w )maps Λ a,b to itself. Then ( u , w ) := ( ST − n u , ST − n w ) and w = (cid:98) n , n , . . . (cid:101) a,b .Also (1 /w , /u ) = R − a (cid:48) ,b (cid:48) (1 /w , /u ) = ( T n S (1 /w ) , T n S (1 /u )) , hence 1 /u = (cid:98) n , n − , n − , . . . (cid:101) a (cid:48) ,b (cid:48) . (cid:3) Remark . Under conditions of Theorem 5.6, if γ projects to a closed geodesicon M , then its coding sequence is periodic, and w = (cid:98) n , n , . . . , n m (cid:101) a,b , 1 /u = (cid:98) n m , . . . , n , n (cid:101) a (cid:48) ,b (cid:48) . Theorem 5.8.
If an ( a, b ) -expansion admits a dual expansion, then the symbolicspace ( X a,b , σ ) is a sofic shift. PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 15
Proof.
The “natural” (topological) partition of the set Λ a,b related to the alphabet N is Λ a,b = ∪ n ∈N Λ n , where Λ n are labeled by the symbols of the alphabet N andare defined by the following condition: Λ n = { ( u, w ) ∈ Λ a,b | n ( u, w ) = n ( w ) = n } . In order to prove that the space ( X a,b , σ ) is sofic one needs to find a topologicalMarkov chain ( M a.b , τ ) and a surjective continuous map h : M a,b → X a,b such that h ◦ τ = σ ◦ h .Notice that the elements Λ n are rectangles for large n ; in fact, at most twoelements in the upper part and at most two elements in the lower part of Λ a,b areincomplete rectangles (see Figure 7). APPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 15
Proof.
The “natural” (topological) partition of the set Λ a,b related to the alphabet N is Λ a,b = ∪ n ∈N Λ n , where Λ n are labeled by the symbols of the alphabet N andare defined by the following condition: Λ n = { ( u, w ) ∈ Λ a,b | n ( u, w ) = n ( w ) = n } . In order to prove that the space ( X a,b , σ ) is sofic one needs to find a topologicalMarkov chain ( M a.b , τ ) and a surjective continuous map h : M a,b → X a,b such that h ◦ τ = σ ◦ h .Notice that the elements Λ n are rectangles for large n ; in fact, at most twoelements in the upper part and at most two elements in the lower part of Λ a,b areincomplete rectangles (see Figure 7). -0.3 0.3 0.6-5-4-3-2-1123434-42 Λ Λ Λ Λ Λ − Λ − -0.3 0.3 0.6-5-4-3-2-11234 R (Λ − ) R (Λ − ) R (Λ − ) R (Λ ) R (Λ ) R (Λ ) R (Λ ) R (Λ ) R (Λ ) Figure 7.
The partition of Λ a,b and its image through R a,b .Since Λ a,b has finite rectangular structure, we can sub-divide horizontally theseincomplete rectangles into rectangles, and extend the alphabet N by adding sub-scripts to the corresponding elements of N . For example, if Λ is subdivided intotwo rectangles, Λ = ∪ i =1 Λ i , the “digit” 2 will give rise to two digits, 2 , inthe extended alphabet N " (see Figure 7). We denote the new partition of Λ a,b by ∪ n ∈N " M n . Notice that it consists of rectangles with horizontal and vertical sides.Since the first return R to Λ a,b corresponds to the left shift of the coding sequence x associated to the geodesic ( u, w ), we see that x = { n k } ∞−∞ , where n k is definedby R k ( u, w ) ∈ Λ n k . Now we define the symbolic space M a,b as follows: to eachsequence x ∈ X a,b we associate a geodesic ( u, w ) by (4.1), and define a new codingsequence y = { m k } ∞−∞ , where m k is defined by R k ( u, w ) ∈ M m k , and τ is the leftshift. Figure 7.
The partition of Λ a,b and its image through R a,b .Since Λ a,b has finite rectangular structure, we can sub-divide horizontally theseincomplete rectangles into rectangles, and extend the alphabet N by adding sub-scripts to the corresponding elements of N . For example, if Λ is subdivided intotwo rectangles, Λ = ∪ i =1 Λ i , the “digit” 2 will give rise to two digits, 2 , inthe extended alphabet N (cid:48) (see Figure 7). We denote the new partition of Λ a,b by ∪ n ∈N (cid:48) M n . Notice that it consists of rectangles with horizontal and vertical sides.Since the first return R to Λ a,b corresponds to the left shift of the coding sequence x associated to the geodesic ( u, w ), we see that x = { n k } ∞−∞ , where n k is definedby R k ( u, w ) ∈ Λ n k . Now we define the symbolic space M a,b as follows: to eachsequence x ∈ X a,b we associate a geodesic ( u, w ) by (4.1), and define a new codingsequence y = { m k } ∞−∞ , where m k is defined by R k ( u, w ) ∈ M m k , and τ is the leftshift. We will prove that ( M a,b , τ ) is a topological Markov chain. For this, in accor-dance to [2, Theorem 7.9], it is sufficient to prove that for any pair of distinct symbol n, m ∈ N (cid:48) , R ( M n ) and M m either do not intersect, or intersect “transversally” i.e.their intersection is a rectangle with two horizontal sides belonging to the horizon-tal boundary of M m and two vertical sides belonging to the vertical boundary of R ( M n ). Let us recall that − ≤ a ≤ ≤ b ≤ M n = Λ n is a complete rectangle, it is, in fact, a 1 × R is an infinite vertical rectangle intersecting all M m transversally. If M n isobtained by subdivision of some Λ k and belongs to the lower part of Λ a,b , its hori-zontal boundaries are the levels of the step-function defining the lower componentof D a,b , and by Proposition 5.4, since the lower boundary of D a,b does not have y -levels with a < y <
0, its image is a vertical rectangle intersecting only the lowercomponent of D a,b whose horizontal boundaries are the levels of the step-functiondefining the lower component of D a,b . Therefore, all possible intersections with M m are transversal. A similar argument applies to the case when M n belongs to theupper part of Λ a,b . The map h : M a,b → X a,b is obviously continuous, surjective,and, in addition, h ◦ τ = σ ◦ h . (cid:3) Invariant measures and ergodic properties
Based on the finite rectangular geometric structure of the domain D a,b and theconnections with the geodesic flow on the modular surface, we study some of themeasure-theoretic properties of the Gauss-type map ˆ f a,b : [ a, b ) → [ a, b ),(6.1) ˆ f a,b ( x ) = − x − (cid:22) − x (cid:25) a,b , ˆ f a,b (0) = 0 . Notice that the associated natural extension map ˆ F a,b (6.2) ˆ F a,b ( x, y ) = (cid:18) ˆ f a,b ( x ) , − y − (cid:98)− /x (cid:101) a,b (cid:19) is obtained from the map F a,b induced on the set Λ a,b by the change of coordinates(6.3) x = − /w, y = u (or, equivalently, on the set D a,b ∩ { ( u, w ) | a ≤ w < b } by the change of coordinates x = w, y = − /u ). Therefore the domain ˆΛ a,b of ˆ F a,b is easily identified knowingΛ a,b and may be considered as its “compactification”.Many of the measure-theoretic properties of ˆ f a,b and ˆ F a,b (existence of an ab-solutely continuous invariant measure, ergodicity) follow from the fact that thegeodesic flow ϕ t on the modular surface M can be represented as a special flow( R a,b , Λ a,b , g a,b ) on the spaceΛ g a,b a,b = { ( u, w, t ) : ( u, w ) ∈ Λ a,b , ≤ t ≤ g a,b ( u, w ) } (see Section 2). We recall that R a,b = F a,b | Λ a,b and g a,b is the ceiling function (thetime of the first return to the cross-section C a,b ) parametrized by ( u, w ) ∈ Λ a,b .We start with the fact that the geodesic flow { ϕ t } preserves the smooth (Liou-ville) measure dm = dudwdt ( w − u ) (see, e.g., [3]), hence R a,b preserves the absolutely PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 17 continuous measure dρ = dudw ( w − u ) . Using the change of coordinates (6.3), the mapˆ F a,b preserves the absolutely continuous measure dν = dxdy (1 + xy ) .The set Λ a,b has finite measure dρ if a (cid:54) = 0 and b (cid:54) = 0, since it is uniformlybounded away from the line ∆ = { ( u, w ) : u = w } ⊂ R (see relations (3.5) and(3.6)). In this situation, we can normalize the measure dρ to obtain the smoothprobability measure(6.4) dρ a,b = dρK a,b = dudwK a,b ( w − u ) where K a,b = ρ (Λ a,b ). Similarly, if a (cid:54) = 0 and b (cid:54) = 0, the map ˆ F a,b preserves thesmooth probability measure(6.5) dν a,b = dxdyK a,b (1 + xy ) and K a,b = ρ (Λ a,b ) = ν (ˆΛ a,b ).Returning to the Gauss-type map, ˆ f a,b , one can obtain explicitly a Lebesgueequivalent invariant probability measure µ a,b by projecting the measure ν a,b ontothe x -coordinate (push-forward); this is equivalent to integrating ν a,b over ˆΛ a,b withrespect to the y -coordinate as explained in [4].We can immediately conclude that the systems ( ˆ F a,b , ν a,b ) and ( ˆ f a,b , µ a,b ) areergodic from the fact that the geodesic flow { ϕ t } is ergodic with respect to dm .By using some well-known results about one dimensional maps that are piecewisemonotone and expanding, and the implications for their natural extension maps,we can establish stronger measure-theoretic properties: ( ˆ f a,b , µ a,b ) is exact, and( ˆ F a,b , ν a,b ) is a Bernoulli shift. Here we follow the presentation from [23] based on[21, 18]. Theorem 6.1.
For any a (cid:54) = 0 and b (cid:54) = 0 , the system ( ˆ f a,b , µ a,b ) is exact and itsnatural extension ( ˆ F a,b , ν a,b ) is a Bernoulli shift.Proof. Let us consider first the case − < a < < b <
1. The interval ( a, b ) admitsa countable partition ξ = { X i } i ∈ Z \{ } of open intervals and the map ˆ f a,b satisfiesconditions (A), (F), (U) listed in [23]. Condition (A) is Adler’s distortion estimate:( A ) : ˆ f (cid:48)(cid:48) a,b / ( ˆ f (cid:48) a,b ) is bounded on X = ∪ i ∈ Z \{ } X i , condition (F) requires the finite image property of the partition ξ ,( F ) : ˆ f a,b ( ξ ) = { ˆ f a,b ( X i ) } i ∈ Z \{ } is finite , while condition (U) is a uniformly expanding condition( U ) : | ˆ f (cid:48) a,b | ≥ τ > X. Let m ≥ n ≥ a − m ≤ − /b < a − m − b + n ≤ − /a X . Zweim¨uller [23] showed that any one-dimensional map for which conditions(A), (F), (U) hold is exact and satisfies Rychlik’s conditions described in [18], henceits natural extension map is Bernoulli.We analyze now the case b ≥
1. Let
K > b ( a + 1) K <
1. We will show that there exists γ > x ∈ (cid:84) Ki =0 ˆ f − ia,b ( X ), some iterate ˆ f na,b ( x ) with n ≤ K +1 is expanding, i.e. | ( ˆ f na,b ) (cid:48) ( x ) | ≥ γ .(For the rest of the proof, we simplify the notations and let ˆ f denote the map ˆ f a,b .)Notice that if x ∈ (cid:84) n − i =0 ˆ f − i ( X ), then ˆ f n is differentiable at x and ddx ˆ f n ( x ) = 1( x ˆ f ( x ) · · · ˆ f n − ( x )) . Assume that ab > −
1. We look at the following cases:(i) If a < x <
0, then b − ≤ ˆ f ( x ) ≤ b , and | x ˆ f ( x ) | ≤ | ab | < < x < b , then a ≤ ˆ f ( x ) ≤ a + 1. Let K be such that b ( a + 1) K < ≤ n ≤ K such that 0 < ˆ f i ( x ) < a + 1 for i =1 , , . . . , n − a < ˆ f n ( x ) <
0, or 0 < ˆ f i ( x ) < a + 1 for i = 1 , , . . . , K .In the former case we have that(6.6) | x ˆ f ( x ) · · · ˆ f n ( x ) | ≤ | ab ( a + 1) n − | < , while in the latter case(6.7) | x ˆ f ( x ) · · · ˆ f K ( x ) | ≤ | b ( a + 1) K | < . In the case ab = −
1, let τ, (cid:15) > b < − / ( a + τ ) < b + 1 and a − < − / ( b − (cid:15) ) < a . We have:(i) If a < x < a + τ , then b − < ˆ f ( x ) < − / ( a + τ ), and | x ˆ f ( x ) | ≤ | a/ ( a + τ ) | <
1. If a + τ ≤ x <
0, then | x ˆ f ( x ) | ≤ | b ( a + τ ) | < b − (cid:15) < x < b , then 0 < ˆ f ( x ) < a + 1 and one has either (6.6) with n ≥ < x ≤ b − (cid:15) , then one has (6.6) or (6.7) where b is replaced by b − (cid:15) .In conclusion, there exists a constant γ > x ∈ (cid:84) Ki =0 ˆ f − ia,b ( X )some iterate ˆ f na,b ( x ) with n ≤ K + 1 satisfies the condition | ( ˆ f na,b ) (cid:48) ( x ) | ≥ γ . Thisimplies that the iterate ˆ f Na,b , with N = ( K + 1)!, is uniformly expanding, i.e. itsatisfies property (U). Since properties (A) and (F) are automatically satisfied byany iterate of ˆ f a,b (see [23]), we have that ˆ F Na,b is Bernoulli. Using one of Ornstein’sresults [17, Theorem 4, p. 39], it follows that ˆ F a,b is Bernoulli. (cid:3) The next result gives a formula of the measure theoretic entropy of ( ˆ F a,b , ν a,b ). PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 19
Theorem 6.2.
The measure-theoretic entropy of ( ˆ F a,b , ν a,b ) is given by (6.8) h ν a,b ( ˆ F a,b ) = 1 K a,b π Proof.
To compute the entropy of this two-dimensional map, we use Abramov’sformula [1]: h ˜ m ( { φ t } ) = h ρ a,b ( R a,b ) (cid:82) Λ a,b g a,b dρ a,b , where ˜ m is the normalized Liouville measure d ˜ m = dmm ( SM ) . It is well-known that m ( SM ) = π / h ˜ m ( { φ t } ) = 1 (see, e.g., [20]). The measure d ˜ m can berepresented by the Ambrose-Kakutani theorem [5] as a smooth probability measureon the space Λ g a,b a,b (6.9) d ˜ m = dρ a,b dt (cid:82) Λ a,b g a,b dρ a,b where dρ a,b is the probability measure on the cross-section Λ a,b given by (6.4). Thisimplies that d ˜ m = dρdtK a,b (cid:82) Λ a,b g a,b dρ a,b = dmK a,b (cid:82) Λ a,b g a,b dρ a,b . Therefore K a,b (cid:82) Λ a,b g a,b dρ a,b = m ( SM ) = π / h ν a,b ( ˆ F a,b ) = h ρ a,b ( R a,b ) = (cid:90) Λ a,b g a,b dρ a,b = 1 K a,b π . (cid:3) Since ( ˆ F a,b , ν a,b ) is the natural extension of ( ˆ f a,b , µ a,b ), the measure-theoreticentropies of the two systems coincide, hence(6.10) h µ a,b ( ˆ f a,b ) = 1 K a,b π . As an immediate consequence of the above entropy formula we derive a growthrate relation for the denominators of the partial quotients p n /q n of ( a, b )-continuedfraction expansions, similar to the classical cases. Proposition 6.3.
Let { q n ( x ) } be the sequence of the denominators of the partialquotients p n /q n associated to the ( a, b ) -continued fraction expansion of x ∈ [ a, b ) .Then (6.11) lim n →∞ log q n ( x ) n = 12 h µ a,b ( ˆ f a,b ) = 1 K a,b π for a.e. x. Proof.
The proof is similar to the classical case: using the Birkhoff’s ergodic theo-rem one has lim n →∞ log q n ( x ) n = − (cid:90) ba log | x | dµ a,b . At the same time, Rokhlin’s formula tells us that h µ a,b ( ˆ f a,b ) = (cid:90) ba log | ˆ f (cid:48) a,b | dµ a,b = − (cid:90) ba log | x | dµ a,b , hence the conclusion. (cid:3) Some explicit formulas for the invariant measure µ a,b In order to obtain explicit formulas for µ a,b and h µ a,b ( ˆ f a,b ), one obviously needsan explicit description of the domain D a,b . In [16] we describe an algorithmicapproach for finding the boundaries of D a,b for all parameter pairs ( a, b ) outside ofa negligible exceptional parameter set E . Let us point out that the set D a,b mayhave an arbitrary large number of horizontal boundary segments. The qualitativestructure of D a,b is given by the cycle properties of a and b . This structure remainsunchanged for all pairs ( a, b ) having cycles with similar combinatorial complexity.For a large part of the parameter set the cycle descriptions are relatively simple(see [16, Section 4]) and we discuss it herein.In what follows, we focus our attention on the situation − ≤ a ≤ ≤ b ≤ a = − b we assume that a ≤ − b .We treat the case 1 ≤ − a ≤ b + 1 and a ≤ − b + m ≤ a + 1 (for some m ≥ D a,b ∩ { ( u, w ) | u < , a ≤ w ≤ b } are given by( − , b − , (cid:16) − , T − S ( b − (cid:17) , . . . , (cid:16) − m + 1 m , ( T − S ) ( m − ( b − (cid:17) , (cid:16) − , − a − (cid:17) while the corners of the boundary segments in the lower region D a,b ∩ { ( u, w ) | u > , a ≤ w ≤ b } are given by (cid:16) m, − b + m (cid:17) , ( m + 1 , a + 1) . Therefore the set ˆΛ a,b is given byˆΛ a,b = m − (cid:91) p =1 [( T − S ) p − ( b − , ( T − S ) p ( b − × [0 , pp + 1 ] ∪ [( T − S ) m − ( b − , − a − × [0 , mm + 1 ] ∪ [ − a − , b ] × [0 , ∪ [ a, − b + m ] × [ − m , ∪ [ − b + m, a + 1] × [ − m + 1 , Theorem 7.1. If ≤ − a ≤ b + 1 and a ≤ − b + m ≤ a + 1 , then µ a,b = 1 K a,b h a,b ( x ) dx , where K a,b = log[( m − a )(1 + b ) − m ] and h a,b ( x ) = h + a,b ( x ) + h − a,b ( x ) with h + a,b ( x ) = x + p +1 p if ( T − S ) p − ( b − ≤ x < ( T − S ) p ( b − , p = 1 , . . . , m − x + m +1 m if ( T − S ) m − ( b − ≤ x < − a − x + 1 if − a − ≤ x < b PPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 21APPLICATIONS OF ( a, b )-CONTINUED FRACTION TRANSFORMATIONS 21 ! ! ! ! Figure 8.
Typical domain ˆΛ a,b for the case studied and h − a,b ( x ) = m − x if a ≤ x < − b + m m + 1 − x if − b + m ≤ x < a + 1 . Proof.
The density formulas are obtained from the simple integration result(7.2) % dc xy ) dy = − x &
11 + dx −
11 + cx ’ = d dx − c cx . For the density in the upper part of ˆΛ a,b , y ≥
0, all integrals have the lowerboundary c = 0, hence the result of (7.2) becomes 1 / ( x + 1 /d ). This gives thedescription of h + a,b ( x ). For the density in the lower part of ˆΛ a,b , y ≤
0, all integralshave the upper boundary d = 0, hence the result − / ( − /c − x ) and the descriptionof h − a,b ( x ). By a somewhat tedious computation, we get K a,b = % Λ a,b h a,b ( x ) dx = log[( m − a )(1 + b ) − m ] , and this completes the proof. ! References [1] L. M. Abramov,
On the entropy of a flow , Sov. Math. Doklady. (1959), no. 5, 873–875.[2] R. Adler,
Symbolic dynamics and Markov partitions , Bull. Amer. Math. Soc. (1998),no. 1, 1–56.[3] R. Adler, L. Flatto, Cross section maps for geodesic flows, I (The Modular surface) ,Birkh¨auser, Progress in Mathematics (ed. A. Katok) (1982), 103–161.[4] R. Adler, L. Flatto,
Geodesic flows, interval maps, and symbolic dynamics , Bull. Amer.Math. Soc. (1991), no. 2, 229–334.[5] W. Ambrose, S. Kakutani, Structure and continuity of measurable flows , Duke Math. J., (1942), 25–42.[6] E. Artin, Ein Mechanisches System mit quasiergodischen Bahnen , Abh. Math. Sem. Univ.Hamburg (1924), 170–175. Figure 8.
Typical domain ˆΛ a,b for the case studied and h − a,b ( x ) = m − x if a ≤ x < − b + m m + 1 − x if − b + m ≤ x < a + 1 . Proof.
The density formulas are obtained from the simple integration result(7.2) (cid:90) dc xy ) dy = − x (cid:18)
11 + dx −
11 + cx (cid:19) = d dx − c cx . For the density in the upper part of ˆΛ a,b , y ≥
0, all integrals have the lowerboundary c = 0, hence the result of (7.2) becomes 1 / ( x + 1 /d ). This gives thedescription of h + a,b ( x ). For the density in the lower part of ˆΛ a,b , y ≤
0, all integralshave the upper boundary d = 0, hence the result − / ( − /c − x ) and the descriptionof h − a,b ( x ). By a somewhat tedious computation, we get K a,b = (cid:90) Λ a,b h a,b ( x ) dx = log[( m − a )(1 + b ) − m ] , and this completes the proof. (cid:3) References [1] L. M. Abramov,
On the entropy of a flow , Sov. Math. Doklady. (1959), no. 5, 873–875.[2] R. Adler,
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