Structure of attractors for boundary maps associated to Fuchsian groups
SSTRUCTURE OF ATTRACTORS FOR BOUNDARY MAPSASSOCIATED TO FUCHSIAN GROUPS
SVETLANA KATOK AND ILIE UGARCOVICI
Dedicated to the memory of Roy Adler
Abstract.
We study dynamical properties of generalized Bowen-Series boundarymaps associated to cocompact torsion-free Fuchsian groups. These maps are definedon the unit circle (the boundary of the Poincar´e disk) by the generators of the groupand have a finite set of discontinuities. We study the two forward orbits of eachdiscontinuity point and show that for a family of such maps the cycle property holds:the orbits coincide after finitely many steps. We also show that for an open set ofdiscontinuity points the associated two-dimensional natural extension maps possessglobal attractors with finite rectangular structure . These two properties belong tothe list of “good” reduction algorithms, equivalence or implications between whichwere suggested by Don Zagier [11]. Introduction
Let Γ be a finitely generated Fuchsian group of the first kind acting on the hyperbolicplane. We will use either the upper half-plane model H or the unit disk model D , andwill denote the Euclidean boundary for either model by S : for the upper half plane S = ∂ ( H ) = P ( R ), and for the unit disk S = ∂ ( D ) = S .Let F be a fundamental domain for Γ with an even number N of sides identified bythe set of generators G = { T , . . . , T N } of Γ, and τ : S → G be a surjective map locallyconstant on S \ J , where J = { x , . . . , x N } is an arbitrary set of jumps. A boundarymap f : S → S is defined by f ( x ) = τ ( x ) x . It is a piecewise fractional-linear map whoseset of discontinuities is J . Let ∆ = { ( x, x ) | x ∈ S } ⊂ S × S be the diagonal of S × S ,and F : S × S \ ∆ → S × S \ ∆ be given by F ( x, y ) = ( τ ( y ) x, τ ( y ) y ) . This is a (natural) extension of f , and if we identify ( x, y ) ∈ S × S \ ∆ with an orientedgeodesic from x to y , we can think of F as a map on geodesics ( x, y ) which we will alsocall a reduction map .Several years ago Don Zagier[11] proposed a list of possible notions of “good” re-duction algorithms associated to Fuchsian groups and conjectured equivalences or im-plications between them. In this paper we consider two of these notions, namely theproperties that “good” reduction algorithms should (i) satisfy the cycle property, and(ii) have an attractor with finite rectangular structure. We prove that for each co-compact torsion-free Fuchsian group there exist families of reduction algorithms whichsatisfy these properties . Thus our results are contributions towards Zagier’s conjecture. Date : September 30, 2016; revised May 1, 2017; accepted for publication in Geometriae Dedicata.2010
Mathematics Subject Classification.
Key words and phrases.
Fuchsian groups, reduction theory, boundary maps, attractor.The second author is partially supported by a Simons Foundation Collaboration Grant. a r X i v : . [ m a t h . D S ] M a y SVETLANA KATOK AND ILIE UGARCOVICI
Although the statement that each Fuchsian group admits a “good” reduction algo-rithm is not part of Zagier’s conjecture, it is certainly related to it, and for the purposesof this paper, we state it here.
Reduction Theory Conjecture for Fuchsian groups.
For every Fuchsian groupΓ there exist F , G as above, and an open set of J (cid:48) s in S N such that(1) The map F possesses a bijectivity domain Ω having a finite rectangular struc-ture , i.e., bounded by non-decreasing step-functions with a finite number ofsteps.(2) Every point ( x, y ) ∈ S × S \ ∆ is mapped to Ω after finitely many iterations of F . Remark . If property (2) holds, then Ω is a global attractor for the map F , i.e.(1.1) Ω = ∞ (cid:92) n =0 F n ( S × S \ ∆) . This conjecture was proved by the authors in [6] for Γ = SL (2 , Z ) and boundary mapsassociated to ( a, b )-continued fractions. Notice that for some classical cases of continuedfraction algorithms property (2) holds only for almost every point, while property (1.1)remains valid.In this paper we address the conjecture for surface groups. In the Poincar´e unit diskmodel D endowed with the hyperbolic metric(1.2) 2 | dz | − | z | , let Γ be a Fuchsian group, i.e. a discrete group of orientation preserving isometriesof D , acting freely on D with Γ \ D compact domain. Such Γ is called a surface group ,and the quotient Γ \ D is a compact surface of constant negative curvature − g >
1. A classical (Ford) fundamental domain for Γ is a 4 g -sided regularpolygon centered at the origin (see a sketch of the construction in [5] in the mannerof [4], and for the complete proof see [8]). A more suitable for our purposes (8 g − F was described by Adler and Flatto in [1]. They showedthat all angles of F are equal to π and, therefore, its sides are geodesic segments whichsatisfy the extension condition of Bowen and Series [3]: the geodesic extensions of thesesegments never intersect the interior of the tiling sets γ F , γ ∈ Γ. Figure 1 shows sucha construction for g = 2.Using notations similar to [1], we label the sides of F in a counterclockwise orderby numbers 1 ≤ i ≤ g −
4, as they are arcs of the corresponding isometric circlesof generators T i . We denote the corresponding vertices of F by V i , so that the side i connects the vertices V i and V i +1 (mod 8 g − σ ( i ) = (cid:40) g − i mod (8 g −
4) for odd i − i mod (8 g −
4) for even i .
The generators T i associated to this fundamental domain are M¨obius transformationssatisfying the following properties: T σ ( i ) T i = Id (1.3) T i ( V i ) = V ρ ( i ) , where ρ ( i ) = σ ( i ) + 1(1.4) T ρ ( i ) T ρ ( i ) T ρ ( i ) T i = Id (1.5) OUNDARY MAPS FOR FUCHSIAN GROUPS 3 P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q F V V V V V V V V V V V V Figure 1.
The fundamental domain F for a genus 2 surfaceWe denote by P i Q i +1 the oriented (infinite) geodesic that extends the side i to theboundary of the fundamental domain F . It is important to remark that P i Q i +1 is theisometric circle for T i , and T i ( P i Q i +1 ) = Q σ ( i )+1 P σ ( i ) is the isometric circle for T σ ( i ) sothat the inside of the former isometric circle is mapped to the outside of the latter.The counter-clockwise order of theses points on S is(1.6) P , Q , P , Q , . . . , P g − , Q g − , P . Bowen and Series [3] defined the boundary map f ¯ P : S → S (1.7) f ¯ P ( x ) = T i ( x ) if P i ≤ x < P i +1 . with the set of jumps J = ¯ P = { P , . . . , P g − } . They showed that such a map isMarkov with respect to the partition (1.6), expanding, and satisfies R´enyi’s distor-tion estimates, hence it admits a unique finite invariant ergodic measure equivalent toLebesgue measure.Adler and Flatto [1] proved the existence of an invariant domain for the correspond-ing natural extension map F ¯ P , Ω ¯ P ⊂ S × S . Moreover, the set Ω ¯ P they identified hasa regular geometric structure, what we call finite rectangular (see Figure 2, with Ω ¯ P shown as a subset of [ − π, π ] ). The maps F ¯ P and f ¯ P are ergodic . Both Series [9] andAdler-Flatto [1] explain how the boundary map can be used for coding symbolicallythe geodesic flow on D / Γ. Notations.
For
A, B ∈ S , the various intervals on S between A and B (with the coun-terclockwise order) will be denoted by [ A, B ] , ( A, B ] , [ A, B ) and (
A, B ). The geodesic(segment) from a point C ∈ S (or D ) to D ∈ S (or D ) will be denoted by CD . More precisely, F ¯ P is a K -automorphism, property that is equivalent to f ¯ P being an exact endo-morphism. SVETLANA KATOK AND ILIE UGARCOVICI P P P P P P P P P P P P Figure 2.
Domain of the Bowen-Series map F ¯ P as a subset of [ − π, π ] Our object of study is a generalization of the Bowen-Series boundary map. Weconsider an open set of jumps J = ¯ A = { A , . . . , A g − } with the only condition A i ∈ ( P i , Q i ), and define f ¯ A : S → S by(1.8) f ¯ A ( x ) = T i ( x ) if A i ≤ x < A i +1 , and the corresponding two-dimensional map:(1.9) F ¯ A ( x, y ) = ( T i ( x ) , T i ( y )) if A i ≤ y < A i +1 . A key ingredient in analyzing map F ¯ A is what we call the cycle property of thepartition points { A , . . . , A g − } . Such a property refers to the structure of the orbitsof each A i that one can construct by tracking the two images T i A i and T i − A i of thesepoints of discontinuity of the map f ¯ A . It happens that some forward iterates of thesetwo images T i A i and T i − A i under f ¯ A coincide. This is another property from Zagier’slist of “good” reduction algorithms.We state the cycle property result below and provide a proof in Section 3. Theorem 1.2 (Cycle Property) . Each partition point A i ∈ ( P i , Q i ) , ≤ i ≤ g − ,satisfies the cycle property, i.e., there exist positive integers m i , k i such that f m i ¯ A ( T i A i ) = f k i ¯ A ( T i − A i ) . If a cycle closes up after one iteration(1.10) f ¯ A ( T i A i ) = f ¯ A ( T i − A i ) , we say that the point A i satisfies the short cycle property . Under this condition, weprove the following: Theorem 1.3 (Main Result) . If each partition point A i satisfies the short cycle prop-erty (1.10) , then there exists a set Ω ¯ A ⊂ S × S with the following properties: (1) Ω ¯ A has a finite rectangular structure, and F ¯ A is (essentially) bijective on Ω ¯ A . OUNDARY MAPS FOR FUCHSIAN GROUPS 5 (2)
Almost every point ( x, y ) ∈ S × S \ ∆ is mapped to Ω ¯ A after finitely manyiterations of F ¯ A , and Ω A is a global attractor for the map F ¯ A , i.e., Ω A = ∞ (cid:92) n =0 F n ¯ A ( S × S \ ∆) . A A A A A A A A A A A A Figure 3.
Domain (and attractor) of the generalized Bowen-Series map F ¯ A Notice that the set of partitions satisfying the short cycle property contains an openset with this property, as explained in Remark 3.11. Thus we prove the ReductionTheory Conjecture. We believe that this result is true in greater generality, i.e., for allpartitions ¯ A = { A i } with A i ∈ ( P i , Q i ). Organization of the paper.
In Section 2 we prove properties (1) and (2) of theReduction Theory Conjecture for the classical Bowen-Series case when the partitionpoints are given by the set ¯ P = { P i } . In Section 3 we prove the cycle property for anypartition ¯ A = { A i } with A i ∈ ( P i , Q i ). In Section 4 we determine the structure of theset Ω ¯ A in the case when the partition ¯ A satisfies the short cycle property and provethe bijectivity of the map F ¯ A on Ω ¯ A . In Section 5 we identify the trapping region forthe map F ¯ A and prove that every point in S × S \ ∆ is mapped to it after finitely manyiterations of the map F ¯ A . And finally, in Section 6 we prove that almost every point S × S \ ∆ is mapped to Ω ¯ A after finitely many iterations of the map F ¯ A and completethe proof of Theorem 1.3. In Section 7 we apply our results to calculate the invariantprobability measures for the maps F ¯ A and f ¯ A .2. Bowen-Series case
In this section we prove properties (1) and (2) of the Reduction Theory Conjecturefor the Bowen-Series classical case, where the partition ¯ A is given by the set of points¯ P = { P , . . . , P g − } . Theorem 2.1.
The two-dimensional Bowen-Series map F ¯ P satisfies properties (1) and(2) of the Reduction Theory Conjecture. SVETLANA KATOK AND ILIE UGARCOVICI
Before we prove this theorem, we state a useful proposition that can be easily derivedusing the isometric circles and the conformal property of M¨obius transformations (seealso Theorem 3.4 of [1]).
Proposition 2.2. T i maps the points P i − , P i , Q i , P i +1 , Q i +1 , Q i +2 respectively to P σ ( i )+1 , Q σ ( i )+1 , Q σ ( i )+2 , P σ ( i ) − , P σ ( i ) , Q σ ( i ) .Proof of Theorem 2.1. In this case the set Ω ¯ P is determined by the corner points lo-cated in each horizontal strip { ( x, y ) ∈ S × S | y ∈ [ P i , P i +1 ) } (see Figure 4) with coordinates( P i , Q i ) (upper part) and ( Q i +2 , P i +1 ) (lower part) . P i -1 P i -1 Q i +2 P i (P i ,Q i )Q i P i +1 P i Figure 4.
Strip y ∈ [ P i , P i +1 ] of Ω ¯ P This set obviously has a finite rectangular structure. One can also verify immediatelythe essential bijectivity, by investigating how different regions of Ω ¯ P are mapped by F ¯ P . More precisely we look at the strip S i of Ω ¯ P given by y ∈ [ P i , P i +1 ], and its imageunder F ¯ P , in this case T i .We consider the following decomposition of this strip: ˜ S i = [ Q i +2 , P i − ] × [ P i , Q i ] (redrectangular horizontal piece), ˆ S i = [ Q i +2 , P i ] × [ Q i , P i +1 ] (green horizontal rectangularpiece). Now T i ( ˜ S i ) = [ T i Q i +2 , T i P i − ] × [ T i P i , T i Q i ] = [ Q σ ( i ) , P σ ( i )+1 ] × [ Q σ ( i )+1 , Q σ ( i )+2 ] T i ( ˆ S i ) = [ T i Q i +2 , T i P i ] × [ T i Q i , T i P i +1 ] = [ Q σ ( i ) , Q σ ( i )+1 ] × [ Q σ ( i )+2 , P σ ( i ) − ]Therefore T i ( S i ) is a complete vertical strip in Ω ¯ P , with Q σ ( i ) ≤ x ≤ Q σ ( i )+1 . Thiscompletes the proof of the property (1).We now prove property (2) for the set Ω P .Consider ( x, y ) ∈ S × S \ ∆. Notice that there exists n ( x, y ) > x n , y n obtained from the n th iterate of F ¯ P , ( x n , y n ) = F n ¯ P ( x, y ), are not insidethe same isometric circle; in other words, ( x n , y n ) (cid:54)∈ X i = [ P i , Q i +1 ] × [ P i , P i +1 ) forall 1 ≤ i ≤ g −
4. Indeed, if one assumes that both coordinates ( x n , y n ) = F n ¯ P ( x, y )belong to such a set X i for all n ≥
0, each time we iterate the pair ( x n , y n ) we applyone of the maps T i which is expanding in the interior of its isometric circle. Thusthe distance between x n and y n would grow sufficiently for the points to be insidedifferent isometric circles. Therefore, there exists n > y n is in some interval[ P i , P i +1 ) ⊂ [ P i , Q i +1 ] and x n (cid:54)∈ [ P i , Q i +1 ]. OUNDARY MAPS FOR FUCHSIAN GROUPS 7 P P P P P P P P P P P P Q Figure 5.
Bijectivity of the Bowen-Series map F ¯ P Notice that, from the definition of Ω ¯ P , in order to prove the attracting prop-erty, we need to analyze the situations ( x n , y n ) ∈ [ P i − , P i ] × [ P i , Q i ] and ( x n , y n ) ∈ [ Q i +1 , Q i +2 ] × [ P i , P i +1 ) and show that a forward iterate lands in Ω ¯ P . Case I.
If ( x n , y n ) ∈ [ Q i +1 , Q i +2 ] × [ P i , P i +1 ), then F ¯ P ( x n , y n ) ∈ [ T i Q i +1 , T i Q i +2 ] × [ T i P i , T i P i +1 ) = [ P σ ( i ) , Q σ ( i ) ] × [ Q σ ( i )+1 , P σ ( i ) − ) . The subset [ P σ ( i ) , Q σ ( i ) ] × [ Q σ ( i )+1 , P σ ( i ) − ] is included in Ω ¯ P so we only need to analyzethe situation ( x n +1 , y n +1 ) ∈ [ P k +2 , Q k +2 ] × [ P k , P k +1 ), where k = σ ( i ) −
2. Then( x n +2 , y n +2 ) = F P ( x n , y n ) = T k T i ( x n , y n ) ∈ [ T k P k +2 , Q σ ( k ) ] × [ Q σ ( k )+1 , P σ ( k ) − ) . Notice that T k P k +2 ∈ [ P σ ( k ) , Q σ ( k ) ]. The subset [ T k P k +2 , Q σ ( k ) ] × [ Q σ ( k )+1 , P σ ( k ) − ] isincluded in Ω ¯ P so we only need to analyze the situation( x n +2 , y n +2 ) ∈ [ T k P k +2 , Q σ ( k ) ] × [ P σ ( k ) − , P σ ( k ) − ) ⊂ [ P σ ( k ) , Q σ ( k ) ] × [ P σ ( k ) − , P σ ( k ) − ) . Notice that σ ( k ) − σ ( σ ( i ) − − i (direct verification), so we are back toanalyzing the situation ( x n +2 , y n +2 ) ∈ [ P i +2 , Q i +2 ] × [ P i , P i +1 ). The boundary map f ¯ P is expanding, so it is not possible for the images of the interval ( y n , P i +1 ) (on the y -axis) to alternate indefinitely between the intervals [ P i , P i +1 ] and [ P σ ( i ) − , P σ ( i ) − ],where T i P i +1 = P σ ( i ) − .This means that either some even iterate F m ( x n , y n ) ∈ [ P i +2 , Q i +2 ] × [ Q i +3 , P i ) ⊂ Ω ¯ P or some odd iterate F m +1 ( x n , y n ) ∈ [ P σ ( i ) , Q σ ( i ) ] × [ Q σ ( i )+1 , P σ ( i ) − ] ⊂ Ω ¯ P . Case II.
If ( x n , y n ) ∈ [ P i − , P i ] × [ P i , Q i ], then F ¯ P ( x n , y n ) ∈ [ T i P i − , T i P i ] × [ T i P i , T i Q i ] = [ P σ ( i )+1 , Q σ ( i )+1 ] × [ Q σ ( i )+1 , Q σ ( i )+2 ] . SVETLANA KATOK AND ILIE UGARCOVICI
There are two subcases that we need to analyze:( a ) ( x n +1 , y n +1 ) ∈ [ P k , Q k ] × [ Q k , P k +1 ) ( b ) ( x n +1 , y n +1 ) ∈ [ P k , Q k ] × [ P k +1 , Q k +1 ] , where k = σ ( i ) + 1. Case (a)
If ( x n +1 , y n +1 ) ∈ [ P k , Q k ] × [ Q k , P k +1 ], then( x n +2 , y n +2 ) ∈ T k ([ P k , Q k ] × [ Q k , P k +1 )) = [ Q σ ( k )+1 , Q σ ( k )+2 ] × [ Q σ ( k )+2 , P σ ( k ) − ] . Notice that σ ( k ) + 1 = σ ( σ ( i ) + 1) + 1 = 4 g + i − x n +2 , y n +2 ) ∈ [ Q g + i − , Q g + i − ] × [ Q g + i − , P g + i − ) the onlyproblematic region is ( x n +2 , y n +2 ) ∈ [ P g + i − , Q g + i − ] × [ Q g + i − , Q g + i ]. Case (b)
If ( x n +1 , y n +1 ) ∈ [ P k , Q k ] × [ P k +1 , Q k +1 ], then( x n +2 , y n +2 ) ∈ T k +1 ([ P k , Q k ] × [ P k +1 , Q k +1 ])= [ P σ ( k +1)+1 , T k +1 Q k ] × [ Q σ ( k +1)+1 , Q σ ( k +1)+2 ] . Notice that T k +1 Q k ∈ [ P σ ( k +1)+1 , Q σ ( k +1)+1 ] and σ ( k +1)+1 = i − x n +2 , y n +2 ) ∈ [ P i − , Q i − ] × [ Q i − , Q i ].To summarize, we started with ( x n +1 , y n +1 ) ∈ [ P σ ( i )+1 , Q σ ( i )+1 ] × [ Q σ ( i )+1 , Q σ ( i )+2 ]and found two situations that need to be analyzed: ( x n +2 , y n +2 ) ∈ [ P i − , Q i − ] × [ Q i − , Q i ] and ( x n +2 , y n +2 ) ∈ [ P g + i − , Q g + i − ] × [ Q g + i − , Q g + i ].We prove in what follows that it is not possible for all future iterates F m ( x n , y n ) tobelong to the sets of type [ P k , Q k ] × [ Q k , Q k +1 ]. First, it is not possible for all F m ( x n , y n )(starting with some m >
0) to belong only to type-a sets [ P k m , Q k m ] × [ Q k m , P k m +1 ],where the sequence { k m } is defined recursively as k m = σ ( k m − ) + 2, because such aset is included in the isometric circle X k m , and the argument at the beginning of theproof disallows such a situation.Also, it is not possible for all F m ( x n , y n ) (starting with some m >
0) to belong onlyto type-b sets [ P k m , Q k m ] × [ P k m +1 , Q k m +1 ], where k m = σ ( k m − + 1) + 1: this wouldimply that the pairs of points ( y n + m , Q k n + m +1 ) (on the y -axis) will belong to the sameinterval [ P k n + m +1 , Q k n + m +1 ] which is impossible due to expansiveness property of themap f ¯ P . Therefore, there exists a pair ( x l , y l ) in the orbit of F m ( x n , y n ) such that( x l , y l ) ∈ [ P j , Q j ] × [ P j +1 , Q j +1 ] (type-b)for some 1 ≤ j ≤ g − x l +1 , y l +1 ) ∈ [ P j (cid:48) , T j +1 Q j ] × [ Q j (cid:48) , P j (cid:48) +1 ] ⊂ [ P j (cid:48) , Q j (cid:48) ] × [ Q j (cid:48) , P j (cid:48) +1 ] (type-a) , where j (cid:48) = σ ( j + 1) + 1. Then( x l +2 , y l +2 ) ∈ T j (cid:48) ([ P j (cid:48) , T j +1 Q j ] × [ Q j (cid:48) , P j (cid:48) +1 ]) = [ Q j (cid:48)(cid:48) , T j (cid:48) T j +1 Q j ] × [ Q j (cid:48)(cid:48) +1 , P j (cid:48)(cid:48) − ]where j (cid:48)(cid:48) = σ ( j (cid:48) ) + 1.Using the results of the Appendix (Corollary 8.3), we have that the arc lengthdistance (cid:96) ( P j (cid:48) , T j +1 Q j ) = (cid:96) ( T j +1 P j , T j +1 Q j ) < (cid:96) ( P j (cid:48) , Q j (cid:48) ) . Now we can use Corollary 8.2 (ii) applied to the point T j +1 Q j ∈ [ P j (cid:48) , Q j (cid:48) ] to concludethat T j (cid:48) T j +1 Q j ∈ [ Q j (cid:48)(cid:48) , P j (cid:48)(cid:48) +1 ]. Therefore ( x l +2 , y l +2 ) ∈ Ω ¯ P . This completes the proofof the property (2). (cid:3) Remark . One can prove along the same lines that if the partition ¯ A is given bythe set ¯ Q = { Q , . . . , Q g − } , the properties (1) and (2) of the Reduction TheoryConjecture also hold. OUNDARY MAPS FOR FUCHSIAN GROUPS 9 The cycle property
The map f ¯ A is discontinuous at x = A i , 1 ≤ i ≤ g −
4. We associate to each point A i two forward orbits: the upper orbit O u ( A i ) = { f n ¯ A ( T i A i ) } n ≥ , and the lower orbit O (cid:96) ( A i ) = { f n ¯ A ( T i − A i ) } n ≥ . We use the convention that if an orbit hits one of thediscontinuity points A j , then the next iterate is computed according to the left or rightlocation: for example, if the lower orbit of A i hits some A j , then the next iterate willbe T j − A j , and if the upper orbit of A i hits some A j then the next iterate is T j A j .Now we explore the patterns in the above orbits. The following property plays anessential role in studying the maps f ¯ A and F ¯ A . Definition 3.1.
We say that the point A i has the cycle property if for some non-negative integers m i , k i f m i ¯ A ( T i A i ) = f k i ¯ A ( T i − A i ) =: c A i . We will refer to the set { T i A i , f ¯ A T i A i , . . . , f m i − A T i A i } as the upper side of the A i -cycle , the set { T i − A i , f ¯ A T i − A i , . . . , f k i − A T i − A i } as the lower side of the A i -cycle , and to c A i as the end of the A i -cycle .The main goal of this section is to prove Theorem 1.2 (cycle property) stated in theIntroduction. First, we prove some preliminary results. Lemma 3.2.
The following identity holds (3.1) T σ ( i )+1 T i = T σ ( i − − T i − Proof.
Using relation (1.5) stated in the Introduction, we have that T ρ ( i ) T i = T − ρ ( i ) T − ρ ( i ) (where ρ ( i ) = σ ( i ) + 1), so it is enough to show that T − ρ ( i ) = T σ i − − and T − ρ ( i ) = T i − .For that we analyze the two parity cases. If i is odd , we have the following identities mod (8 g − ρ ( i ) = σ ( i ) + 1 = 4 g − i + 1 (even) ρ ( i ) = σ (4 g − i + 1) + 1 = 2 − (4 g − i + 1) + 1 = 2 − g + i = 4 g − i (odd) ρ ( i ) = σ (2 − g + i ) + 1 = 4 g − (2 − g + i ) + 1 = 8 g − − i = 3 − i (even)Since σ ( i −
1) = 3 − i = ρ ( i ), one has T − ρ ( i ) = T i − by using (1.4). Also, σ ( i − − − ( i − − − i and σ ( ρ ( i )) = 2 − i , hence T − ρ ( i ) = T σ ( i − − . If i is even , we have the following identities mod (8 g − ρ ( i ) = σ ( i ) + 1 = 3 − i (odd) ρ ( i ) = σ (3 − i ) + 1 = 4 g − (3 − i ) + 1 = 4 g − i (even) ρ ( i ) = σ (4 g − i ) + 1 = 2 − (4 g − i ) + 1 = 5 − g − i = 4 g + 1 − i (odd)Since σ ( i −
1) = 4 g − ( i −
1) = ρ ( i ), one has T − ρ ( i ) = T i − by using (1.4). Also, σ ( i − − g − i and σ ( ρ ( i )) = 4 g − i , hence T − ρ ( i ) = T σ ( i − − .Identity (3.1) has been proved for both cases. (cid:3) Remark . By introducing the notation θ ( i ) = σ ( i ) −
1, relation (3.1) can be written(3.2) T ρ ( i ) T i = T θ ( i − T i − , which will simplify further calculations. Lemma 3.4.
For any ≤ i ≤ g − , θ ( θ ( i − − i and ρ ( ρ ( i ) + 1) + 1 = i .Proof. Immediate verification. (cid:3)
Lemma 3.5.
The relations f A ( P i ) = P i and f A ( Q i ) = Q i hold for all i . In addition, f ¯ A ( P i ) = P i if i ∈ { , g, g − , g − } , and f ¯ A ( Q i ) = Q i if i ∈ { , g + 1 , g, g − } .Proof. We have f A ( P i ) = f ¯ A ( T i − P i ) = f ¯ A ( P θ ( i − ) = P θ ( θ ( i − − = P i and f A ( Q i ) = f ¯ A ( T i Q i ) = f ¯ A ( Q ρ ( i )+1 ) = Q ρ ( ρ ( i )+1)+1 = Q i by Lemma 3.4. The second part follows easily, too. (cid:3) Proof of Theorem 1.2.
Let us analyze the upper and lower orbits of A i . By Proposition2.2 and the orientation preserving property of the M¨obius transformations, we have(3.3) T i [ P i , Q i ] = [ Q ρ ( i ) , Q ρ ( i )+1 ] , T i − [ P i , Q i ] = [ P θ ( i − , P θ ( i − ] , therefore(3.4) T i A i ∈ (cid:0) Q ρ ( i ) , Q ρ ( i )+1 (cid:1) , T i − A i ∈ (cid:0) P θ ( i − , P θ ( i − (cid:1) Depending on whether T i A i ∈ ( Q ρ ( i ) , A ρ ( i )+1 ) or T i A i ∈ [ A ρ ( i )+1 , Q ρ ( i )+1 ) we haveeither f ¯ A ( T i A i ) = T ρ ( i ) T i A i or f ¯ A ( T i A i ) = T ρ ( i )+1 T i A i . Also, depending on whether T i − A i ∈ ( P θ ( i − , A θ ( i − ] or T i − A i ∈ ( A θ ( i − , P θ ( i − )we have either f ¯ A ( T i − A i ) = T θ ( i − − T i − A i or f ¯ A ( T i − A i ) = T θ ( i − T i A i . Notice that in the case when T i A i ∈ ( Q ρ ( i ) , A ρ ( i )+1 ) and T i − A i ∈ ( A θ ( i − , P θ ( i − )the cycle property holds immediately with m i = k i = 1, by using relation (3.2).We are left to analyze the cases T i A i ∈ [ A ρ ( i )+1 , Q ρ ( i )+1 ) or T i − A i ∈ ( P θ ( i − , A θ ( i − ]. Lemma 3.6.
Given x ∈ ( P i , Q i ) then one cannot have T i x ∈ [ A ρ ( i )+1 , Q ρ ( i )+1 ) and T i − x ∈ ( P θ ( i − , A θ ( i − ] simultaneously.Proof. Let M i be the midpoint of ( P i , Q i ). By Corollary 8.2 of the Appendix, thereexists a i ∈ ( M i , Q i ) such that T i ( a i ) = P ρ ( i )+1 and b i ∈ ( P i , M i ) such that T j − ( b j ) = Q θ ( j − .Since A ρ ( i )+1 ∈ ( P ρ ( i )+1 , Q ρ ( i )+1 ) and A θ ( i − ∈ ( P θ ( i − , Q θ ( i − , in order for T i x ∈ [ A ρ ( i )+1 , Q ρ ( i )+1 ), x must be in ( a i , Q i ), and in order for T i − x ∈ ( P θ ( i − , A θ ( i − ], x must be in ( P i , b i ). The lemma follows from the fact that these intervals are disjoint. (cid:3) Lemma 3.7. (i)
Assume x ∈ [ A j , Q j ) and T j − ( x ) ∈ ( P θ ( j − , A θ ( j − ] , then T θ ( j − − T j − ( x ) ∈ ( x, P j +1 ) . (ii) Assume x ∈ ( P j , A j ] and T j ( x ) ∈ [ A ρ ( j )+1 , Q ρ ( j )+1 ) , then T ρ ( j )+1 T j ( x ) ∈ ( Q j − , x ) . OUNDARY MAPS FOR FUCHSIAN GROUPS 11 A i T i T i − T i A i T ρ ( i )+1 T θ ( i − T ρ ( i ) T ρ ( ρ ( i )+1) T θ ( ρ ( i )) − T ρ ( θ ( ρ ( i )))+1 T θ ( θ ( ρ ( i )) − T θ ( ρ ( i )) T ρ ( θ ( ρ ( i ))) f ¯ A ( T i A i ) T i − A i f A ( T i − A i ) f A ( T i A i ) f A ( T i − A i ) Figure 6.
The first iterates of the upper and lower orbits of A i Proof. (i) Notice that T θ ( j − − T j − ( P j ) = f A ( P j ) = P j by Lemma 3.5. Also T j − ( x ) ∈ ( P θ ( j − , Q θ ( j − ) therefore T θ ( j − − T j − ( x ) ∈ T θ ( j − − ( P θ ( j − , Q θ ( j − ) = ( P j , P j +1 )by (3.4) and the fact that θ ( θ ( j − −
1) = j by Lemma 3.4. It follows that( T θ ( j − − T j − )[ P j , x ] = [ P j , T θ ( j − − T j − ( x )] ⊂ [ P j , P j +1 ] . Since T θ ( j − − T j − expands [ P j , x ] we get T θ ( j − − T j − ( x ) ∈ ( x, P j +1 ).Part (ii) can be proved similarly. (cid:3) We continue the proof of the theorem and assume the situation T i A i ∈ [ A ρ ( i )+1 , Q ρ ( i )+1 ) . Lemma 3.6 implies that T i − A i / ∈ ( P θ ( i − , A θ ( i − ], i.e. T i − A i ∈ ( A θ ( i − , P θ ( i − ).Notice that f ¯ A ( T i − A i ) can be rewritten as T ρ ( i ) T i A i by Lemma 3.1, and the beginningof the two orbits of A i are given by O u ( A i ) = { T i A i , T ρ ( i )+1 T i A i , . . . } , O l ( A i ) = { T i − A i , T ρ ( i ) T i A i , . . . } . We can now apply Lemma 3.7 part (ii) for x = A i to obtain that f ¯ A ( T i A i ) = T ρ ( i )+1 T i A i ∈ ( Q i − , A i ) , therefore f A ( T i A i ) = T ρ ( ρ ( i )+1) T ρ ( i )+1 ( T i A i ) (recalling that ρ ( ρ ( i ) + 1) = i − T ρ ( i ) T i A i ∈ (cid:0) P θ ( ρ ( i )) , P θ ( ρ ( i ))+1 (cid:1) . Depending on whether T ρ ( i ) T i A i ∈ (cid:0) P θ ( ρ ( i )) , A θ ( ρ ( i )) (cid:3) or T ρ ( i ) T i A i ∈ (cid:0) A θ ( ρ ( i )) , P θ ( ρ ( i ))+1 (cid:1) we have that f ¯ A ( T ρ ( i ) T i A i ) = T θ ( ρ ( i )) − ( T ρ ( i ) T i A i ) or f ¯ A ( T ρ ( i ) T i A i ) = T θ ( ρ ( i )) ( T ρ ( i ) T i A i ) . In the latter case, the cycle property holds, by using relation (3.2): we have f A ( T i A i ) = f A ( T i − A i ), i.e. T ρ ( ρ ( i )+1) T ρ ( i )+1 ( T i A i ) = T θ ( ρ ( i )) T ρ ( i ) ( T i A i ) . We have O u ( A i ) = { T i A i , T ρ ( i )+1 T i A i , T θ ( ρ ( i )) ( T ρ ( i ) T i A i ) . . . }O l ( A i ) = { T i − A i , T ρ ( i ) T i A i , T θ ( ρ ( i ) − ( T ρ ( i ) T i A i ) . . . } . Proposition 3.8.
Assume that T i A i ∈ [ A ρ ( i )+1 , Q ρ ( i )+1 ) , and A i does not satisfy thecycle property up to iteration M + 2 . Let ψ n = ( θ ◦ ρ ) n . Then, for any ≤ n ≤ M , f n ¯ A ( T i A i ) ∈ [ A ρ ( ψ n ( i ))+1 , Q ρ ( ψ n ( i ))+1 ) f n +1¯ A ( T i A i ) = T ρ ( ψ n ( i ))+1 ( f n ¯ A ( T i A i )) f n +1¯ A ( T i − A i ) = T θ ( ψ n ( i ) − ( f n ¯ A ( T i − A i )) = T ρ ( ψ n ( i )) ( f n ¯ A ( T i A i ))(3.5) f n +1¯ A ( T i − A i ) ∈ ( P ψ n +1 ( i ) , A ψ n +1 ( i ) ] f n +2¯ A ( T i A i ) = T ρ ( ψ n ( i ))+1 ( f n +1¯ A ( T i A i )) = T ψ n +1 ( i ) ( f n +1¯ A ( T i − A i )) f n +2¯ A ( T i − A i ) = T ψ n +1 ( i ) − ( f n +1¯ A ( T i − A i ))(3.6) f n ¯ A ( T i A i ) T ρ ( ψ n ( i ))+1 T θ ( ψ n ( i ) − T ρ ( ψ n ( i )) T ρ ( ρ ( ψ n ( i ))+1) T ψ n +1 ( i ) − T ρ ( ψ n +1 ( i ))+1 T θ ( ψ n +1 ( i ) − T ψ n +1 ( i ) T ρ ( ψ n +1 ( i )) f n +1¯ A ( T i A i ) f n ¯ A ( T i − A i ) f n +2¯ A ( T i − A i ) f n +3¯ A ( T i A i ) f n +3¯ A ( T i − A i ) Figure 7.
Iterates of upper and lower orbits of A i Proof.
We prove this by induction. The case n = 0 has been already presented above( ψ ( i ) = i ). Assume now that the relations are true for k = 1 , , . . . , n < M . Weanalyze the case k = n + 1. Let (cid:96) = ψ n ( i ). First, notice that f n +2¯ A ( T i − A i ) = T ψ n +1 ( i ) − ( f n +1¯ A ( T i − A i )) = T ψ n +1 ( i ) − T ρ ( ψ n ( i )) ( f n ¯ A ( T i A i ))= T θ ( ρ ( (cid:96) )) − T ρ ( (cid:96) ) ( f n ¯ A ( T i A i ))Since f n ¯ A ( T i A i ) ∈ [ A ρ ( ψ n ( i ))+1 , Q ρ ( ψ n ( i ))+1 ) = [ A ρ ( (cid:96) )+1 , Q ρ ( (cid:96) )+1 )and T ρ ( (cid:96) ) ( f n ¯ A ( T i A i )) = f n +1¯ A ( T i − A i ) ∈ ( P θ ( ρ ( l )) , A θ ( ρ ( l )) ] OUNDARY MAPS FOR FUCHSIAN GROUPS 13 we can apply Lemma 3.7 part (i) for x = f n ¯ A ( T i A i ), j = ρ ( (cid:96) ) + 1 to conclude that f n +2¯ A ( T i − A i ) ∈ ( A ρ ( (cid:96) )+1 , P ρ ( (cid:96) )+2 ) and(3.7) f n +3¯ A ( T i − A i ) = T ρ ( (cid:96) )+1 ( f n +2¯ A ( T i − A i )) = T θ ( ψ n +1 ( i ) − ( f n +2¯ A ( T i − A i ))because ρ ( (cid:96) ) + 1 = θ ( θ ( ρ ( (cid:96) )) −
1) = θ ( ψ n +1 ( i ) − f n +2¯ A ( T i A i ) = T ψ n +1 ( i ) ( f n +1¯ A ( T i − A i ))and f n +1¯ A ( T i − A i ) ∈ ( P ψ n +1 ( i ) , A ψ n +1 ( i ) ]we have that f n +2¯ A ( T i A i ) ∈ ( Q ρ ( ψ n +1 ( i )) , Q ρ ( ψ n +1 ( i ))+1 ). Using relations (3.2), (3.6),(3.7), the following holds: T ρ ( ψ n +1 ( i )) ( f n +2¯ A ( T i A i )) = T ρ ( ψ n +1 ( i )) T ψ n +1 ( i ) ( f n +1¯ A ( T i − A i ))= T θ ( ψ n +1 ( i ) − T ψ n +1 ( i ) − ( f n +1¯ A ( T i − A i ))= f n +3¯ A ( T i − A i ) . For the cycle property not to hold, one has f n +3¯ A ( T i A i ) (cid:54) = f n +3¯ A ( T i − A i ) (= T ρ ( ψ n +1 ( i )) ( f n +2¯ A ( T i A i ))) . Hence, f n +2¯ A ( T i A i ) ∈ ( Q ρ ( ψ n +1 ( i )) , Q ρ ( ψ n +1 ( i ))+1 ) \ ( Q ρ ( ψ n +1 ( i )) , A ρ ( ψ n +1 ( i ))+1 )and relations (3.5) are proved for k = n + 1.One proceeds similarly to prove (3.6) for k = n + 1. (cid:3) We can now complete the proof of Theorem 1.2. Assume by contradiction that thecycle property does not hold. Thus relations (3.5) and (3.6) will be satisfied for all n .In particular f n +1¯ A ( T i − A i ) ∈ ( P ψ n +1 ( i ) , A ψ n +1 ( i ) ]. Recall that ψ n ( i ) = ( θ ◦ ρ ) n ( i ). Adirect computation shows that θ ( ρ ( i )) = 4 g − i (mod 8 g − ψ n ( i ) = i + n (4 g −
4) (mod 8 g − . We show that there exists n such that ψ n ( i ) belongs to a congruence class of one ofthe numbers { , g + 1 , g, g − } . More precisely,(1) if i ≡ n such that ψ n ( i ) ≡ g (mod 8 g − i ≡ n such that ψ n ( i ) ≡ g − i ≡ g is even, then there exists n such that ψ n ( i ) ≡ g + 1 (mod 8 g − i ≡ g is odd, then there exists n such that ψ n ( i ) ≡ g − g − i ≡ g is even, then there exists n such that ψ n ( i ) ≡ g − g − i ≡ g is odd, then there exists n such that ψ n ( i ) ≡ g + 1 (mod 8 g − This follows from the fact that for any g ≥ g − g − i = 4 k + 3. Then ψ n ( i ) = 4 k + 3 + 4 n ( g − g is odd, 2 g − g − s forsome integer s . Since g − g − n and m such that k + n ( g −
1) = s + m (2 g − . Multiplying by 4 and adding 3 to both sides, we obtain3 + 4 k + 4 n ( g −
1) = 3 + 4 s + 4 m (2 g −
1) = 2 g − m (2 g −
1) + 3 , and therefore ψ n ( i ) ≡ g + 1 (mod 8 g − . Let n be such an integer, with the property that ψ n ( i ) belongs to the congruenceclass of one of the numbers { , g + 1 , g, g − } . By Lemma 3.5, Q ψ n ( i ) is fixed by T ψ n ( i ) . Using (3.6) we have f n − A ( T i − A i ) ∈ ( P ψ n ( i ) , A ψ n ( i ) ] and f n ¯ A ( T i A i ) = T ψ n ( i ) ( f n − A ( T i − A i )) ∈ ( Q ψ n ( i ) − , T ψ n ( i ) A ψ n ( i ) ] ⊂ ( Q ψ n ( i ) − , Q ψ n ( i ) ) . The interval [ A ψ n ( i ) , Q ψ n ( i ) ) expands under T ψ n ( i ) , so T ψ n ( i ) A ψ n ( i ) ∈ ( Q ψ n ( i ) − , A ψ n ( i ) ).Therefore, f n ¯ A ( T i A i ) ∈ ( Q ψ n ( i ) − , A ψ n ( i ) ), which assures us that the cycle propertyholds since f n +1¯ A ( T i A i ) = T ψ n ( i ) − ( f n ¯ A ( T i A i )) = T ρ ( ψ n ( i )) ( f n ¯ A ( T i A i )) = f n +1¯ A ( T i − A i ) . (cid:3) Remark . In contrast, if ¯ A = ¯ P the upper and lower orbits of all P i are periodic.Specifically, O u ( P i ) = { Q ρ ( i )+1 , Q ρ ( i )+1 , . . . } if i ∈ { , g + 1 , g, g − }O u ( P i ) = { , Q ρ ( i )+1 , Q i , Q ρ ( i )+1 , Q i , . . . } for other i, and O (cid:96) ( P i ) = { P i , P i , . . . } if i ∈ { , g, g − , g − }O (cid:96) ( P i ) = { P θ ( i − , P i , P θ ( i − , . . . } for other i. Notice that these two phenomena have something in common: in both cases the setsof values are finite.We have seen in the proof of Theorem 1.2 that, when T i A i ∈ ( Q ρ ( i ) , A ρ ( i )+1 ) and T i − A i ∈ ( A θ ( i − , P θ ( i − ), the cycle property holds immediately with m i = k i = 1,by using relation (3.2). In this case we have(3.8) f ¯ A ( T i A i ) = f ¯ A ( T i − A i ) . Definition 3.10.
A partition point A i is said to satisfy the short cycle property if (3.8)holds, or, equivalently, if T i A i ∈ ( Q ρ ( i ) , A ρ ( i )+1 ) and T i − A i ∈ ( A θ ( i − , P θ ( i − ) . This notion will be used in the next section.
Remark . The existence of an open set of partitions ¯ A satisfying the short cycleproperty follows from Corollary 8.2 of the Appendix: it is sufficient to take A i ∈ ( b i , a i )for each i . OUNDARY MAPS FOR FUCHSIAN GROUPS 15 Construction of Ω ¯ A According to the philosophy of the SL (2 , Z ) situation treated in [6] we expect the y -levels of the attractor set of F ¯ A , Ω ¯ A , to be comprised from the values of the cyclesof { A i } . If the cycles are short, the situation is rather simple: y -levels of the upperconnected component of Ω ¯ A are B i := T σ ( i − A σ ( i − , and y -levels of the lower connected component of Ω ¯ A are C i := T σ ( i +1) A σ ( i +1)+1 . The x -levels in this case are the same as for the Bowen-Series map F ¯ P , and the set Ω ¯ A is determined by the corner points located in the strip { ( x, y ∈ S × S | y ∈ [ A i , A i +1 ) } (see Figure 8) with coordinates( P i , B i ) (upper part ) and ( Q i +1 , C i ) (lower part) . This set obviously has a finite rectangular structure. P i -1 A i A i +1 P i -1 Q i +1 P i Q i +2 (P i ,B i ) (Q i+1 ,C i )Q i P i +1 Figure 8.
Strip y ∈ [ A i , A i +1 ] of Ω ¯ A We will prove the desired properties of the set Ω ¯ A stated in Theorem 1.3: property(1) (Theorem 4.2) and property (2) (Theorem 6.1). Remark . Alternatively, the domain of bijectivity of F ¯ A can be constructed usingan approach first described by of I. Smeets in her thesis [10]: start with the knowndomain Ω ¯ P of the Bowen-Series map F ¯ P and modify it by an infinite “quilting process”by adding and deleting rectangles where the maps F ¯ A and F ¯ P differ. In the case ofshort cycles the “quilting process” gives exactly the region Ω ¯ A , but unfortunately, itdoes not work when the cycles are longer. Since in the short cycles case the domainΩ ¯ A can be described explicitly, we do not go into the details of the “quilting process”here. Theorem 4.2.
The map F ¯ A : Ω ¯ A → Ω ¯ A is one-to-one and onto.Proof. We investigate how different regions of Ω ¯ A are mapped by F ¯ A . More preciselywe look at the strip S i of Ω ¯ A given by y ∈ [ A i , A i +1 ], and its image under F ¯ A , inthis case T i . See Figure 9. We consider the following decomposition of this strip:˜ S i = [ Q i +2 , P i − ] × [ A i , A i +1 ] (red rectangular piece), S (cid:96)i = [ Q i +1 , Q i +2 ] × [ A i , C i ] (blue lower corner) and S ui = [ P i − , P i ] × [ B i , A i +1 ] (green upper corner). Now T i ( ˜ S i ) = T i ([ Q i +2 , P i − ] × [ A i , A i +1 ]) = [ Q σ ( i ) , P σ ( i )+1 ] × [ B σ ( i )+1 , C σ ( i ) − ](4.1) T i ( S (cid:96)i ) = T i ([ Q i +1 , Q i +2 ] × [ A i , C i ]) = [ P σ ( i ) , Q σ ( i ) ] × [ B σ ( i )+1 , T i C i ](4.2) T i ( S ui ) = T i ([ P i − , P i ] × [ B i , A i +1 ]) = [ P σ ( i )+1 , Q σ ( i )+1 ] × [ T i B i , C σ ( i ) − ](4.3) A A A A A A A A A A A A Figure 9.
Bijectivity of the F ¯ A mapNotice that • T i ( ˜ S i ) is a complete vertical strip in Ω ¯ A , Q σ ( i ) ≤ x ≤ P σ ( i )+1 ; • T i ( S ui ) together with T j ( S (cid:96)j ) (where σ ( j + 1) = σ ( i − −
1) form a completevertical strip in Ω ¯ A , P σ ( i )+1 ≤ x ≤ Q σ ( i )+1 . (We are using here the short cycleproperty T i T σ ( i − A σ ( i − = T j T σ ( j +1) A σ ( j +1)+1 .) • T i ( S (cid:96)i ) together with T k ( S uk ) (where σ ( k ) + 1 = σ ( i )) form a complete verticalstrip in Ω ¯ A , P σ ( i ) ≤ x ≤ Q σ ( i ) .This proves the bijectivity property of F ¯ A on Ω ¯ A . (cid:3) We showed that the ends of the cycles do not appear as y -levels of the boundary ofΩ ¯ A . We state this important property as a corollary. Corollary 4.3.
For i and j related via σ ( j + 1) = σ ( i − − , we have (4.4) T j C j = T i B i ∈ [ B ρ ( i )+1 , C θ ( i ) ] = [ B ρ ( j ) , C θ ( j ) − ] . Trapping region
In order to prove property (2) of Ω ¯ A , we enlarge it and prove the trapping propertyfor the enlarged region first. Let Ψ ¯ A = Ω ¯ A ∪ D , where D = g − (cid:91) i =1 R i and R i = [ P i − , P i ] × [ Q i , B i ] . OUNDARY MAPS FOR FUCHSIAN GROUPS 17
Notice that Ψ ¯ A can be also expressed as Ψ ¯ A = Ω ¯ P ∪ A , where A = ∪ g − i =1 [ Q i +1 , Q i +2 ] × [ P i , C i ]. The y -levels of the upper part of Ψ ¯ A are given by the Q i ’s and the y -levels ofthe lower part of Ψ ¯ A are given by the C i ’s. A A A A A A A A A A A A Figure 10.
Trapping region Ψ ¯ A consisting of the set Ω ¯ A (grey) andthe added set D (purple) Theorem 5.1.
The set Ψ ¯ A is a trapping region for the map F ¯ A , i.e., • given any ( x, y ) ∈ S × S \ ∆ , there exists n ≥ such that F n ¯ A ( x, y ) ∈ Ψ ¯ A ; • F ¯ A (Ψ ¯ A ) ⊂ Ψ ¯ A .Proof. We start with ( x, y ) ∈ S × S \ ∆ and show that there exists n ≥ F n ¯ A ( x, y ) ∈ Ψ ¯ A . We have Q i ∈ [ A i , P i +1 ) ⊂ [ A i , A i +1 ), and by the short cycle condition, C i ∈ [ A i , P i +1 ) ⊂ [ A i , A i +1 ).Consider ( x, y ) ∈ S × S \ ∆. Notice that there exists n ( x, y ) > x n , y n obtained from the n th iterate of F ¯ A , ( x n , y n ) = F n ¯ A ( x, y ), are not insidethe same isometric circle; in other words, ( x n , y n ) (cid:54)∈ X i = [ P i , Q i +1 ] × [ A i , A i +1 ) for all1 ≤ i ≤ g − x n , y n ) ∈ Y i = [ P i − , P i ] × [ A i , Q i ) (orange set), and ( x n , y n ) ∈ Z i = [ Q i +1 , Q i +2 ] × ( C i , A i +1 ](green set), and show that a forward iterate lands in Ψ ¯ A . Case (I)
If ( x n , y n ) ∈ Y i = [ P i − , P i ] × [ A i , Q i ), then F ¯ A ( x n , y n ) ∈ [ T i P i − , T i P i ] × [ T i A i , T i Q i ) = [ P ρ ( i ) , Q ρ ( i ) ] × [ B ρ ( i ) , Q ρ ( i )+1 ) . Since B ρ ( i ) ∈ [ Q ρ ( i ) , A ρ ( i )+1 ], we need to analyze the regions[ P ρ ( i ) , Q ρ ( i ) ] × [ B ρ ( i ) , A ρ ( i )+1 ] and [ P ρ ( i ) , Q ρ ( i ) ] × [ A ρ ( i )+1 , Q ρ ( i )+1 ) . (a) If ( x n +1 , y n +1 ) ∈ [ P k , Q k ] × [ B k , A k +1 ], where k = ρ ( i ), then( x n +2 , y n +2 ) = T k ( x n +1 , y n +1 ) ∈ [ Q ρ ( k ) , Q ρ ( k )+1 ] × [ T k B k , T k A k +1 ] . P i -1 A i A i +1 P i -1 Q i +1 P i Q i +2 ( P i ,B i ) Q i P i +1 R i Z i Y i X i ( Q i +1 ,C i ) Figure 11.
The strip y ∈ [ A i , A i +1 ] of the trapping region Ψ ¯ A togetherwith the sets Y i = [ P i − , P i ] × [ A i , Q i ) (orange) and Z i = [ Q i +1 , Q i +2 ] × ( C i , A i +1 ] (green) outside of it that require special considerationsSince T k A k +1 = C θ ( k ) , and T k B k ∈ [ B ρ ( k )+1 , C θ ( k ) ] the only part of the vertical stripabove where ( x n +2 , y n +2 ) might still lie outside of Ψ ¯ A is a subset of [ P ρ ( k )+1 , Q ρ ( k )+1 ] × [ B ρ ( k )+1 , Q ρ ( k )+2 ).Notice that ρ ( k ) = σ ( σ ( i ) + 1) + 1 = 4 g + i − x n +2 , y n +2 ) ∈ [ P g + i − , Q g + i − ] × [ B g + i − , Q g + i ). (b) If ( x n +1 , y n +1 ) ∈ [ P k , Q k ] × [ A k +1 , Q k +1 ), then( x n +2 , y n +2 ) = T k +1 ( x n +1 , y n +1 ) ∈ [ P ρ ( k +1) , T k +1 Q k ] × [ B ρ ( k +1) , Q ρ ( k +1)+1 ) . Notice that T k +1 Q k ∈ [ P ρ ( k +1) , Q ρ ( k +1) ] and ρ ( k + 1) = ρ ( ρ ( i ) + 1) = i − x n +2 , y n +2 ) ∈ [ P i − , Q i − ] × [ B i − , Q i ).To summarize, we started with ( x n +1 , y n +1 ) ∈ [ P ρ ( i ) , Q ρ ( i ) ] × [ B ρ ( i ) , Q ρ ( i )+1 ) andfound two situations that need to be analyzed ( x n +2 , y n +2 ) ∈ [ P i − , Q i − ] × [ B i − , Q i )or ( x n +2 , y n +2 ) ∈ [ P g + i − , Q g + i − ] × [ B g + i − , Q g + i ).We prove in what follows that it is not possible for all future iterates F m ( x n , y n ) tobelong to the sets of type [ P k , Q k ] × [ B k , Q k +1 ).First, it is not possible for all F m ( x n , y n ) (starting with some m >
0) to belong onlyto type-a sets [ P k m , Q k m ] × [ B k m , A k m +1 ], where k m +1 = ρ ( k m ) + 1 because such a setis included in the isometric circle X k m , and the argument at the beginning of the proofdisallows such a situation.Also, it is not possible for all F m ( x n , y n ) (starting with some m >
0) to belong onlyto type-b sets [ P k m , Q k m ] × [ A k m +1 , Q k m +1 ), where k m +1 = ρ ( k m + 1): this would implythat the pairs of points ( y n + m , A k n + m +1 ) (on the y-axis) will belong to the same interval[ A k n + m , Q k n + m +1 ) which is impossible due to expansiveness property of the map f ¯ A .Therefore, there exists a pair ( x l , y l ) in the orbit of F m ( x n , y n ) such that( x l , y l ) ∈ [ P j , Q j ] × [ A j +1 , Q j +1 ) (type-b)for some 1 ≤ j ≤ g − x l +1 , y l +1 ) ∈ [ P j (cid:48) , T j +1 Q j ] × [ T j +1 A j +1 , P j (cid:48) +1 ] ⊂ [ P j (cid:48) , Q j (cid:48) ] × [ Q j (cid:48) , P j (cid:48) +1 ] (type-a) , where j (cid:48) = ρ ( j + 1). Then( x l +2 , y l +2 ) ∈ T j (cid:48) ([ P j (cid:48) , T j +1 Q j ] × [ Q j (cid:48) , P j (cid:48) +1 ]) = [ Q j (cid:48)(cid:48) , T j (cid:48) T j +1 Q j ] × [ Q j (cid:48)(cid:48) +1 , P j (cid:48)(cid:48) − ]where j (cid:48)(cid:48) = ρ ( j (cid:48) ). OUNDARY MAPS FOR FUCHSIAN GROUPS 19
Using the results of the Appendix (Corollary 8.3), we have that the arc lengthdistance satisfies (cid:96) ( P j (cid:48) , T j +1 Q j ) = (cid:96) ( T j +1 P j , T j +1 Q j ) < (cid:96) ( P j (cid:48) , Q j (cid:48) ) . Now we can use Corollary 8.2 (ii) applied to the point T j +1 Q j ∈ [ P j (cid:48) , Q j (cid:48) ] to concludethat T j (cid:48) T j +1 Q j ∈ [ Q j (cid:48)(cid:48) , P j (cid:48)(cid:48) +1 ]. Therefore ( x l +2 , y l +2 ) ∈ Ψ ¯ A . Case (II)
If ( x n , y n ) ∈ Z i = [ Q i +1 , Q i +2 ] × ( C i , A i +1 ], then F ¯ A ( x n , y n ) ∈ T i ([ Q i +1 , Q i +2 ] × ( C i , A i +1 ]) = [ P σ ( i ) , Q σ ( i ) ] × ( T i C i , C θ ( i ) ] . Since T i C i ∈ [ B ρ ( i ) , C θ ( i ) − ] by (4.4) and the set [ P σ ( i ) , Q σ ( i ) ] × [ B ρ ( i ) , C θ ( i ) − ] is in Ψ ¯ A ,we are left with analyzing the situation( x n +1 , y n +1 ) ∈ [ P σ ( i ) , Q σ ( i ) ] × ( C θ ( i ) − , C θ ( i ) ] . This requires two subcases depending on y n +1 ∈ ( C k − , A k ) or y n +1 ∈ [ A k , C k ], where k = θ ( i ). (a) If ( x n +1 , y n +1 ) ∈ [ P k +1 , Q k +1 ] × ( C k − , A k ), then( x n +2 , y n +2 ) = T k − ( x n +1 , y n +1 ) ∈ [ T k − P k +1 , Q σ ( k − ] × ( T k − C k − , T k − A k ) . Notice that σ ( k −
1) = σ ( θ ( i ) −
1) = i + 2 (direct verification). Since T k − P k +1 ∈ [ P σ ( k − , Q σ ( k − ) = [ P i +2 , Q i +2 ) ,T k − A k = C θ ( k − = C i +1 and T k − C k − ∈ [ B ρ ( k − , C θ ( k − − ) = [ B i +3 , C i ), we havethat ( x n +2 , y n +2 ) ∈ [ P i +2 , Q i +2 ) × [ B i +3 , C i +1 ). The only part of this vertical stripwhere ( x n +2 , y n +2 ) might still lie outside of Ψ ¯ A is a subset of [ P i +2 , Q i +2 ] × ( C i , C i +1 ),and that is the situation we need to analyze. (b) If ( x n +1 , y n +1 ) ∈ [ P k +1 , Q k +1 ] × [ A k , C k ], then( x n +2 , y n +2 ) ∈ T k ([ P k +1 , Q k +1 ] × [ A k , C k ]) = [ P σ ( k ) − , P σ ( k ) ] × [ B ρ ( k ) , T k C k ] . Since T k C k ∈ [ B ρ ( k ) , C θ ( k ) − ] by (4.4) and σ ( k ) = σ ( θ ( i )) = 4 g + i −
1, then( x n +2 , y n +2 ) ∈ [ P g + i − , P g + i − ] × [ B g + i , C g + i − ]and the only part of this vertical strip where ( x n +2 , y n +2 ) might still lie outside of Ψ ¯ A is a subset of [ P g + i − , Q g + i − ] × [ A g + i − , C g + i − ].To summarize, we started with ( x n +1 , y n +1 ) ∈ [ P σ ( i ) , Q σ ( i ) ] × ( C θ ( i ) − , C θ ( i ) ] andfound two situations that need to be analyzed ( x n +2 , y n +2 ) ∈ [ P i +2 , Q i +2 ] × ( C i , C i +1 ]or ( x n +2 , y n +2 ) ∈ [ P g + i − , Q g + i − ] × [ A g + i − , C g + i − ].We prove that it is not possible for all future iterates F m ( x n , y n ) to belong to thesets of type [ P k +1 , Q k +1 ] × [ C k − , C k ].First, it is not possible for all F m ( x n , y n ) (starting with some m >
0) to belong onlyto type-a sets [ P k m +1 , Q k m +1 ] × ( C k m − , A k m ), where k m +1 = σ ( k m − y n + m , A k n + m ) (on the y -axis) will belong to the sameinterval ( C k n + m − , A k n + m ] ⊂ [ A k n + m − , A k n + m ] which is impossible due to expansivenessproperty of the map f ¯ A on such intervals.From the discussion of Case (b), if an iterate F m ( x n , y n ) belongs to a type-b set,then F m +1 ( x n , y n ) either belongs to Ψ ¯ A or to another type-b set. However, it is notpossible for all iterates F m ( x n , y n ) (starting with some m >
0) to belong to type-b sets[ P k m +1 , Q k m +1 ] × [ A k m , C k m ], where k m +1 = σ ( k m ) − X k m , and the argument at the beginning of the proof disallows such a situation. Thus, once an iterate F m ( x n , y n ) belongs to a type-b set, then it willeventually belong to Ψ ¯ A .We showed that any point ( x, y ) that belongs to a set [ P k +1 , Q k +1 ] × ( C k − , C k ]will have a future iterate in Ψ ¯ A . This completes the proof of Case II and, hence, thetheorem. (cid:3) Reduction theory
We can now complete the proof of Theorem 1.3.
Theorem 6.1.
For almost every point ( x, y ) ∈ S × S \ ∆ , there exists K > such that F K ¯ A ( x, y ) ∈ Ω ¯ A , and the set Ω A is a global attractor for F ¯ A , i.e., Ω A = ∞ (cid:92) n =0 F n ¯ A ( S × S \ ∆) . Proof.
By Theorem 5.1, every point ( x, y ) ∈ S × S \ ∆ is mapped to the trapping regionΨ ¯ A = Ω ¯ A ∪ D by some iterate F n ¯ A . Therefore, it suffices to track the set D = (cid:83) g − i =1 R i .The image of each rectangle R i = [ P i − , P i ] × [ Q i , B i ] under F ¯ A , F ¯ A ( R i ) = T i ( R i ), is arectangular set(6.1) F ¯ A ( R i ) = [ T i P i − , T i P i ] × [ T i Q i , T i B i ] = [ P ρ ( i ) , Q ρ ( i ) ] × [ Q ρ ( i )+1 , T i B i ] . The “top” of this rectangle, [ P ρ ( i ) , Q ρ ( i ) ] ×{ T i B i } is inside Ω ¯ A , since T i B i ∈ [ B ρ ( i )+1 , C θ ( i ) ].Moreover,(6.2) F ¯ A ( R i ) \ Ω ¯ A = [ P ρ ( i ) , Q ρ ( i ) ] × [ Q ρ ( i )+1 , B ρ ( i )+1 ] ⊂ R ρ ( i )+1 , so, by letting j = ρ ( i ) + 1, F ¯ A ( D ) \ Ω ¯ A = g − (cid:91) j =1 [ P j − , Q j − ] × [ Q j , B j ]and F ¯ A (Ω ¯ A ∪ D ) = Ω ¯ A ∪ g − (cid:91) j =1 [ P j − , Q j − ] × [ Q j , B j ] . Now the image of the rectangular set [ P j − , Q j − ] × [ Q j , B j ] under F ¯ A (= T j ) is F ¯ A ([ P j − , Q j − ] × [ Q j , B j ]) = [ P ρ ( j ) , T j Q j − ] × [ Q ρ ( j )+1 , T j B j ] , hence(6.3) F ¯ A (cid:0) F ¯ A ( D )) \ Ω ¯ A = g − (cid:91) j =1 [ P ρ ( j ) , T j Q j − ] × [ Q ρ ( j )+1 , B ρ ( j )+1 ] . Corollary 8.3 tells us that the length of the segment [ P ρ ( j ) , T j Q j − ] = T j ([ P j − , Q j − ])is less than of [ P ρ ( j ) , Q ρ ( j ) ]. If we let k = ρ ( j ) + 1, and denote T j Q j − by S (2) k , then(6.3) becomes F A ( D ) \ Ω ¯ A = g − (cid:91) k =1 [ P k − , S (2) k − ] × [ Q k , B k ] OUNDARY MAPS FOR FUCHSIAN GROUPS 21 with the length of the segment [ P k − , S (2) k − ] being less than of [ P k − , Q k − ]. Induc-tively, it follows that:(6.4) F n ¯ A ( D ) \ Ω ¯ A = g − (cid:91) k =1 [ P k − , S ( n ) k − ] × [ Q k , B k ]where the length of the segment [ P k − , S ( n ) k − ] is less than n − of [ P k − , Q k − ]. Thus, F n ¯ A (Ω ¯ A ∪ D ) = Ω ¯ A ∪ g − (cid:91) k =1 [ P k − , S ( n ) k − ] × [ Q k , B k ]and ∞ (cid:92) n =0 F n ¯ A ( S × S \ ∆) = ∞ (cid:92) n =0 F n ¯ A (Ω A ∪ D ) = Ω ¯ A ∪ ∞ (cid:92) n =0 (cid:32) g − (cid:91) k =1 [ P k − , S nk − ] × [ Q k , B k ] (cid:33) = Ω ¯ A ∪ g − (cid:91) k =1 { P k − } × [ Q k , B k ] = Ω ¯ A In what follows, we will show that any point ( x, y ) ∈ D (see Figure 10) is actuallymapped to Ω ¯ A after finitely many iterations with the exception of the Lebesgue measurezero set consisting of the union of horizontal segments (cid:83) g − i =1 [ P i − , P i ] × { Q i } and theirpreimages. For that, let ( x, y ) ∈ R i with y (cid:54) = Q i and assume that F n ¯ A ( x, y ) = ( x n , y n ) ∈ F n ¯ A ( D ) \ Ω ¯ A . Using (6.4), this means that the sequence of points y n ∈ ( Q k n , B k n ] for all n ≥
1. But y n +1 = T k n y n , Q k n +1 = T k n Q k n and the map T k n is (uniformly) expandingon [ Q k n , B k n ] (a subset of the isometric circle of T k n ), which contradicts the assumption y n ∈ ( Q k n , B k n ]. (cid:3) Invariant measures
It is a standard computation that the measure dν = | dx | | dy || x − y | is preserved by M¨obiustransformations applied to unit circle variables x and y , hence by F ¯ A . Therefore, F ¯ A preserves the smooth probability measure(7.1) dν ¯ A = 1 K ¯ A dν, where K ¯ A = (cid:90) Ω ¯ A dν. Alternatively, by considering F ¯ A as a reduction map acting on geodesics, the invariantmeasure can be derived more elegantly by using the geodesic flow on the hyperbolicdisk and the Poincar´e cross-section maps, but we are not pursuing that direction here.In what follows, we compute K ¯ A for the case when ¯ A satisfies the short cycle prop-erty. Recall that the domain Ω ¯ A was described in the proof of Theorem 4.2 as:(7.2) Ω ¯ A = g − (cid:91) i =1 [ Q i +2 , P i − ] × [ A i , A i +1] ∪ [ Q i +1 , Q i +2 ] × [ A i , C i ] ∪ [ P i − , P i ] × [ B i , A i +1 ] . Proposition 7.1.
If the points A i satisfy the short cycle property and p i , q i , b i , c i rep-resent the angular coordinates of P i , Q i , B i = T i A i , and C i = T i − A i , respectively,then (7.3) ν (Ω A ) = K A = ln g − (cid:89) i =1 | sin (cid:16) c i − q i +2 (cid:17) || sin (cid:16) b i − p i − (cid:17) || sin (cid:16) b i − p i (cid:17) || sin (cid:16) c i − q i +1 (cid:17) | . Proof.
Since Ω A is given by (7.2), we have K ¯ A = (cid:90) Ω ¯ A dν = g − (cid:88) i =1 (cid:32)(cid:90) P i − Q i +2 (cid:90) A i +1 A i dν + (cid:90) Q i +2 Q i +1 (cid:90) C i A i dν + (cid:90) P i P i − (cid:90) A i +1 B i dν (cid:33) . In order to compute each of the three integrals above, we use angular coordinates θ and φ corresponding to x = e iθ , y = e iφ , and write for some arbitrary values A, B, C, D : I A,B,C,D := (cid:90) BA (cid:90) DC | dx || dy || x − y | = (cid:90) ba (cid:90) dc dθdφ | exp( iθ ) − exp( iφ ) | = (cid:90) ba (cid:90) dc dθdφ − θ − φ ) =: I a,b,c,d , where a, b, c, d are the angular coordinates corresponding to A, B, C, D : A = e ia , B = e ib , C = e ic , D = e id . The double integral (which we denoted by I a,b,c,d ) can be computed explicitly. First(7.4) (cid:90) ba dθ − θ − φ ) = −
12 cot (cid:18) θ − φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ = bθ = a = 12 (cid:18) cot (cid:18) a − φ (cid:19) − cot (cid:18) b − φ (cid:19)(cid:19) . Then, using the fact that the antiderivative (cid:82) cot xdx = ln | sin x | we obtain I a,b,c,d = 12 (cid:90) dc (cid:18) cot (cid:18) a − φ (cid:19) − cot (cid:18) b − φ (cid:19)(cid:19) dφ = (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) φ − b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) φ − a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) φ = dφ = c = ln (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) d − b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) c − a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) c − b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − ln (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) d − a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = ln | sin (cid:0) d − b (cid:1) || sin (cid:0) c − a (cid:1) || sin (cid:0) c − b (cid:1) || sin (cid:0) d − a (cid:1) | . Now, using the angular coordinates p i , q i , a i , b i , c i corresponding to the points P i , Q i , A i , B i , C i , we obtain K ¯ A = g − (cid:88) i =1 ( I q i +2 ,p i − ,a i ,a i +1 + I q i +1 q i +2 ,a i ,c i + I p i − ,p i ,b i ,a i +1 )= ln g − (cid:89) i =1 | sin (cid:16) a i +1 − p i − (cid:17) || sin (cid:16) a i − q i +2 (cid:17) || sin (cid:16) a i − p i − (cid:17) || sin (cid:16) a i +1 − q i +2 (cid:17) | + ln g − (cid:89) i =1 | sin (cid:16) c i − q i +2 (cid:17) || sin (cid:16) a i − q i +1 (cid:17) || sin (cid:16) a i − q i +2 (cid:17) || sin (cid:16) c i − q i +1 (cid:17) | + ln g − (cid:89) i =1 | sin (cid:16) a i +1 − p i (cid:17) || sin (cid:16) b i − p i − (cid:17) || sin (cid:16) b i − p i (cid:17) || sin (cid:16) a i +1 − p i − (cid:17) | = ln g − (cid:89) i =1 | sin (cid:16) c i − q i +2 (cid:17) || sin (cid:16) b i − p i − (cid:17) || sin (cid:16) b i − p i (cid:17) || sin (cid:16) c i − q i +1 (cid:17) | . The last equality is obtained due to cancellations. (cid:3)
OUNDARY MAPS FOR FUCHSIAN GROUPS 23
The circle map f ¯ A is a factor of F ¯ A (projecting on the y -coordinate), so one canobtain its smooth invariant probability measure dµ ¯ A by integrating dν ¯ A over Ω ¯ A withrespect to the u -coordinate. Thus, from the exact shape of the set Ω ¯ A , we can calculatethe invariant measure precisely. Proposition 7.2. dµ ¯ A = 1 K ¯ A g − (cid:88) i =1 (cid:18) cot (cid:18) q i +1 − φ (cid:19) − cot (cid:18) p i − φ (cid:19)(cid:19) dφ. Proof. dµ ¯ A = 1 K ¯ A g − (cid:88) i =1 (cid:32)(cid:90) P i − Q i +2 | dx || x − y | + (cid:90) Q i +2 Q i +1 | dx || x − y | + (cid:90) P i P i − | dx || x − y | (cid:33) | dy | . Using the calculations (7.4) we obtain dµ ¯ A = 1 K ¯ A g − (cid:88) i =1 (cid:32) cot (cid:18) q i +2 − φ (cid:19) − cot (cid:18) p i − − φ (cid:19) + cot (cid:18) q i +1 − φ (cid:19) − cot (cid:18) q i +2 − φ (cid:19) + cot (cid:18) p i − − φ (cid:19) − cot (cid:18) p i − φ (cid:19) (cid:33) dφ = 1 K ¯ A g − (cid:88) i =1 (cid:18) cot (cid:18) q i +1 − φ (cid:19) − cot (cid:18) p i − φ (cid:19)(cid:19) dφ. (cid:3) Appendix
In this section we use the explicit description of the fundamental domain F given inthe Introduction to obtain certain estimates used in the proofs.The fundamental domain F is a regular (8 g − π . Let us labelthe vertices of F by V , . . . , V g − , where V i is the intersection of the geodesics P i − Q i and P i Q i +1 (see Figure 12 for g = 3). We first prove the following geometric lemma. Lemma 8.1.
Consider five consecutive isometric circles of F : P i − Q i − , P i − Q i , P i Q i +1 , P i +1 Q i +2 , and P i +2 Q i +3 . Then (i) the angle between geodesics V i +1 P i +2 and V i +1 Q i +1 is greater than π , (ii) the angle between geodesics V i Q i − and V i P i is greater than π .Proof. Let the Euclidean distance from the center of the unit disk D , O , to the centerof each isometric circle be d , the Euclidean radius of each isometric circle by R , and v be the distance from O to the vertex V i +1 (see Figure 12). The angle between theimaginary axis and the ray from the origin to V i +1 is equal to t = π g − . The anglebetween geodesics V i +1 P i +2 and V i +1 Q i +1 is equal to the angle between the radii of theEuclidean circles (of centers O i , O (cid:48) i +1 ) representing these geodesics, i.e., ∠ O i V i +1 O (cid:48) i +1 .Our goal is to express it as a function of t , ω ( t ).Let ϕ = ∠ O i OQ i +1 . We have sin ϕ = | O i Q i +1 | /d , and sin t = | O i H | /d , where O i H ⊥ OH . Since the angle of F at V i +1 is equal to π , | O i H | = | O i V i +1 | / √
2, andsince | O i V i +1 | = | O i Q i +1 | = R , we obtain(8.1) sin ϕ = √ t, yRd v ? φ γ αβ δ φ t Q i+2 P i-1 Q i+3 O V i-1 V i Q i-1 P i+2 P i Q i O i P i+1 H Q i+1 V i+2 V i+1 O' i+1 P i-2 Figure 12.
Calculation of angle ∠ O i V i +1 O (cid:48) i +1 and therefore(8.2) cos ϕ = (cid:112) cos(2 t ) . In the right triangle ∆ O i OH we have | OH | = v + R √ and | O i H | = R √ , hence by thePythagorean Theorem, (cid:18) v + R √ (cid:19) + R d = R t , which implies v + R √ R √ t, and hence v = R √ (cid:18) cos t sin t − (cid:19) . Using that R = √ t ) d and d = 1cos ϕ = 1 (cid:112) cos(2 t ) , we obtain R and v as functionsof t ,(8.3) R ( t ) = √ t (cid:112) cos(2 t ) , v ( t ) = (cid:114) cos t − sin t cos t + sin t , and we now can express all further quantities as functions of t .In the triangle ∆ OO i V i +1 , let ∠ OO i V i +1 = β ( t ) and ∠ OV i +1 O i = δ ( t ). In thetriangle ∆ OP i +2 V i +1 , let | V i +1 P i +2 | = y ( t ), ∠ OP i +2 V i +1 = α ( t ), ∠ OV i +1 P i +2 = γ ( t ). OUNDARY MAPS FOR FUCHSIAN GROUPS 25
One can easily see that ∠ V i +1 OP i +2 = 3 t − ϕ ( t ). Using the Rule of Cosines, we have y ( t ) = 1 + v ( t ) − v ( t ) cos(3 t − ϕ ) . Using the Rule of Sines in the triangles ∆ OP i +2 V i +1 and ∆ OO i V i +1 we obtainsin( α ( t )) = v ( t ) sin(3 t − ϕ ) y ( t ) , sin( β ( t )) = v ( t ) sin( t ) R ( t ) = cos t − sin t √ , and the last equation implies β = π − t .The angle ω ( t ) = ∠ O i V i +1 O (cid:48) i +1 in question is calculated as ω ( t ) = 2 π − γ ( t ) − δ ( t ) − (cid:16) π − α ( t ) (cid:17) . Expressing γ ( t ) and δ ( t ) from these triangles we obtain(8.4) ω ( t ) = 4 t − ϕ ( t ) + 2 α ( t ) + β ( t ) − π
2= 4 t − ϕ ( t ) + 2 α ( t ) + π − t − π
2= 3 t − ϕ ( t ) + 2 α ( t ) − π . We see that the desired inequality(8.5) ω ( t ) > π t − ϕ ( t ) + 2 α ( t ) > π , and since from ∆ OV i +1 P i +2 we have3 t − ϕ ( t ) + α ( t ) + γ ( t ) = π, (8.5) is equivalent to(8.6) γ ( t ) − α ( t ) < π . Recall that γ ( t ) and α ( t ) are the angles of the triangle ∆ OV i +1 P i +2 , with γ ( t ) > π and α ( t ) < π , hence 0 < γ ( t ) − α ( t ) < π . In order to prove (8.6), we need to show that(8.7) cos( γ ( t ) − α ( t )) > . Using the Rule of Sines we obtainsin γ ( t ) = sin α ( t ) v ( t ) . Using the Rule of Cosines we obtaincos γ ( t ) = y ( t ) + v ( t ) − y ( t ) v ( t ) and cos α = 1 + y ( t ) − v ( t )2 y ( t ) . In what follows we will suppress dependence of all functions on t . Thuscos( γ − α ) = cos γ cos α + sin γ sin α = ( y + v − y − v )4 y v + sin αv = 8 v − v (1 + v ) cos(3 t − ϕ )4 vy . Since v and y are positive, it is sufficient to prove the positivity of the function g ( t ) = 2 v (1 + v ) − cos(3 t − ϕ ) = cos ϕ cos t − cos(3 t − ϕ )= cos ϕ cos t − cos((3 t − ϕ ) + ϕ )= cos ϕ cos t − (cos(3 t − ϕ ) cos ϕ − sin(3 t − ϕ ) sin ϕ )= cos ϕ (cid:18) t − cos(3 t − ϕ ) (cid:19) + sin(3 t − ϕ ) sin ϕ. The first term is positive since cos ϕ , cos t and cos(3 t − ϕ ) are less than 1. The secondterm is positive since(8.8) 3 t − ϕ > . The latter follows from the fact that the function h ( t ) = 3 t − ϕ ( t ) = 3 t − √ t )has second derivative h (cid:48)(cid:48) ( t ) = − √ t cos / (2 t )negative on (0 , π/ h (cid:48) ( t ) = 3 − √ t cos / (2 t )is decreasing on (0 , π/ h (cid:48) ( t ) ≥ h (cid:48) ( π/
12) = 3 − √ √ / > t ∈ (0 , π/ h is strictly increasing on (0 , π/ h ( t ) > h (0) = 0 for any t ∈ (0 , π/
12] whichimplies (8.8). Thus (8.5) follows. The second inequality follows from the symmetry ofthe fundamental domain F . (cid:3) In what follows (cid:96) will denote the arc length on the unit circle S . Corollary 8.2. (i)
There exist a j , b j ∈ ( P j , Q j ) such that d ( P j , a j ) > (cid:96) ( P j , Q j ) and (cid:96) ( b j , Q j ) > (cid:96) ( P j , Q j ) such that T j ( a j ) = P ρ ( j )+1 and T j − ( b j ) = Q θ ( j − . (ii) For any point x ∈ [ P j , Q j ] such that (cid:96) ( P j , x ) ≤ (cid:96) ( P j , Q j ) , we have T j ( x ) ∈ [ Q σ ( j )+1 , P σ ( j )+2 ] . (iii) For any point x ∈ [ P j , Q j ] such that (cid:96) ( x, Q j ) ≤ (cid:96) ( P j , Q j ) , we have T j − ( x ) ∈ [ Q θ ( j − , P θ ( j − ] .Proof. (i) Let M j be the midpoint of [ P j , Q j ]. Since the angle at each V j is equal to π , the angle between the geodesic segments V j P j and V j M j is equal π . Recall that T j ([ P j , Q j ]) = [ Q ρ ( j ) , Q ρ ( j )+1 ]. Since, by Lemma 8.1 (i) for i = σ ( j ), the angle betweenthe geodesic segments V ρ ( j ) P ρ ( j )+1 and V ρ ( j ) Q ρ ( j ) is > π , and T j is conformal, theexistence of a j ∈ ( M j , Q j ) such that T j ( a j ) = P ρ ( j )+1 follows. Similarly, we knowthat T j − ([ P j , Q j ]) = [ P θ ( j − , P θ ( j − ]. Since by Lemma 8.1 (ii) with i = σ ( j − V σ ( j − Q θ ( j − and V σ ( j − P θ ( j − is greaterthan π and T j − is conformal, the existence of b j ∈ ( P j , M j ) such that T j − ( b j ) = Q θ ( j − follows. Parts (ii) and (iii) follow immediately from (i). (cid:3) OUNDARY MAPS FOR FUCHSIAN GROUPS 27
Corollary 8.3.
The arc length of the interval T k ([ P k +2 , Q k +2 ]) is less than of [ P σ ( k ) , Q σ ( k ) ] and the length of the interval T k ([ P k − , Q k − ]) is less than of [ P σ ( k )+1 , Q σ ( k )+1 ] .Proof. By Proposition 2.2, we have T k ( Q k +1 ) = P σ ( k ) and T k ( Q k +2 ) = Q σ ( k ) . The factthat the length of T k ([ P k +2 , Q k +2 ]) < (cid:96) ( P σ ( k ) , Q σ ( k ) ) is equivalent to the fact that T k ( P k +2 ) ∈ [ M σ ( k ) , Q σ ( k ) ], where M σ ( k ) is the middle of [ P σ ( k ) , Q σ ( k ) ]. But the laststatement follows from the fact that the angle between the geodesic V k +1 P k +2 and thegeodesic V k +1 Q k +2 is less then π , a direct consequence of the fact that the angle inthe part (i) of Lemma 8.1 is greater that π . The second statement follows immediatelyfrom the part (ii) of Lemma 8.1. (cid:3) Acknowledgments.
We thank the anonymous referee for several useful commentsand suggestions.
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