Boundary maps for Fuchsian groups

# Structure of attractors for boundary maps associated to Fuchsian groups

Svetlana Katok Department of Mathematics, The Pennsylvania State University, University Park, PA 16802  and  Ilie Ugarcovici Department of Mathematical Sciences, DePaul University, Chicago, IL 60614
September 30, 2016; revised May 1, 2017; accepted for publication in Geometriae Dedicata
###### Abstract.

We study dynamical properties of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups. These maps are defined on the unit circle (the boundary of the Poincaré disk) by the generators of the group and have a finite set of discontinuities. We study the two forward orbits of each discontinuity 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 of discontinuity points the associated two-dimensional natural extension maps possess global attractors with finite rectangular structure. These two properties belong to the list of “good” reduction algorithms, equivalence or implications between which were suggested by Don Zagier [11].

###### Key words and phrases:
Fuchsian groups, reduction theory, boundary maps, attractor
###### 2010 Mathematics Subject Classification:
37D40
The second author is partially supported by a Simons Foundation Collaboration Grant

## 1. Introduction

Let be a finitely generated Fuchsian group of the first kind acting on the hyperbolic plane. We will use either the upper half-plane model or the unit disk model , and will denote the Euclidean boundary for either model by : for the upper half plane , and for the unit disk .

Let be a fundamental domain for with an even number of sides identified by the set of generators of , and be a surjective map locally constant on , where is an arbitrary set of jumps. A boundary map is defined by . It is a piecewise fractional-linear map whose set of discontinuities is . Let be the diagonal of , and be given by

 F(x,y)=(τ(y)x,τ(y)y).

This is a (natural) extension of , and if we identify with an oriented geodesic from to , we can think of as a map on geodesics which we will also call a reduction map.

Several years ago Don Zagier[11] proposed a list of possible notions of “good” reduction algorithms associated to Fuchsian groups and conjectured equivalences or implications between them. In this paper we consider two of these notions, namely the properties that “good” reduction algorithms should (i) satisfy the cycle property, and (ii) have an attractor with finite rectangular structure. We prove that for each cocompact torsion-free Fuchsian group there exist families of reduction algorithms which satisfy these properties. Thus our results are contributions towards Zagier’s conjecture.

Although the statement that each Fuchsian group admits a “good” reduction algorithm is not part of Zagier’s conjecture, it is certainly related to it, and for the purposes of this paper, we state it here.

### Reduction Theory Conjecture for Fuchsian groups

For every Fuchsian group there exist as above, and an open set of in such that

1. The map possesses a bijectivity domain having a finite rectangular structure, i.e., bounded by non-decreasing step-functions with a finite number of steps.

2. Every point is mapped to after finitely many iterations of .

###### Remark 1.1.

If property (2) holds, then is a global attractor for the map , i.e.

 (1.1) Ω=∞⋂n=0Fn(S×S∖Δ).

This conjecture was proved by the authors in [6] for and boundary maps associated to -continued fractions. Notice that for some classical cases of continued fraction 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 Poincaré unit disk model endowed with the hyperbolic metric

 (1.2) 2|dz|1−|z|2,

let be a Fuchsian group, i.e. a discrete group of orientation preserving isometries of , acting freely on with compact domain. Such is called a surface group, and the quotient is a compact surface of constant negative curvature of a certain genus . A classical (Ford) fundamental domain for is a -sided regular polygon centered at the origin (see a sketch of the construction in [5] in the manner of [4], and for the complete proof see [8]). A more suitable for our purposes -sided fundamental domain was described by Adler and Flatto in [1]. They showed that all angles of are equal to and, therefore, its sides are geodesic segments which satisfy the extension condition of Bowen and Series [3]: the geodesic extensions of these segments never intersect the interior of the tiling sets , . Figure 1 shows such a construction for .

Using notations similar to [1], we label the sides of in a counterclockwise order by numbers , as they are arcs of the corresponding isometric circles of generators . We denote the corresponding vertices of by , so that the side connects the vertices and . The identification of the sides is given by the pairing rule:

 σ(i)={4g−imod(8g−4) for odd i\par2−imod(8g−4) for even i.

The generators associated to this fundamental domain are Möbius transformations satisfying the following properties:

 (1.3) Tσ(i)Ti=Id (1.4) Ti(Vi)=Vρ(i), where ρ(i)=σ(i)+1 (1.5) Tρ3(i)Tρ2(i)Tρ(i)Ti=Id

We denote by the oriented (infinite) geodesic that extends the side to the boundary of the fundamental domain . It is important to remark that is the isometric circle for , and is the isometric circle for so that the inside of the former isometric circle is mapped to the outside of the latter.

The counter-clockwise order of theses points on is

 (1.6) P1,Q1,P2,Q2,…,P8g−4,Q8g−4,P1.

Bowen and Series [3] defined the boundary map

 (1.7) f¯P(x)=Ti(x)if Pi≤x

with the set of jumps . They showed that such a map is Markov with respect to the partition (1.6), expanding, and satisfies Rényi’s distortion estimates, hence it admits a unique finite invariant ergodic measure equivalent to Lebesgue measure.

Adler and Flatto [1] proved the existence of an invariant domain for the corresponding natural extension map , . Moreover, the set they identified has a regular geometric structure, what we call finite rectangular (see Figure 2, with shown as a subset of ). The maps and are ergodic111More precisely, is a -automorphism, property that is equivalent to being an exact endomorphism.. Both Series [9] and Adler-Flatto [1] explain how the boundary map can be used for coding symbolically the geodesic flow on .

Notations. For , the various intervals on between and (with the counterclockwise order) will be denoted by and . The geodesic (segment) from a point (or ) to (or will be denoted by .

Our object of study is a generalization of the Bowen-Series boundary map. We consider an open set of jumps

 J=¯A={A1,…,A8g−4}

with the only condition , and define by

 (1.8) f¯A(x)=Ti(x)if Ai≤x

and the corresponding two-dimensional map:

 (1.9) F¯A(x,y)=(Ti(x),Ti(y))if Ai≤y

A key ingredient in analyzing map is what we call the cycle property of the partition points . Such a property refers to the structure of the orbits of each that one can construct by tracking the two images and of these points of discontinuity of the map . It happens that some forward iterates of these two images and under coincide. This is another property from Zagier’s list 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 , , satisfies the cycle property, i.e., there exist positive integers such that

 fmi¯A(TiAi)=fki¯A(Ti−1Ai).

If a cycle closes up after one iteration

 (1.10) f¯A(TiAi)=f¯A(Ti−1Ai),

we say that the point satisfies the short cycle property. Under this condition, we prove the following:

###### Theorem 1.3 (Main Result).

If each partition point satisfies the short cycle property (1.10), then there exists a set with the following properties:

1. has a finite rectangular structure, and is (essentially) bijective on .

2. Almost every point is mapped to after finitely many iterations of , and is a global attractor for the map , i.e.,

 ΩA=∞⋂n=0Fn¯A(S×S∖Δ).

Notice that the set of partitions satisfying the short cycle property contains an open set with this property, as explained in Remark 3.11. Thus we prove the Reduction Theory Conjecture. We believe that this result is true in greater generality, i.e., for all partitions with .

### Organization of the paper

In Section 2 we prove properties (1) and (2) of the Reduction Theory Conjecture for the classical Bowen-Series case when the partition points are given by the set . In Section 3 we prove the cycle property for any partition with . In Section 4 we determine the structure of the set in the case when the partition satisfies the short cycle property and prove the bijectivity of the map on . In Section 5 we identify the trapping region for the map and prove that every point in is mapped to it after finitely many iterations of the map . And finally, in Section 6 we prove that almost every point is mapped to after finitely many iterations of the map and complete the proof of Theorem 1.3. In Section 7 we apply our results to calculate the invariant probability measures for the maps and .

## 2. Bowen-Series case

In this section we prove properties (1) and (2) of the Reduction Theory Conjecture for the Bowen-Series classical case, where the partition is given by the set of points .

###### Theorem 2.1.

The two-dimensional Bowen-Series map satisfies properties (1) and (2) of the Reduction Theory Conjecture.

Before we prove this theorem, we state a useful proposition that can be easily derived using the isometric circles and the conformal property of Möbius transformations (see also Theorem 3.4 of [1]).

###### Proposition 2.2.

maps the points , , , , , respectively to , , , , , .

###### Proof of Theorem 2.1.

In this case the set is determined by the corner points located in each horizontal strip

 {(x,y)∈S×S∣y∈[Pi,Pi+1)}

(see Figure 4) with coordinates

 (Pi,Qi) (upper part) and (Qi+2,Pi+1) (% lower part).

This set obviously has a finite rectangular structure. One can also verify immediately the essential bijectivity, by investigating how different regions of are mapped by . More precisely we look at the strip of given by , and its image under , in this case .

We consider the following decomposition of this strip: (red rectangular horizontal piece), (green horizontal rectangular piece). Now

 Ti(~Si) =[TiQi+2,TiPi−1]×[TiPi,TiQi]=[Qσ(i),Pσ(i)+1]×[Qσ(i)+1,Qσ(i)+2] Ti(^Si) =[TiQi+2,TiPi]×[TiQi,TiPi+1]=[Qσ(i),Qσ(i)+1]×[Qσ(i)+2,Pσ(i)−1]

Therefore is a complete vertical strip in , with . This completes the proof of the property (1).

We now prove property (2) for the set .

Consider . Notice that there exists such that the two values obtained from the th iterate of , , are not inside the same isometric circle; in other words, for all . Indeed, if one assumes that both coordinates belong to such a set for all , each time we iterate the pair we apply one of the maps which is expanding in the interior of its isometric circle. Thus the distance between and would grow sufficiently for the points to be inside different isometric circles. Therefore, there exists such that is in some interval and .

Notice that, from the definition of , in order to prove the attracting property, we need to analyze the situations and and show that a forward iterate lands in .

Case I. If , then

 F¯P(xn,yn)∈[TiQi+1,TiQi+2]×[TiPi,TiPi+1)=[Pσ(i),Qσ(i)]×[Qσ(i)+1,Pσ(i)−1).

The subset is included in so we only need to analyze the situation , where . Then

 (xn+2,yn+2)=F2P(xn,yn)=TkTi(xn,yn)∈[TkPk+2,Qσ(k)]×[Qσ(k)+1,Pσ(k)−1).

Notice that . The subset is included in so we only need to analyze the situation

 (xn+2,yn+2)∈[TkPk+2,Qσ(k)]×[Pσ(k)−2,Pσ(k)−1)⊂[Pσ(k),Qσ(k)]×[Pσ(k)−2,Pσ(k)−1).

Notice that (direct verification), so we are back to analyzing the situation . The boundary map is expanding, so it is not possible for the images of the interval (on the -axis) to alternate indefinitely between the intervals and , where .

This means that either some even iterate

 F2m(xn,yn)∈[Pi+2,Qi+2]×[Qi+3,Pi)⊂Ω¯P

or some odd iterate

 F2m+1(xn,yn)∈[Pσ(i),Qσ(i)]×[Qσ(i)+1,Pσ(i)−2]⊂Ω¯P.

Case II. If , then

 F¯P(xn,yn)∈[TiPi−1,TiPi]×[TiPi,TiQi]=[Pσ(i)+1,Qσ(i)+1]×[Qσ(i)+1,Qσ(i)+2].

There are two subcases that we need to analyze:

 (a)(xn+1,yn+1)∈[Pk,Qk]×[Qk,Pk+1)(b)(xn+1,yn+1)∈[Pk,Qk]×[Pk+1,Qk+1],

where .

Case (a) If , then

 (xn+2,yn+2)∈Tk([Pk,Qk]×[Qk,Pk+1))=[Qσ(k)+1,Qσ(k)+2]×[Qσ(k)+2,Pσ(k)−1].

Notice that (direct verification), so when analyzing the situation the only problematic region is .

Case (b) If , then

 (xn+2,yn+2) ∈Tk+1([Pk,Qk]×[Pk+1,Qk+1]) =[Pσ(k+1)+1,Tk+1Qk]×[Qσ(k+1)+1,Qσ(k+1)+2].

Notice that and (direct verification) so we are left to investigate .

To summarize, we started with and found two situations that need to be analyzed: and .

We prove in what follows that it is not possible for all future iterates to belong to the sets of type . First, it is not possible for all (starting with some ) to belong only to type-a sets , where the sequence is defined recursively as , because such a set is included in the isometric circle , and the argument at the beginning of the proof disallows such a situation.

Also, it is not possible for all (starting with some ) to belong only to type-b sets , where : this would imply that the pairs of points (on the -axis) will belong to the same interval which is impossible due to expansiveness property of the map . Therefore, there exists a pair in the orbit of such that

 (xl,yl)∈[Pj,Qj]×[Pj+1,Qj+1] (type-b)

for some and

 (xl+1,yl+1)∈[Pj′,Tj+1Qj]×[Qj′,Pj′+1]⊂[Pj′,Qj′]×[Qj′,Pj′+1] (type-a),

where . Then

 (xl+2,yl+2)∈Tj′([Pj′,Tj+1Qj]×[Qj′,Pj′+1])=[Qj′′,Tj′Tj+1Qj]×[Qj′′+1,Pj′′−2]

where .

Using the results of the Appendix (Corollary 8.3), we have that the arc length distance

 ℓ(Pj′,Tj+1Qj)=ℓ(Tj+1Pj,Tj+1Qj)<12ℓ(Pj′,Qj′).

Now we can use Corollary 8.2 (ii) applied to the point to conclude that . Therefore . This completes the proof of the property (2). ∎

###### Remark 2.3.

One can prove along the same lines that if the partition is given by the set , the properties (1) and (2) of the Reduction Theory Conjecture also hold.

## 3. The cycle property

The map is discontinuous at , . We associate to each point two forward orbits: the upper orbit , and the lower orbit . We use the convention that if an orbit hits one of the discontinuity points , then the next iterate is computed according to the left or right location: for example, if the lower orbit of hits some , then the next iterate will be , and if the upper orbit of hits some then the next iterate is .

Now we explore the patterns in the above orbits. The following property plays an essential role in studying the maps and .

###### Definition 3.1.

We say that the point has the cycle property if for some non-negative integers

 fmi¯A(TiAi)=fki¯A(Ti−1Ai)=:cAi.

We will refer to the set

 {TiAi,f¯ATiAi,…,fmi−1¯ATiAi}

as the upper side of the -cycle, the set

 {Ti−1Ai,f¯ATi−1Ai,…,fki−1¯ATi−1Ai}

as the lower side of the -cycle, and to as the end of the -cycle.

The main goal of this section is to prove Theorem 1.2 (cycle property) stated in the Introduction. First, we prove some preliminary results.

###### Lemma 3.2.

The following identity holds

 (3.1) Tσ(i)+1Ti=Tσ(i−1)−1Ti−1
###### Proof.

Using relation (1.5) stated in the Introduction, we have that

 Tρ(i)Ti=T−1ρ2(i)T−1ρ3(i)

(where ), so it is enough to show that and . For that we analyze the two parity cases.

If is odd, we have the following identities :

 ρ(i) =σ(i)+1=4g−i+1 (even) ρ2(i) =σ(4g−i+1)+1=2−(4g−i+1)+1=2−4g+i=4g−2+i (odd) ρ3(i) =σ(2−4g+i)+1=4g−(2−4g+i)+1=8g−1−i=3−i (even)

Since , one has by using (1.4). Also, and , hence .

If is even, we have the following identities :

 ρ(i) =σ(i)+1=3−i (odd) ρ2(i) =σ(3−i)+1=4g−(3−i)+1=4g−2+i (even) ρ3(i) =σ(4g−2+i)+1=2−(4g−2+i)+1=5−4g−i=4g+1−i (odd)

Since , one has by using (1.4). Also, and , hence .

Identity (3.1) has been proved for both cases. ∎

###### Remark 3.3.

By introducing the notation , relation (3.1) can be written

 (3.2) Tρ(i)Ti=Tθ(i−1)Ti−1,

which will simplify further calculations.

For any , and .

###### Proof.

Immediate verification. ∎

###### Lemma 3.5.

The relations and hold for all . In addition, if , and if .

###### Proof.

We have

 f2¯A(Pi)=f¯A(Ti−1Pi)=f¯A(Pθ(i−1))=Pθ(θ(i−1)−1)=Pi

and

 f2¯A(Qi)=f¯A(TiQi)=f¯A(Qρ(i)+1)=Qρ(ρ(i)+1)+1=Qi

by Lemma 3.4. The second part follows easily, too. ∎

###### Proof of Theorem 1.2.

Let us analyze the upper and lower orbits of . By Proposition 2.2 and the orientation preserving property of the Möbius transformations, we have

 (3.3) Ti[Pi,Qi]=[Qρ(i),Qρ(i)+1],Ti−1[Pi,Qi]=[Pθ(i−1),Pθ(i−1)+1],

therefore

 (3.4) TiAi∈(Qρ(i),Qρ(i)+1),Ti−1Ai∈(Pθ(i−1),Pθ(i−1)+1)

Depending on whether or we have either

 f¯A(TiAi)=Tρ(i)TiAi or f¯A(TiAi)=Tρ(i)+1TiAi.

Also, depending on whether or we have either

 f¯A(Ti−1Ai)=Tθ(i−1)−1Ti−1Ai or f¯A(Ti−1Ai)=Tθ(i−1)TiAi.

Notice that in the case when