Flat Surfaces with Finite Holonomy Groups

# Flat Surfaces with Finite Holonomy Groups

###### Abstract

We prove that flow of a generic geodesic on a flat surface with finite holonomy group is ergodic. We use this result to prove that flows of generic billiards on certain flat surfaces with boundary are also ergodic.

## 1 Introduction

A surface is called flat if it has a flat metric having finitely many singularities of conical type. These surfaces may not be orientable and they may have boundary components. Ergodicity of a generic geodesic (or billiard trajectory), existence of closed geodesics (or billiard trajectories) are the main problems studied in dynamics of flat surfaces and billiards.

If a flat surface is orientable, closed and its holonomy group is trivial, then it is called very flat. Above problems admit satisfactory and complete answers for very flat surfaces. See  for ergodicity of geodesic flows and , ,  for counting problem of closed geodesics. Also, see , ,  for introductory presentations about ergodic theory of very flat surfaces. Note that ergodicity of the geodesic flow immediately implies existence of asymptotic cycles. See  for a definition of asymptotic cycle.

If we are given a rational polygon, we can obtain a very flat surface by gluing a finite number of its copies along their edges. See ,, . Since directional flow at almost all directions of a very flat surface is ergodic, this implies that flow of a generic billiard in such a polygon is ergodic. See ,  for much information about ergodicity of billiard flows in rational polygons.

Our aim is to prove that flow of a generic geodesic on a closed, orientable flat surface with finite holonomy group is ergodic. We call these surfaces really flat. Note that we say flow of a geodesic on such a surface is ergodic if at each region of it spends a time proportional to the volume of the region. Equivalently, this means that for each continuous on , the following limit exists:

 limK→∞1K∫K0f(gt)dt =1A(S)∫Sfdμ,

where is the area of .

We show that these surfaces can be covered by a very flat surfaces, where the covering is branched, Galois and respects flat metrics on the surfaces. Indeed, this immediately implies that flow of a generic geodesic on a really flat surface is ergodic. Finally, we use these results to prove ergodicity of flows of generic billiards on certain flat surfaces with boundary. We call these surfaces also really flat. Note that these surfaces may be non-orientable.

#### Convention:

Surfaces that we consider and maps between them are of class . They are also compact. They may have boundary components and be non-orientable.

## 2 Flat coverings

In this section, we introduce a notion of covering for flat surfaces. From the topological point of view, such a covering is nothing else than a branched covering. But, we also require them to respect flat metrics.

Given a branched covering between two surfaces, we denote set of its branched points and ramification points by and , respectively.

Let and be two flat surfaces.

###### Definition.

A map is called a flat (covering) map, if it is a branched covering, and

 ψ↾S∗−ψ−1(b): S∗−ψ−1(b)→S−b

is a local isometry.

We describe two ways to obtain flat maps.

• Let be a finite isometry group of so that each element except identity fixes a finite number of points. Consider (topological) orbifold . induces a flat metric on , and is a flat map with respect to the metric on and the induced metric on .

• Let be a branched cover. Observe that induces a flat metric on even if it is not Galois, and this makes a flat map.

###### Remark.

If is a flat map and is a point with ramification index , then we have

 θ(x)=mθ(ϕ(x)),

where and are the angles at and , respectively.

## 3 Really flat surfaces

In this section, we introduce a new family of flat surfaces. We do not exclude non-oriantable surfaces and surfaces with boundary from our discussion.

Recall that an orientable flat surface without boundary is called very flat if it has trivial holonomy group. Surfaces defined below are natural generalizations of the very flat surfaces.

###### Definition.

A flat surface is called really flat if it has the following properties:

1. Its holonomy group is finite.

2. For any two non-singular points on the boundary of , for any curve joining them and for any non-zero vector we have

 ty(v′)−tx(v) is π−% rational,

where is parallel transport of through , is the angle between and its boundary component, is the angle between and the boundary component that is in.

Observe that last item of the above definition makes sense: rationality of the change in angles does not depend on orientation of the boundary component.

###### Remark.
1. If the surface does not have boundary component then it is really flat if and only if its holonomy group is finite.

2. A planer polygon is rational if and only it is really flat. See Figure 1.

3. Angles between boundary components of a really flat surface are -rational.

#### Double of a flat surface

Let be a flat surface with at least one boundary component. Take another copy of , call it . For each , let be the corresponding point in . Glue and so that each boundary point comes together with . We will call resulting flat surface double of and denote as .

###### Remark.

is really flat if and only if is really flat. Figure 1: A polygon in the plane. It is really flat if and only if α and angles at vertices of the polygon are π-rational.

## 4 Holonomy representation

In this section, we assume that flat surfaces are closed, oriented and flat maps are orientation preserving. Our aim is to relate holonomy representations and flat maps.

Let be a flat surface, be the set of singular points of and be a finite subset of that contains . Let . We denote holonomy representation as

 hol:Π1(S−c,y)→Ty=T={z∈C:|z|=1},

where is the rotations of the unit circle in the tangent space at of , and is the fundamental group of based at .

###### Definition.

A curve on is called polygonal if it consists of finitely many geodesic segments and does not pass through any point in .

Note that we do not assume a polygonal curve is not self-intersecting. Even it is possible that a geodesic segment of a polygonal curve is self-intersecting.

Observe that we can define (oriented) angle between two non-zero vectors at a given point of . We will denote the angle by .

Let be a polygonal loop based at . Let be the number of its vertices, and denote these vertices by so that ordering is compatible with orientation of . Let be the edge of the geodesic loop which originates from . Let be the unit vector at which is in direction of and be the unit vector at which is in direction of . Note that indices are given modulo . See Figure 2.

###### Lemma 1.

Let

 Θ=n∑i=1θ(Ωi,Ω′i).

We have

 hol(P)=eıΘ.
###### Proof.

Let be a non-zero vector based at and be the parallel transport of through . Since parallel transport of through is , we have

 θ(β′,Ω′i+1)=θ(β,Ω′i)+θ(Ωi+1,Ω′i+1).

Let be a vector based at . We denote its parallel transport at (through ) by . Above equality implies the followings:

 θ(α(1),Ω′2)=θ(α,Ω′1)+θ(Ω2,Ω′2)
 θ(α(2),Ω′3)=θ(α(1),Ω′2)+θ(Ω3,Ω′3)
 …
 θ(α(n−1),Ω′n)=θ(α(n−2),Ω′n−1)+θ(Ωn,Ω′n)
 θ(α(n),Ω′1)=θ(α(n−1),Ω′n)+θ(Ω1,Ω′1)

Note that we obtained the last equality by counting indices modulo , and is parallel transport of through . Adding above equalities we get

 θ(α(n),Ω′1)=θ(α,Ω′1)+Θ,

which means that . ∎

###### Proposition 1.

Let be a flat covering, and , where and are the sets of singular points of and , respectively.

The following diagram is commutative.

 Π1(S∗−ψ−1(s∪ψ(s∗)),x)\lx@stackrelhol⟶T⏐⏐↓ψ∗⇑‖⇓Π1(S−(s∪ψ(s∗)),y)\lx@stackrelhol⟶T
###### Proof.

Take a polygonal loop in which is based at . By Lemma 1, depends only sum of angles between edges of . Observe that is a polygonal loop in which is based at . Since is a local isometry, it is angle preserving. This implies that .

Now we relate really flat surfaces to very flat surfaces. Let be a really flat surface. For each , let , where and are coprime positive integers. Let be the set of singular points of and . Let be the order of the holonomy group of at . Consider as a differentiable orbifold with a divisor .

###### Theorem 1.
1. There exists a flat Galois covering so that is very flat.

2. Assume that is a flat covering. is really flat if and only if is really flat.

###### Proof.
1. For each , is divisible by . Therefore, there exists a homomorphism so that the triangle on the right of the following diagram is commutative:

Consider . Observe that is a finite index normal subgroup of . Consider Galois orbifold covering which corresponds to . Observe that this covering is finite. Let be a point so that . Commutativity of the square in the above diagram implies that the diagram is commutative. Consider the metric induced on from the one on by the map . Observe that is a flat map for the metrics on and . Proposition 1 implies that the following diagram is also commutative:

Since is trivial homomorhism, commutativity of the first diagram implies that is also trivial. By the commutativity of above diagram, the map

 hol:Π1(S∗−ψ−1(s),y∗)→T

is trivial, which means that is very flat.

2. Assume that is really flat. By the first item of the present theorem, we may assume that is very flat. Let be set of branched points of . Let and be set of singular points of and . Let . Let and . Consider the below diagram which is commutative by Proposition 1.

Let . We have that

 α∗(Π1(S′′−α−1(A),y′′))⊂K⊂Π1(S′−A,y′).

Observe that , since is a finite covering. This implies that is a finite index subgroup of . Hence image of is finite. Thus, is really flat.

Other implication of the last part of the present proposition immediately follows from Proposition 1.

## 5 Ergodicity

In this section, we study behavior of a typical geodesic and billiard trajectory on a really flat surface. Note that if the holonomy group of the surface is not trivial, we can not talk about directional flows. Therefore, we consider flow of a generic geodesics on a really flat surfaces. First, we recall notion of ergodicity, and then state a main theorem which is proved by Kerckhoff, Masur and Smillie .

Let be a set, be a algebra and be a finite measure on .

###### Definition.

A measurable semiflow on is called ergodic if for each measurable set of satisfying for all , either or is equal to .

###### Definition.

A measurable semiflow on a measurable space is called uniquely ergodic if there exists a unique invariant measure (up to scaling with a positive number) for which is ergodic.

### 5.1 Ergodicity of geodesic flows on very flat surfaces

Let be a very flat surface. Observe that triviality of the holonomy group implies that for each point on , any vector on tangent space of can be carried to any non-singular point of by a parallel transport and resulting vector is independent of the chosen path. That is, directional flow exists for any direction on the surface.

###### Theorem (Kerchoff, Masur, Smillie).

For almost all directions, directional flow on is uniquely ergodic with respect to the area measure of .

Let’s denote geodesic flow in a direction by and area measure of by . Note that we can state the result above as follows. For almost all directions and for any continuous function on

 1K∫K0f(gt(x))dt  \textupconvergesuniformlyto 1μ(S′)∫S′fdμ.

### 5.2 Ergodicity of geodesic flows on really flat surfaces without boundary

Let be an orientable really flat surface without boundary. Let be the flat Galois cover which is constructed in Theorem 1 and corresponds to the kernel of the homomorphism . Let be the group of deck transformations of the cover . Note that acts transitively on the fiber of each point and is cyclic. Also observe that acts on by isometries. Let be the order of , or equivalently, degree of the cover. Let and be the area measures on and , respectively. We assume that , thus .

Now, we state a lemma whose proof is based on Theorem 1.

###### Lemma 2.

For almost all and for almost all , flow of the geodesic in direction of is ergodic.

###### Proof.

Consider the flat covering . Let be the set of singular points of . For each , , let be the geodesic based at with initial direction and be its lift so that . Let so that . It is clear that initial direction of is .

Observe that for almost all and , flow of the geodesic is ergodic and does not pass through . See Section 5.1. Thus for each such and , the following the following limit exists

 limK→∞1K∫K0f(ϕ(g∗t(x∗)))dt  =1N∫S∗f∘ϕ dμ∗,

for any continuous function on . Since

 f(ϕ(g∗t(x∗)))=f(gt(x))\textupand∫Xf∘ϕ dμ∗=N∫Sf dμ,

the following limit also exists:

 limK→∞1K∫K0f(gt(x))dt  =∫Sf dμ.

We continue to use notation of the above proof. Assume that directional flow on with respect to is ergodic. Let so that is obtained from parallel transport of through a path joining to .

###### Corollary 1.

For almost all , flow of the geodesic with initial direction is ergodic.

###### Proof.

Consider directional flow on obtained from . For each having above properties, there is such that , and is obtained by parallel transport of . Since directional flow is ergodic, we see that for almost all , geodesic flow with respect to is ergodic. As in the proof of the Lemma 2, this implies that for almost all , geodesic flow with respect to is ergodic. ∎

Now we generalize result of Lemma 2 to non-orientable flat surfaces.

###### Theorem 2.

Let be a really flat surface without boundary. For almost all and for almost all , flow of the geodesic in direction of is ergodic.

###### Proof.

If is orientable, then the statement is true. See Lemma 2. Assume that is not orientable. Consider orientable double cover . The map induces a metric on by the one on , and it is a flat map with respect to these metrics. Observe that flow of a geodesic on is ergodic if and only if flow of one of its lifts is ergodic. Since is really flat and orientable, Lemma 2 implies the result. ∎

### 5.3 Ergodicity of billiard flows on really flat surfaces

We use Theorem 2 to obtain a similar result about ergodicity of billiard flows on really flat surfaces with boundary.

###### Theorem 3.

Let be a really flat surface with boundary. For almost all and for almost all , flow of the billiard in direction of is ergodic.

###### Proof.

Observe that flow of a billiard on is ergodic if and only if flow of the corresponding geodesic in double of , , is ergodic. Since is really flat, result follows from Theorem 2. ∎

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