Conclusiones del capítulo III

In this section we discuss an important condition on the function r : X_A+ → R+ that we require in order to prove the main results in this thesis.

Recall the definition of cohomologous functions from subsection 2.2.3, then we have the following definition of what we call non-lattice functions.

Definition 3.1.1. We say that a function f : X_A+ → R is non-lattice if f is not cohomologous to a constant plus a function valued in a discrete subgroup of

R.

In other words, the function r : X_A+ → _R is non-lattice if there are no continuous functionsψ:X_A+ →R,M :X_A+→dZand c∈Rsuch that r=c+

M+ψ◦σ−ψ. The importance of this condition for the functionr is that we are going to use transfer operators (see subsection 2.2.3 for definition)Lsr, s∈C, to

carry out the analysis for studying pairs of closed geodesics with the constraints in the word length and geometric lengths. The fact that r is non-lattice is going to control the spectral properties of the transfer operatorLsr, s∈C(see

Remark 2.3.3). Furthermore, as we are going to see in the next subsection, this condition onrwill imply that the functionRis not cohomologous to a constant. This property of R will be useful to analyse a pressure functionP(sR), s∈R.

To guarantee that r is non-lattice we need the following assumption on the geometric lengths of closed geodesics:

The non-lattice condition: {`(γ)−c|γ|:γ closed geodesics} 6⊂ dZ, for any

c, d∈_R.

We can easily see that the non-lattice condition implies thatr is non-lattice.

Lemma 3.1.4. If the non-lattice condition holds, then r is non-lattice.

Proof. Suppose thatr is not non-lattice. Then, by definition, there arec, d∈R

and a continuous function M : X_A+ → dZ such that, whenever σnx = x, we

havern(x)−nc=Mn(x). Hence

{rn(x)−nc:σnx=x, n≥1} ⊂dZ.

Sincern(x) =`(γ), whereγ is the closed geodesic corresponding to the periodic orbit of x, this is equivalent to {`(γ)−c|γ|: γ is a closed geodesic} ⊂ dZ, so

the non-lattice condition fails to hold.

Recall that consideringRas vector space overQ, we letLbe the smallest

subspace containing all the lengths of closed geodesics, i.e. L = span_Q{`(γ) : γis closed geodesic}. The following lemma shows that the non-lattice condition is implied by requiring the dimension of dim_Q(L) to be at least 3.

Lemma 3.1.5. If dim_Q(L)≥3, then the non-lattice condition holds.

Proof. Suppose that the non-lattice condition does not hold. Then, this means there arec, d∈Rsuch that

{`(γ)−c|γ|:γ closed geodesics} ⊂dZ.

Another way to write this is that there arec, d∈_Rsuch that

`(γ) =c|γ|+dM,

whereM ∈_Z. This implies that l(γ)∈cZ⊕dZ⊂cQ⊕dQ. This implies that

dim_Q(L)≤2

We are going to give three examples here. The first one shows a case when the non-lattice condition cannot be true for any choice of geometric lengths on the edges and so any geometric length of closed geodesic. While the second one shows that if the non-lattice condition holds, then it is not nec- essarily true that dim_Q(L)≥3. The third example shows a metric graph when the non-lattice condition is satisfied.

Example 3.1.1. Consider a graph of one vertex and two edges (which are loops in this graph). To see why the non-lattice condition cannot be satisfied here, let us denote the edges (or the loops) a and b with lengths la and lb,

respectively (see Figure 3.1). Consider a closed geodesicsγ with|γ|=N. Then

`(γ) =nla+ (N−n)lb for some 0≤n≤N. Therefore,

`(γ) =n(la−lb)∈(la−lb)Z+lbZ.

Figure 3.1

la lb

Example 3.1.2. Consider a graph of one vertex and three loops with edge lengths1,√2,1 +√2 as shown in Figure 3.2, so have thatdim_Q(L) = 2. But in this metric graph the non-lattice condition holds. To see this, suppose that the non-lattice condition fails to hold, then there arec, d∈_Rsuch that`(γ)−c|γ| ∈

dZ, for every closed geodesic γ. Just considering three loops (where |γ| = 1), this gives us the equations:

1−c∈dZ, (3.1.1)

√

2−c∈dZ, (3.1.2)

1 +√2−c∈dZ. (3.1.3)

Subtracting (3.1.2) from (3.1.3) we get 1 ∈dZ. Sod∈ Q and hence dZ⊂Q. Also, subtracting (3.1.1) from (3.1.3) we get√2∈dZ⊂Q, i.e.

√

2 is rational, which is a contradiction. Therefore, the non-lattice condition holds.

Figure 3.2

1

√

2 1 +√2

Example 3.1.3. For the third example, to make sure that the non-lattice is sat- isfied we are going to draw a metric graph where we make sure thatdim_Q(L)≥3

and hence by Lemma 3.1.5 the non-lattice condition holds. Consider a graph that has three vertices and five edges and assign lengths to the edges as shown in Figure 3.3. For the values of the edge lengths we have in this metric graph, clearlydim(span_Q{`(γ) :γ is closed geodesic})≥3.

Figure 3.3 1 π √ 3 2 √ 2