# A Note on the Growth of Periodic Points for Commuting Toral Automorphisms

Authors: Mark Pollicott
Journal: ISRN Geometry
 (2012)

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## Abstract

In this note we study the growth of the number of periodic points for non-degenerate actions of commuting hyperbolic toral automorophisms.


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## Body

## 1. Introduction

There are well-known formulae for the number of fixed points for powers of a given orientation preserving hyperbolic linear toral automorphismT:𝕋d→𝕋d. In particular, if A∈SL(d,ℝ) is the associated hyperbolic matrix, then the number of fixed points for Tn is given by
(1.1)Card{Tnx=x}=|det⁡(An-1)|.
Since the hyperbolicity of T is equivalent to the fact that the eigenvalues for A do not lie on the unit circle, it is easy to see that the number of fixed points for Tn grows exponentially fast in n. We then recall that the growth rate of the number of periodic points for T is given by
(1.2)h(T)=lim⁡⁡k→+∞1klog⁡⁡Card{x:Tkx=x}>0,
where h(T) is the topological entropy of T:𝕋d→𝕋d (or, equivalently, the sum of the logarithms of the eigenvalue of the matrix A with absolute value at least 1).In this note, we want to consider fixed points for commuting hyperbolic toral automorphisms. This necessarily requires the torus to have dimensiond≥3, and for simplicity of exposition we shall initially assume that d=3. Let us, therefore, consider a pair of commuting hyperbolic matrices A1,A2∈SL(3,ℤ) (i.e., A1A2=A2A1 and neither matrix has an eigenvalue of modulus one) and associate the natural ℤ2-action on the three dimensional torus 𝕋3=ℝ3/ℤ3 defined by
(1.3)A:Z2×T3⟶T3givenbyA(n1,n2,x)=A1n1A2n2x+Z3.
We will also ask for this action to be nondegenerate, that is, if n1,n2∈ℤ satisfy A1n1A2n2=I, then this necessarily implies that n1=n2=0. We say that A1 and A2 are independent.We can now consider the growth of the number of fixed points for the action associated to any element(n1,n2)∈ℤ2.Definition 1.1.
We denote the number of fixed points of byA1n1A2n2 on 𝕋3 by
(1.4)N(n1,n2)=Card{x∈T3:A(n1,n2,x)=x}.We want to give uniform estimates on the rate of growth of the number of fixed points for the actions𝒜(n1,n2,·):𝕋3→𝕋3 in terms of (n1,n2)∈ℤ2. In particular, we want to give a lower bound on the growth of the fixed points in terms as ∥(n1,n2)∥2=n12+n22→+∞. In the present context, we can assume without loss of generality that the eigenvalues α1,α2,α3 of A1 and the eigenvalues β1,β2,β3 of A2 are real.Definition 1.2.
We denote(1.5)λ¯:=sup⁡⁡0≤θ≤2π{max⁡⁡i=1,2,3{cos⁡⁡θlog⁡⁡|αi|+sin⁡⁡θlog⁡⁡|βi|}},λ_:=inf⁡⁡0≤θ≤2π{max⁡⁡i=1,2,3{cos⁡⁡θlog⁡⁡|αi|+sin⁡⁡θlog⁡⁡|βi|}}.Our main result, in the particular cased=3, is the following.Theorem 1.3.
LetA1,A2∈SL(3,ℤ) be commuting independent hyperbolic matrices. The growth rates of the fixed points
(1.6)λ¯=limsup⁡‖(n1,n2)‖2→+∞1‖(n1,n2)‖2log⁡⁡N(n1,n2),λ_=liminf⁡‖(n1,n2)‖2→+∞1‖(n1,n2)‖2log⁡⁡N(n1,n2)>0
satisfy 0<λ_<λ¯<+∞.Related problems have been studied forℤd-actions in algebraic and symbolic examples by Miles and Ward [1]. Interestingly, whereas their analysis relies on deep results in diophantine approximation, in the present context the required analysis is completely elementary.The quantityλ_ is related to the supremum of the sum of the Lyapunov exponents for the action. In particular, the bound λ_>0 can then be deduced from ([2], Lemma 4.3 (a)).Remark 1.4.
The valuesθ¯ and θ_ realizing the supremum and infimum, respectively, in (1.6) can be understood as giving the “approximate directions” of largest and smallest growth in the number of fixed points points.Remark 1.5.
There is no analogous result for rates of mixing. The reason for this is simply because any hyperbolic toral automorphism mixes superexponentially with respect to the Haar measure andC∞ test functions. In particular, the rate of mixing is infinite and there is no useful way to distinguish between the actions. By the same token, there is no analogous result for rates of equidistribution for closed orbits [3, Theorem 1.6].The calculations in this paper were inspired by a lecture by Tom Ward, who presented tables similar to those in this note in the context ofℤ2-subshifts of finite type.


## 2. Examples

Let us consider some examples that illustrate Theorem1.3.Example 2.1.
Consider the commuting matricesA1,A2∈SL(3,ℤ) given by
(2.1)A1=(1-10-12-10-12),A2=(20-1011-112).
The number of fixed points N(n1,n2) for |n1|,|n2|≤4 is presented in Table 1.Table 1
The number of fixed pointsN(n1,n2) for |n1|,|n2|≤4. The columns correspond to n1 and the rows correspond to n2.
−4
−3
−2
−1
0
1
2
3
4
4
533
27
203
377
533
448
1261
11857
68411
3
377
91
13
64
91
27
559
3913
21463
2
203
64
13
7
13
13
203
1247
6656
1
27
13
7
1
1
7
64
377
2009
0
533
91
13
1
∞
1
13
91
533
−1
2009
377
64
7
1
1
7
13
27
−2
6656
1247
203
13
13
7
13
64
203
−3
21463
3913
559
27
91
64
13
91
377
−4
68411
11857
1261
448
533
377
203
27
533The eigenvalues ofA1 are α1=3.24698…, α2=1.55496…, and α3=0.198062…, and the eigenvalues of A2 are β1=0.198062…, β2=3.24698…, and β3=1.55496… (which happen to be a permutation of those for A1). Corresponding to these eigenvalues are the common eigenvectors
(2.2)e1=(-0.327985…0.736976…-0.591009…),e2=(-0.591009…0.327985…0.736976…),e3=(0.736976…0.591009…0.327985…).
Using these eigenvalues we can now plot the function θ↦{maxi=1,2,3{cosθlog|αi|+sinθlog|βi|} (cf. Figure 1) and then read off the values of λ¯ and λ_ as the maximum and minimum values, respectively.Figure 1
A plot of{maxi=1,2,3{cosθlog|αi|+sinθlog|βi|} as a function of 0≤θ<2π.In this example, we see thatλ_=0.60501… (occurring at θ_=4.07742…) and λ¯=2.00219… (occurring at θ¯=5.34124…).Example 2.2 (cf. [4]).
We can let(2.3)A1=(0100011-115),A2=(2-1002-1-111-5).
The number of fixed pointsN(n1,n2) for |n1|,|n2|≤4 is presented in Table 2.Table 2
The number of fixed pointsN(n1,n2) for |n1|,|n2|≤4. The columns correspond to n1 and the rows correspond to n2.
−4
−3
−2
−1
0
1
2
3
4
4
25600
8132
464
4652
10880
21388
40592
75356
133120
3
5476
1928
460
296
988
2008
3700
6152
7004
2
1040
452
128
4
80
172
256
76
2000
1
68
152
20
8
4
8
20
232
1404
0
640
148
16
4
∞
4
16
148
640
−1
1404
232
20
8
4
8
20
152
68
−2
2000
76
256
172
80
4
128
452
1040
−3
7004
6152
3700
2008
988
296
460
1928
5476
−4
133120
75356
40592
21388
10880
4652
464
8132
25600The eigenvalues forA are α1=4.70928, α2=0.0967881, and α3=2.19394, and the eigenvalues for A2 are β1=-2.70928, β2=1.90321, and β3=-0.193937. These correspond to the eigenvectors
(2.4)(-0.0440649-0.207514-0.977239),(-0.995305-0.0963337-0.00932395),(0.1857540.4075320.894099).In this case, we can computeλ_=0.689643… (occurring at θ_=4.17448…) and λ¯=2.2481… (occurring at θ¯=5.43348…) (cf. Figure 2).Figure 2
A plot of{maxi=1,2,3{cosθlog|αi|+sinθlog|βi|} as a function of 0≤θ<2π.


## 3. Proof of Theorem1.3

We begin by fixing our notation. LetA1,A2∈SL(3,ℤ) be commuting hyperbolic matrices (i.e., none of the eigenvalues has modulus unity. In this particular case, it is not possible to have ergodic nonhyperbolic toral automorphisms.) We shall assume the associated action is nondegenrate (i.e., A1n1A2n2=I implies (n1,n2)=(0,0)).We next recall the following standard results.Lemma 3.1.
Under the above hypotheses,(1)
the eigenvaluesα1,α2,α3 of A1 are real, and the eigenvalues β1,β2,β3 of A2 are real;(2)
each of the common eigenvectorse1,e2,e3 for A1 and A2 has irrational slope (i.e., each ℝvi+ℤ3 is dense in 𝕋3);(3)
each of the real numberslog|αi|/log|βi|, i=1,2,3, is irrational.Proof.
The first result is a consequence of a standard general result for more general Cartan actions, applied in the particular case ofℤ2-actions [4].
For the second part, we can restrict to the casee1, with the other cases being similar. It is easy to see that we can make an appropriate choice of n,m∈ℤ3 such that matrix A1nA2m either has |α1nβ1m|>1>|α2nβ2m|,|α3nβ3m| or |α1nβ1m|<1<|α2nβ2m|,|α3nβ3m|. In particular, T=𝒜(n1,n2,·) corresponds to a linear hyperbolic toral automorphism T for which ℝvi+ℤ3 is a leaf of either the one-dimensional stable or one-dimensional unstable manifold foliation. In particular, this is dense by the well known minimality of the stable and unstable manifolds.
Finally, for the last part, the irrationality of the ratio of the logarithm of the eigenvalues is a consequence of the nontriviality assumption and part 2. More precisely, if we assume for a contradiction thatlogαi/logβi is a rational p/q, say, then by comparing the actions of A1 and A2 on the dense ℝe1+ℤ3 set, we then see from the second part that A1pA2q=I. This contradicts the nondegeneracy condition, completing the proof.In particular, we see from parts 2 and 3 of Lemma3.1 that for all (n1,n2)∈ℤ2-(0,0) we have that A1n1A2n2 has no eigenvalues of modulus 1.We recall the following standard result for the fixed points of the single transformation𝒜(n1,n2,·):𝕋3→𝕋3.Lemma 3.2.
For each(n1,n2)∈ℤ2-{(0,0)}, we can write
(3.1)N(n1,n2)=|det⁡(I-A1n1A2n2)|.Proof.
This is a standard result, which can also be easily deduced from the Lefschetz fixed point theorem.Lemma3.2 is particularly useful in computing the numerical values of fixed points in the tables we have for the examples. We also have the following simple, but useful, corollary.Lemma 3.3.
For each(n1,n2)∈ℤ2-{(0,0)}, we can write
(3.2)N(n1,n2)=|1-(α1n1β1n2+α2n1β2n2+α2n1β2n2)+(α1-n1β1-n1+α2-n2β2-n2+α3-n3β3-n3)-1|.Proof.
The matrixA1n1A2n2 has eigenvalues α1n1β1n2, α2n1β2n2, and α3n1β3n2. Multiplying out this expression for N(n1,n2) gives
(3.3)N(n1,n2)=|det⁡(I-A1n1A2n2)|=|(1-α1n1β1n2)(1-α2n1β2n2)(1-α3n1β3n2)|=|1-(α1n1β1n2+α2n1β2n2+α2n1β2n2)X+((α1α2)n1(β1β2)n2+(α1α3)n1(β1β3)n2+(α2α3)n1(β2β3)n2)-1|=|1-(α1n1β1n2+α2n1β2n2+α2n1β2n2)+(α1-n1β1-n2+α2-n1β2-n2+α3-n1β3-n2)-1|,
where we have used the identities α1α2α3=detA1=1 and β1β2β3=detA2=1 for the last line.We want to use this lemma to estimate the growth ofN(n1,n2). In particular, we want to get bounds based on the largest of the terms (in modulus) contributing to the right hand side of (3.2). In order to formulate these estimates, it is convenient to introduce the vectors in ℝ2 defined by
(3.4)v1=(log⁡|α1|log⁡|β1|),v2=(log⁡|α2|log⁡|β2|),v3=(log⁡|α3|log⁡|β3|).
Each of these has irrational slope, by the final part of Lemma 3.1.Lemma 3.4.
All of the vectorsv1, v2, and v3 are nonzero and satisfy v1+v2+v2=0.Proof.
For the first part, we need only observe that ifvi=0, say, then this would require |αi|=|βi|=1, that is, at least one of the eigenvalues for the matrices is of modulus one which would contradict the hyperbolicity assumption.
For the second part, we observe that sinceα1α2α3=detA1=1 and β1β2β3=detA2=1 we immediately see that v1+v2+v2=0.We now parameterize the unit vectors inℝ2 by
(3.5)wθ=(cos⁡θsin⁡θ),for0≤θ<2π.
We can then write that
(3.6)〈vi,wθ〉=cos⁡⁡θlog⁡⁡αi+sin⁡⁡θlog⁡⁡βi,fori=1,2,3.
In particular, if we write (n1,n2)=(Rcosθ,Rsinθ), say, where R=∥(n1,n2)∥2, then we can write
(3.7)|αin1βin2|=exp⁡⁡(R(cos⁡⁡θlog⁡⁡|αi|+sin⁡⁡θlog⁡⁡|βi|)).To prove Theorem1.3, it suffices to show that the vectors v1, v2, v3 are not collinear. Since v1+v2+v3=0 and the vectors v1,v2,v3 are nonzero, and additionally we know that the vectors are noncollinear, it is then easy to see that this is enough to know that for any 0≤θ<2π there is some i such that 〈vi,vθ〉>0 (se Figure 3). For typical θ, there will be a single dominant term of the form (3.7) contributing to the right hand side of (3.2).Figure 3
The projection ofwθ onto one of the vectors v1, v2, v3 must have a strictly positive component.Assume for a contradiction that the vectorsv1, v2, and v3 are collinear. Then we can choose δ≠0 such that
(3.8)δ=log⁡⁡|α1|log⁡⁡|β1|=log⁡⁡|α2|log⁡⁡|β2|=log⁡⁡|α3|log⁡⁡|β3|.
First we observe that δ cannot be irrational since otherwise {nlog|α1|+mlog|β1|:n,m∈ℤ} will be dense on the real line ℝ. However, since ℝ+v1+ℤ3 is dense in 𝕋3 this means that we can choose nk,mk∈ℤ such that Ank,Bmk→I as k→+∞, but with Ank,Bmk≠I. However, this is clearly false in the lattice SL(3,ℤ). On the other hand, if δ=p/q were a rational then by again considering the action on the dense set ℝ+w1+ℤ3 we see that ApBq=I, which contradicts the nondegeneracy hypothesis.


## 4. Generalizations toℤk-Actions

We will consider the more general setting of higher-dimensional actions. The basic results are similar to the case of Theorem1.3.Hypothesis 1.
Let2≤k≤d-1.(1)
We shall assume thatA1,…,Ak∈SL(d,ℤ) are commuting matrices, that is, AiAj=AjAi for 1≤i,j≤k.(2)
We shall assume that each matrixA1n1⋯Aknk, (n1,…,nk)∈ℤk-(0,…,0) is ergodic (i.e., they do not have eigenvalues, which are roots of unity).(3)
We shall assume that the action is nondegenerate, that is, if there existn1,…,nk∈ℤ such that
(4.1)A1n1A2n2⋯Aknk=Ithenn1=⋯=nk=0.(4)
We shall assume that the action is irreducible, that is, no𝒜(n1,…,nk):𝕋d→𝕋d preserves a proper invariant toral subgroup of 𝕋d.(5)
We shall additionally assume, mainly for convenience, that the matrices are semisimple (i.e., they diagonalize over the complex numbers) andAi has complex eigenvalues α1(i),…,αd(i) for i=1,…,k.The special case that bears closest comparison with the special case ofk=2 and d=3 is when k=d-1. In particular, in this case Ai has real eigenvalues α1(i),…,αd(i) for i=1,…,k.We now generalize two definitions from the first section.Definition 4.1.
Let𝒜:ℤk×𝕋d→𝕋d be the action given by 𝒜(n1,…,nd,x)=A1n1⋯Aknkx+ℤd, then we denote
(4.2)N(n1,…,nk)=Card{x∈Td:A(n1,…nk,x)=x}.Definition 4.2.
We can define(4.3)λ_=inf⁡⁡‖v‖2=1{sup⁡⁡w〈v,w〉},λ¯=sup⁡⁡‖v‖2=1{sup⁡⁡w〈v,w〉},
where the supreme ranges over all unit vectors v in ℝk.The natural generalization of Theorem1.3 is the following.Theorem 4.3.
The growth rates of the number of fixed points(4.4)λ¯:=limsup⁡‖(n1,…,nk)‖2→+∞1‖(n1,…,nk)‖2log⁡⁡N(n1,…,nk)},λ_:=liminf⁡‖(n1,…,nk)‖2→+∞1‖(n1,…,nk)‖2log⁡⁡N(n1,…,nk)}>0
satisfy 0<λ_<λ¯<+∞.To begin the proof, we need the following standard generalization of Lemma3.2.Lemma 4.4.
For each(n1,…,nk)∈ℤ2-{(0,…,0)}, we can write
(4.5)N(n1,…,nk)=|det⁡(I-A1n1⋯Aknk)|.Proof.
This is again a standard application of the Lefschetz formula.In particular, we can use Lemma4.4 to write
(4.6)N(n1,…,nk)=|det⁡(I-A1n1⋯Aknk)|=∏j=1d‍|1-∏i=1d-1‍(αj(i))ni|.
It is convenient to use the parameterization (n1,…,nk)=(p1R,…,pkR), where  (1)
0≤p1,…,pk≤1 with p12+⋯+pk2=1;(2)
R=∥(n1,…,nk)∥2.We can now introduce the notationvp=(p1,…,pk) and vj=(log|αj(1)|,…,log|αj(k)|), for j=1,…,k. We can now easily see from (4.6) that for any δ>0, there exists R0=R0(δ) such that
(4.7)N(n1,…,nk)≥∏j:〈vp,vj〉>0‍(exp⁡⁡(R〈vp,vj〉)-1)∏j:〈vp,vj〉<0‍(1-exp⁡(R〈vp,vj〉))≥(1-δ)∏j:〈vp,vj〉≥0‍exp⁡⁡(R〈vp,vj〉)≥(1-δ)exp⁡⁡(R〈vp,(∑j:〈vp,vj〉≥0‍vj)〉),
for R≥R0. In particular, we see that
(4.8)N(n1,…,nk)≥(1-δ)exp⁡⁡(λ_‖(n1,…,nk)‖2),
where
(4.9)λ_=inf⁡⁡p{〈vp,(∑j:〈vp,vj〉≥0‍vj)〉}.
Similarly, we see that for R≥R0,
(4.10)N(n1,…,nk)≤(1+δ)exp⁡⁡(λ¯‖(n1,…,nk)‖2),
where
(4.11)λ¯=sup⁡⁡p{〈vp,(∑j:〈vp,vj〉≥0‍vj)〉}.To see thatλ_>0, we need to know that v1,…,vk are not confined to a codimension one hyperplane in ℝk orthogonal to some vp. Assume for a contradiction that there is a unit vector vp such that 〈vp,vi〉=0 for i=1,…,k. Let vp=(vp(1),…,vp(k)), then by Dirichlet's theorem of simultaneous diophantine approximation, for any ϵ>0, we choose 1≤q≤([1/ϵ]+1)k and (n1,…,nk)∈ℤk with
(4.12)‖(n1,…,nk)-qvp‖∞≤ϵ.
In particular, the eigenvalues (αj1)n1⋯(αjk)nk, j=1,…,k, for the matrix A1n1⋯Aknk∈SL(d,ℝ) satisfy
(4.13)|log⁡⁡|(αj(1))n1⋯(αj(k))nk||=|∑l=1knllog⁡⁡|αj(l)||≤kϵ+q|∑l=1k‍vllog⁡⁡|αj(l)||︸=|〈vp,vi〉|=0.
In particular, the algebraic integers, and its conjugates, occurring as zeros of the characteristic polynomial det(zI-A1n1⋯Aknk)=0 can be arbitrarily close to one. It only remains to show this cannot happen, which we deduce from the following two results.Lemma 4.5 (Krönecker, [5]).
Any algebraic integerα whose conjugate roots α=α1,…,αd all lie on the unit circle must necessarily be a root of unity.Proof.
We include the simple proof for completeness. Let us define a sequence of monomials(4.14)Pn(x):=∏i=1d‍(x-αin)=xd+ad-1(n)xd-1+⋯+ak(n)xk+⋯+a1(n)x+a0(n).
In particular, since
(4.15)|ak(n)|=|∑i1<⋯<id-kαi1(n)⋯αid-k(n)|≤K:=d!,
we see that {Pn(x):n≥1} is a finite set as is the set of roots α of these polynomials. Thus for any such root, the pigeonhole principle applied to {αn:n≥0} shows that there exists 0≤p<q≤K+1 such that αp=αq, and thus αq-p=1.We can also prove the following variant.Lemma 4.6.
Givend≥2, there exists ϵ>0 such that if α is an algebraic number of degree, which is not an algebraic integer, then the conjugate values α=α1,…,αd cannot all be contained in the annulus
(4.16)A(ϵ)={z∈C:1-ϵ≤|z|≤1+ϵ}.Proof.
Since the proof is elementary, we include it for convenience. Assume for a contradiction that for somed≥2 we can find an infinite sequence of monomials
(4.17)Pn(x)=xd+ad-1(n)xd-1+⋯+ak(n)xk+⋯+a1(n)x+a0(n)∈Z[x]],forn≥2,
whose roots α1(n),…,αd(n)∈A(1/n) do not lie on the unit circle. In particular, since Pn(x)=∏i=1d‍(x-αi(n)), we see that
(4.18)|ak(n)|=|∑i1<⋯<id-k‍αi1(n)⋯αid-k(n)|≤K:=(1+1n)d!.
Since for each k, we have ak(n)∈ℤ∩[-K,K], for all n≥1, we can use the pigeonhole principle to choose an infinite subsequence with P(x):=Pn1(x)=Pn2(x)=Pn3(x)=⋯ for which the coefficients all agree. But this contradicts the zeros of each polynomial not lying on the unit circle.Remark 4.7.
In fact, Schinzel and Zassenhaus showed that ifα is not a root of unity, then |α|≥1+1/42+d/2. (cf. [6]).Remark 4.8.
The formula (4.1) and the description of the growth of periodic points for a single hyperbolic matrix were a core ingredient in Manning's famous work on the classification of Anosov toral automorphisms [7].Example 4.9 (cf. [2]).
We can consider the action on𝕋6 defined by the matrices
(4.19)A1=(010000001000000100000010000001-125352),A2=(0-6-6-3-62-24407-22-6-6-2-103-389313-44-11-12-3-175-51414322-7).
The number of fixed points N(n1,n2) for |n1|,|n2|≤4 is presented in Table 3.Table 3
The number of fixed pointsN(n1,n2) for |n1|,|n2|≤4. The columns correspond to n1 and the rows correspond to n2.
−4
−3
−2
−1
0
1
2
3
4
4
13395375
1295405
61440
2645
15
125
48735
6452480
560164815
3
1613760
722000
37500
3920
540
1280
37500
524880
269340
2
141135
182405
3375
1280
15
1805
59535
2415125
79626240
1
48735
3920
15
125
15
405
19440
1245005
56745375
0
10786560
188180
960
20
∞
20
960
188180
10786560
−1
56745375
1245005
19440
405
15
125
15
3920
48735
−2
79626240
2415125
59535
1805
15
1280
3375
182405
141135
−3
269340
524880
37500
1280
540
3920
37500
722000
1613760
−4
560164815
6452480
48735
125
15
2645
61440
1295405
13395375The matrixA1 has eigenvalues
(4.20)α1=3.68631…,α2=-1.32361…,α3=0.0607659…+0.998152i…,α4=0.0607659…-0.998152i…,α5=-0.75551…,α6=0.271274…,
and the matrix A2 has corresponding eigenvalues
(4.21)β1=-0.463258…,β2=-22.1542…,β3=0.910592…-0.413307i…,β4=0.910592…+0.413307i…,β5=-0.0451382…,β6=-2.15862…In this example, we see thatλ_=1.06415… (occurring at θ_=0.258896…) and λ¯=3.11069… (occurring at θ¯=4.62214…) (cf. Figure 4).Figure 4
A plot of{maxi=1,2,3{cosθlog|αi|+sinθlog|βi|} as a function of 0≤θ<2π.


## 5. A Sector Theorem and Directional Growth

A natural refinement is to estimate the number of fixed points for(n1,n2) lying in a sector of the form 𝒮(θ1,θ2):={(n1,n2)∈ℤ2:n2tan(θ1)≤n1≤n2tan(θ2)}, for 0≤θ1<θ2≤2π.Definition 5.1.
We can denote(5.1)λ¯(θ1,θ2)=sup⁡⁡θ1≤θ≤θ2{max⁡⁡i=1,2,3{cos⁡⁡θlog⁡⁡|αi|+sin⁡⁡θlog⁡⁡|βi|}},λ_(θ1,θ2)=inf⁡⁡θ1≤θ≤θ2{max⁡⁡i=1,2,3{cos⁡⁡θlog⁡⁡|αi|+sin⁡⁡θlog⁡⁡|βi|}}.We then have the following natural refinement of Theorem1.3.Theorem 5.2 (Sector Theorem).
LetA1,A2∈SL(3,ℤ) be commuting independent hyperbolic matrices. Let 0≤θ1<θ2≤2π. The growth rates of the fixed points in the sector 𝒮(θ1,θ2)(5.2)λ¯(θ1,θ2):=limsup⁡‖(n1,n2)‖2→+∞,(n1,n2)∈S(θ1,θ2)1‖(n1,n2)‖2log⁡⁡N(n1,n2),λ_(θ1,θ2):=liminf⁡‖(n1,n2)‖2→+∞,(n1,n2)∈S(θ1,θ2)1‖(n1,n2)‖2log⁡⁡N(n1,n2)
satisfy 0<λ_(θ1,θ2)<λ¯(θ1,θ2)<+∞.Proof.
The proof follows easily by modifying the proof of Theorem1.3. Recall that the number  of fixed points of the single transformation 𝒜(n1,n2,·):𝕋3→𝕋3, this time restricting to (n1,n2)∈𝒮, can be written as
(5.3)N(n1,n2)=|det⁡(I-A1n1A2n2)|=|1-(α1n1β1n2+α2n1β2n2+α2n1β2n2)+(α1-n1β1-n1+α2-n2β2-n2+α3-n3β3-n3)-1|.
We can again consider the vectors v1,v2,v3, but this time we only need to consider unit vectors vθ with θ1≤θ≤θ2. We can again write that 〈vi,wθ〉=cosθlogαi+sinθlogβi for i=1,2,3. In particular, if we write (n1,n2)=(Rcosθ,Rsinθ)∈𝒮(θ1,θ2), say, where R=∥(n1,n2)∥2, then we have that
(5.4)|αin1βin2|=exp⁡⁡(R(cos⁡⁡θlog⁡⁡|αi|+sin⁡⁡θlog⁡⁡|βi|)).
We now want to estimate N(n1,n2) in terms of the largest expression of the form (5.4) where (n1,n2)∈𝒮(θ1,θ2). In particular, modifying the proof of Theorem 1.3, we observe that
(5.5)λ¯(θ1,θ2)=sup⁡⁡θ1≤θ≤θ2max⁡⁡i=1,2,3{〈vi,wθ〉}≥inf⁡⁡θ1≤θ≤θ2max⁡⁡i=1,2,3{〈vi,wθ〉}=λ_(θ1,θ2),
as required.Definition 5.3.
Let us denote(5.6)λ(θ):=max⁡⁡i=1,2,3{cos⁡⁡θlog⁡⁡|αi|+sin⁡⁡θlog⁡⁡|βi|}.We then have the following corollary.Corollary 5.4 (Directional growth).
LetA1,A2∈SL(3,ℤ) be commuting independent hyperbolic matrices. Let 0≤θ<2π. The following limits exist and agree:
(5.7)λ¯(θ):=lim⁡ϵ→0limsup⁡‖(n1,n2)‖2→+∞,(n1,n2)∈S(θ-ϵ,θ2+ϵ)1‖(n1,n2)‖2log⁡⁡N(n1,n2),λ_(θ):=lim⁡ϵ→0liminf⁡‖(n1,n2)‖2→+∞,(n1,n2)∈S(θ-ϵ,θ+ϵ)1‖(n1,n2)‖2log⁡⁡N(n1,n2),
and λ¯(θ)=λ_(θ)=λ(θ).Proof.
This follows immediately from Theorem5.2 and continuity of λ(θ).Remark 5.5.
We have that for each fixed choice(n1,n2)∈𝒮(θ1,θ2) that
(5.8)h(A(n1,n2,⋅))=lim⁡k→+∞1klog⁡⁡Card{x:A(kn1,kn2)x=x}.
We see that for any ϵ>0 we have that
(5.9)λ_(θ1,θ2)-ϵ≤h(A(n1,n2))‖(n1,n2)‖2≤λ¯(θ1,θ2)+ϵ
providing ∥(n1,n2)∥2 is sufficiently large. In particular, by continuity we see that we have the limit
(5.10)lim⁡R→+∞h(A([Rcos⁡θ,Rsin⁡θ]))R=λ(θ).



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*Source: 165808-2012-08-16.xml*