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

Abstract

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

1. Introduction

There are well-known formulae for the number of fixed points for powers of a given orientation preserving hyperbolic linear toral automorphism 𝑇∶𝕋^𝑑→𝕋^𝑑. In particular, if 𝐴∈SL(𝑑,ℝ) is the associated hyperbolic matrix, then the number of fixed points for 𝑇^𝑛 is given by Card{𝑇^𝑛𝑥=𝑥}=||det(𝐴^𝑛−1)||.(1.1) 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 𝑇^𝑛 grows exponentially fast in 𝑛. We then recall that the growth rate of the number of periodic points for 𝑇 is given by ℎ(𝑇)=lim_𝑘→+∞1𝑘logCard𝑥∶𝑇^𝑘𝑥=𝑥>0,(1.2) where ℎ(𝑇) is the topological entropy of 𝑇∶𝕋^𝑑→𝕋^𝑑 (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 dimension 𝑑≥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 𝒜∶ℤ²×𝕋³⟶𝕋³givenby𝒜𝑛₁,𝑛₂,𝑥=𝐴^𝑛1₁𝐴^𝑛2₂𝑥+ℤ³.(1.3) We will also ask for this action to be nondegenerate, that is, if 𝑛₁,𝑛₂∈ℤ satisfy 𝐴^𝑛1₁𝐴^𝑛2₂=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 by 𝐴^𝑛1₁𝐴^𝑛2₂ on 𝕋³ by 𝑁𝑛₁,𝑛₂=Card𝑥∈𝕋³∶𝒜𝑛₁,𝑛₂,𝑥=𝑥.(1.4)

We want to give uniform estimates on the rate of growth of the number of fixed points for the actions 𝒜(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=𝑛²₁+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 𝜆∶=sup_{0≤𝜃≤2𝜋}max_𝑖=1,2,3cos𝜃log||𝛼_𝑖||+sin𝜃log||𝛽_𝑖||,𝜆_{_}∶=inf_{0≤𝜃≤2𝜋}max_𝑖=1,2,3cos𝜃log||𝛼_𝑖||+sin𝜃log||𝛽_𝑖||.(1.5)

Our main result, in the particular case 𝑑=3, is the following.

Theorem 1.3. Let 𝐴₁,𝐴₂∈SL(3,ℤ) be commuting independent hyperbolic matrices. The growth rates of the fixed points 𝜆=limsup_{‖‖𝑛1,𝑛2‖‖2→+∞}1‖‖𝑛₁,𝑛₂‖‖₂log𝑁𝑛₁,𝑛₂,𝜆_{_}=liminf_{‖‖𝑛1,𝑛2‖‖2→+∞}1‖‖𝑛₁,𝑛₂‖‖₂log𝑁𝑛₁,𝑛₂>0(1.6) satisfy 0<𝜆_{_}<𝜆<+∞.

Related problems have been studied for ℤ^𝑑-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 quantity 𝜆_{_} 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 values 𝜃 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 and 𝐶^∞ 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 of ℤ²-subshifts of finite type.

2. Examples

Let us consider some examples that illustrate Theorem 1.3.

Example 2.1. Consider the commuting matrices 𝐴₁,𝐴₂∈SL(3,ℤ) given by 𝐴₁=⎛⎜⎜⎜⎜⎝1−10−12−10−12⎞⎟⎟⎟⎟⎠,𝐴₂=⎛⎜⎜⎜⎜⎝20−1011−112⎞⎟⎟⎟⎟⎠.(2.1) The number of fixed points 𝑁(𝑛₁,𝑛₂) for |𝑛₁|,|𝑛₂|≤4 is presented in Table 1.

Table 1 

The number of fixed points 𝑁(𝑛₁,𝑛₂) for |𝑛₁|,|𝑛₂|≤4. The columns correspond to 𝑛₁ and the rows correspond to 𝑛₂.

The eigenvalues of 𝐴₁ 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 𝑒₁=⎛⎜⎜⎜⎜⎝−0.327985…0.736976…−0.591009…⎞⎟⎟⎟⎟⎠,𝑒₂=⎛⎜⎜⎜⎜⎝−0.591009…0.327985…0.736976…⎞⎟⎟⎟⎟⎠,𝑒₃=⎛⎜⎜⎜⎜⎝0.736976…0.591009…0.327985…⎞⎟⎟⎟⎟⎠.(2.2) Using these eigenvalues we can now plot the function 𝜃↦{max_𝑖=1,2,3{cos𝜃log|𝛼_𝑖|+sin𝜃log|𝛽_𝑖|} (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 that 𝜆_{_}=0.60501… (occurring at 𝜃_{_}=4.07742…) and 𝜆=2.00219… (occurring at 𝜃=5.34124…).

Example 2.2 (cf. [4]). We can let 𝐴₁=⎛⎜⎜⎜⎜⎝0100011−115⎞⎟⎟⎟⎟⎠,𝐴₂=⎛⎜⎜⎜⎜⎝2−1002−1−111−5⎞⎟⎟⎟⎟⎠.(2.3)
The number of fixed points 𝑁(𝑛₁,𝑛₂) for |𝑛₁|,|𝑛₂|≤4 is presented in Table 2.

Table 2 

The number of fixed points 𝑁(𝑛₁,𝑛₂) for |𝑛₁|,|𝑛₂|≤4. The columns correspond to 𝑛₁ and the rows correspond to 𝑛₂.

The eigenvalues for 𝐴 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 ⎛⎜⎜⎜⎜⎝−0.0440649−0.207514−0.977239⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎝−0.995305−0.0963337−0.00932395⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎝0.1857540.4075320.894099⎞⎟⎟⎟⎟⎠.(2.4)

In this case, we can compute 𝜆_{_}=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 Theorem 1.3

We begin by fixing our notation. Let 𝐴₁,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., 𝐴^𝑛1₁𝐴^𝑛2₂=𝐼 implies (𝑛₁,𝑛₂)=(0,0)).

We next recall the following standard results.

Lemma 3.1. Under the above hypotheses, (1)the eigenvalues 𝛼₁,𝛼₂,𝛼₃ of 𝐴₁ are real, and the eigenvalues 𝛽₁,𝛽₂,𝛽₃ of 𝐴₂ are real; (2)each of the common eigenvectors 𝑒₁,𝑒₂,𝑒₃ for 𝐴₁ and 𝐴₂ has irrational slope (i.e., each ℝv_𝑖+ℤ³ is dense in 𝕋³); (3)each of the real numbers log|𝛼_𝑖|/log|𝛽_𝑖|, 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 of ℤ²-actions [4].
For the second part, we can restrict to the case 𝑒₁, 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 𝐴^𝑛₁𝐴^𝑚₂ either has |𝛼^𝑛₁𝛽^𝑚₁|>1>|𝛼^𝑛₂𝛽^𝑚₂|,|𝛼^𝑛₃𝛽^𝑚₃| or |𝛼^𝑛₁𝛽^𝑚₁|<1<|𝛼^𝑛₂𝛽^𝑚₂|,|𝛼^𝑛₃𝛽^𝑚₃|. In particular, 𝑇=𝒜(𝑛₁,𝑛₂,⋅) corresponds to a linear hyperbolic toral automorphism 𝑇 for which ℝ𝑣_𝑖+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 that log𝛼_𝑖/log𝛽_𝑖 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 𝐴^𝑝₁𝐴^𝑞₂=𝐼. This contradicts the nondegeneracy condition, completing the proof.

In particular, we see from parts 2 and 3 of Lemma 3.1 that for all (𝑛₁,𝑛₂)∈ℤ²−(0,0) we have that 𝐴^𝑛1₁𝐴^𝑛2₂ has no eigenvalues of modulus 1.

We recall the following standard result for the fixed points of the single transformation 𝒜(𝑛₁,𝑛₂,⋅)∶𝕋³→𝕋³.

Lemma 3.2. For each (𝑛₁,𝑛₂)∈ℤ²−{(0,0)}, we can write 𝑁𝑛₁,𝑛₂=||det𝐼−𝐴^𝑛1₁𝐴^𝑛2₂||.(3.1)

Proof. This is a standard result, which can also be easily deduced from the Lefschetz fixed point theorem.

Lemma 3.2 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 𝑁𝑛₁,𝑛₂=||1−𝛼^𝑛1₁𝛽^𝑛2₁+𝛼^𝑛1₂𝛽^𝑛2₂+𝛼^𝑛1₂𝛽^𝑛2₂+𝛼^−𝑛1₁𝛽^−𝑛1₁+𝛼^−𝑛2₂𝛽^−𝑛2₂+𝛼^−𝑛3₃𝛽^−𝑛3₃−1||.(3.2)

Proof. The matrix 𝐴^𝑛1₁𝐴^𝑛2₂ has eigenvalues 𝛼^𝑛1₁𝛽^𝑛2₁, 𝛼^𝑛1₂𝛽^𝑛2₂, and 𝛼^𝑛1₃𝛽^𝑛2₃. Multiplying out this expression for 𝑁(𝑛₁,𝑛₂) gives 𝑁𝑛₁,𝑛₂=||det𝐼−𝐴^𝑛1₁𝐴^𝑛2₂||=||1−𝛼^𝑛1₁𝛽^𝑛2₁1−𝛼^𝑛1₂𝛽^𝑛2₂1−𝛼^𝑛1₃𝛽^𝑛2₃||=||1−𝛼^𝑛1₁𝛽^𝑛2₁+𝛼^𝑛1₂𝛽^𝑛2₂+𝛼^𝑛1₂𝛽^𝑛2₂+𝛼₁𝛼₂^𝑛1𝛽₁𝛽₂^𝑛2+𝛼₁𝛼₃^𝑛1𝛽₁𝛽₃^𝑛2+𝛼₂𝛼₃^𝑛1𝛽₂𝛽₃^𝑛2−1||=||1−𝛼^𝑛1₁𝛽^𝑛2₁+𝛼^𝑛1₂𝛽^𝑛2₂+𝛼^𝑛1₂𝛽^𝑛2₂+𝛼^−𝑛1₁𝛽^−𝑛2₁+𝛼^−𝑛1₂𝛽^−𝑛2₂+𝛼^−𝑛1₃𝛽^−𝑛2₃−1||,(3.3) 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 of 𝑁(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 𝑣₁=⎛⎜⎜⎝log||𝛼₁||log||𝛽₁||⎞⎟⎟⎠,𝑣₂=⎛⎜⎜⎝log||𝛼₂||log||𝛽₂||⎞⎟⎟⎠,𝑣₃=⎛⎜⎜⎝log||𝛼₃||log||𝛽₃||⎞⎟⎟⎠.(3.4) Each of these has irrational slope, by the final part of Lemma 3.1.

Lemma 3.4. All of the vectors 𝑣₁, 𝑣₂, and 𝑣₃ are nonzero and satisfy 𝑣₁+𝑣₂+𝑣₂=0.

Proof. For the first part, we need only observe that if 𝑣_𝑖=0, say, then this would require |𝛼_𝑖|=|𝛽_𝑖|=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 since 𝛼₁𝛼₂𝛼₃=det𝐴₁=1 and 𝛽₁𝛽₂𝛽₃=det𝐴₂=1 we immediately see that 𝑣₁+𝑣₂+𝑣₂=0.

We now parameterize the unit vectors in ℝ² by 𝑤_𝜃=⎛⎜⎜⎝cos𝜃sin𝜃⎞⎟⎟⎠,for0≤𝜃<2𝜋.(3.5) We can then write that ⟨𝑣_𝑖,𝑤_𝜃⟩=cos𝜃log𝛼_𝑖+sin𝜃log𝛽_𝑖,for𝑖=1,2,3.(3.6) In particular, if we write (𝑛₁,𝑛₂)=(𝑅cos𝜃,𝑅sin𝜃), say, where 𝑅=‖(𝑛₁,𝑛₂)‖₂, then we can write ||𝛼^𝑛1_𝑖𝛽^𝑛2_𝑖||=exp𝑅cos𝜃log||𝛼_𝑖||+sin𝜃log||𝛽_𝑖||.(3.7)

To prove Theorem 1.3, 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 ⟨𝑣_𝑖,𝑣_𝜃⟩>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 of 𝑤𝜃 onto one of the vectors 𝑣1, 𝑣2, 𝑣3 must have a strictly positive component.

Assume for a contradiction that the vectors 𝑣₁, 𝑣₂, and 𝑣₃ are collinear. Then we can choose 𝛿≠0 such that 𝛿=log||𝛼₁||log||𝛽₁||=log||𝛼₂||log||𝛽₂||=log||𝛼₃||log||𝛽₃||.(3.8) 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 𝑛_𝑘,𝑚_𝑘∈ℤ such that 𝐴^𝑛𝑘,𝐵^𝑚𝑘→𝐼 as 𝑘→+∞, but with 𝐴^𝑛𝑘,𝐵^𝑚𝑘≠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 𝐴^𝑝𝐵^𝑞=𝐼, which contradicts the nondegeneracy hypothesis.

4. Generalizations to ℤ^𝑘-Actions

We will consider the more general setting of higher-dimensional actions. The basic results are similar to the case of Theorem 1.3.

Hypothesis 1. Let 2≤𝑘≤𝑑−1.(1)We shall assume that 𝐴₁,…,𝐴_𝑘∈SL(𝑑,ℤ) are commuting matrices, that is, 𝐴_𝑖𝐴_𝑗=𝐴_𝑗𝐴_𝑖 for 1≤𝑖,𝑗≤𝑘.(2)We shall assume that each matrix 𝐴^𝑛1₁⋯𝐴^𝑛𝑘_𝑘, (𝑛₁,…,𝑛_𝑘)∈ℤ^𝑘−(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 exist 𝑛₁,…,𝑛_𝑘∈ℤ such that 𝐴^𝑛1₁𝐴^𝑛2₂⋯𝐴^𝑛𝑘_𝑘=𝐼then𝑛₁=⋯=𝑛_𝑘=0.(4.1)(4)We shall assume that the action is irreducible, that is, no 𝒜(𝑛₁,…,𝑛_𝑘)∶𝕋^𝑑→𝕋^𝑑 preserves a proper invariant toral subgroup of 𝕋^𝑑. (5)We shall additionally assume, mainly for convenience, that the matrices are semisimple (i.e., they diagonalize over the complex numbers) and 𝐴_𝑖 has complex eigenvalues 𝛼^(𝑖)₁,…,𝛼^(𝑖)_𝑑 for 𝑖=1,…,𝑘.

The special case that bears closest comparison with the special case of 𝑘=2 and 𝑑=3 is when 𝑘=𝑑−1. In particular, in this case 𝐴_𝑖 has real eigenvalues 𝛼^(𝑖)₁,…,𝛼^(𝑖)_𝑑 for 𝑖=1,…,𝑘.

We now generalize two definitions from the first section.

Definition 4.1. Let 𝒜∶ℤ^𝑘×𝕋^𝑑→𝕋^𝑑 be the action given by 𝒜(𝑛₁,…,𝑛_𝑑,𝑥)=𝐴^𝑛1₁⋯𝐴^𝑛𝑘_𝑘𝑥+ℤ^𝑑, then we denote 𝑁𝑛₁,…,𝑛_𝑘=Card𝑥∈𝕋^𝑑∶𝒜𝑛₁,…𝑛_𝑘,𝑥=𝑥.(4.2)

Definition 4.2. We can define 𝜆_{_}=inf_{‖𝑣‖2=1}sup_𝑤⟨𝑣,𝑤⟩,𝜆=sup_{‖𝑣‖2=1}sup_𝑤⟨𝑣,𝑤⟩,(4.3) where the supreme ranges over all unit vectors 𝑣 in ℝ^𝑘.

The natural generalization of Theorem 1.3 is the following.

Theorem 4.3. The growth rates of the number of fixed points 𝜆∶=limsup_{‖‖𝑛1,…,𝑛𝑘‖‖2→+∞}1‖‖𝑛₁,…,𝑛_𝑘‖‖₂log𝑁𝑛₁,…,𝑛_𝑘},𝜆_{_}∶=liminf_{‖‖𝑛1,…,𝑛𝑘‖‖2→+∞}1‖‖𝑛₁,…,𝑛_𝑘‖‖₂log𝑁𝑛₁,…,𝑛_𝑘}>0(4.4) satisfy 0<𝜆_{_}<𝜆<+∞.

To begin the proof, we need the following standard generalization of Lemma 3.2.

Lemma 4.4. For each (𝑛₁,…,𝑛_𝑘)∈ℤ²−{(0,…,0)}, we can write 𝑁𝑛₁,…,𝑛_𝑘=||det𝐼−𝐴^𝑛1₁⋯𝐴^𝑛𝑘_𝑘||.(4.5)

Proof. This is again a standard application of the Lefschetz formula.

In particular, we can use Lemma 4.4 to write 𝑁𝑛₁,…,𝑛_𝑘=||det𝐼−𝐴^𝑛1₁⋯𝐴^𝑛𝑘_𝑘||=^𝑑_𝑗=1|||||1−^𝑑−1_𝑖=1𝛼^(𝑖)_𝑗^𝑛𝑖|||||.(4.6) It is convenient to use the parameterization (𝑛₁,…,𝑛_𝑘)=(𝑝₁𝑅,…,𝑝_𝑘𝑅), where (1)0≤𝑝₁,…,𝑝_𝑘≤1 with 𝑝²₁+⋯+𝑝²_k=1; (2)𝑅=‖(𝑛₁,…,𝑛_𝑘)‖₂.

We can now introduce the notation 𝑣_𝑝=(𝑝₁,…,𝑝_𝑘) and 𝑣_𝑗=(log|𝛼⁽¹⁾_𝑗|,…,log|𝛼^(𝑘)_𝑗|), for 𝑗=1,…,𝑘. We can now easily see from (4.6) that for any 𝛿>0, there exists 𝑅₀=𝑅₀(𝛿) such that 𝑁𝑛₁,…,𝑛_𝑘≥_{𝑗∶𝑣}_{𝑝,𝑣𝑗>0}exp𝑅𝑣_𝑝,𝑣_𝑗−1_{𝑗∶𝑣}_{𝑝,𝑣𝑗<0}1−exp𝑅𝑣_𝑝,𝑣_𝑗≥(1−𝛿)_{𝑗∶𝑣}_{𝑝,𝑣𝑗≥0}exp𝑅𝑣_𝑝,𝑣_𝑗≥(1−𝛿)exp⎛⎜⎜⎝𝑅𝑣_𝑝,⎛⎜⎜⎝_{𝑗∶𝑣}_{𝑝,𝑣𝑗≥0}𝑣_𝑗⎞⎟⎟⎠⎞⎟⎟⎠,(4.7) for 𝑅≥𝑅₀. In particular, we see that 𝑁𝑛₁,…,𝑛_𝑘≥(1−𝛿)exp𝜆_{_}‖‖𝑛₁,…,𝑛_𝑘‖‖₂,(4.8) where 𝜆_{_}=inf_𝑝⎧⎪⎨⎪⎩𝑣_𝑝,⎛⎜⎜⎝_{𝑗∶𝑣}_{𝑝,𝑣𝑗≥0}𝑣_𝑗⎞⎟⎟⎠⎫⎪⎬⎪⎭.(4.9) Similarly, we see that for 𝑅≥𝑅₀, 𝑁𝑛₁,…,𝑛_𝑘≤(1+𝛿)exp𝜆‖‖𝑛₁,…,𝑛_𝑘‖‖₂,(4.10) where 𝜆=sup_𝑝⎧⎪⎨⎪⎩𝑣_𝑝,⎛⎜⎜⎝_{𝑗∶𝑣}_{𝑝,𝑣𝑗≥0}𝑣_𝑗⎞⎟⎟⎠⎫⎪⎬⎪⎭.(4.11)

To see that 𝜆_{_}>0, we need to know that 𝑣₁,…,𝑣_𝑘 are not confined to a codimension one hyperplane in ℝ^𝑘 orthogonal to some 𝑣_𝑝. Assume for a contradiction that there is a unit vector 𝑣_𝑝 such that ⟨𝑣_𝑝,𝑣_𝑖⟩=0 for 𝑖=1,…,𝑘. Let 𝑣_𝑝=(𝑣⁽¹⁾_𝑝,…,𝑣^(𝑘)_𝑝), then by Dirichlet's theorem of simultaneous diophantine approximation, for any 𝜖>0, we choose 1≤𝑞≤([1/𝜖]+1)^𝑘 and (𝑛₁,…,𝑛_𝑘)∈ℤ^𝑘 with ‖‖𝑛₁,…,𝑛_𝑘−𝑞𝑣_𝑝‖‖_∞≤𝜖.(4.12) In particular, the eigenvalues (𝛼¹_𝑗)^𝑛1⋯(𝛼^𝑘_𝑗)^𝑛𝑘, 𝑗=1,…,𝑘, for the matrix 𝐴^𝑛1₁⋯𝐴^𝑛𝑘_𝑘∈SL(𝑑,ℝ) satisfy |||log|||𝛼⁽¹⁾_𝑗^𝑛1⋯𝛼^(𝑘)_𝑗^𝑛𝑘||||||=|||||^𝑘_𝑙=1𝑛_𝑙log|||𝛼^(𝑙)_𝑗||||||||≤𝑘𝜖+𝑞||||^𝑘_𝑙∑₌₁𝑣_𝑙log|||𝛼^(𝑙)_𝑗||||||_{=||𝑣𝑝,𝑣𝑖||=0}|.(4.13) In particular, the algebraic integers, and its conjugates, occurring as zeros of the characteristic polynomial det(𝑧𝐼−𝐴^𝑛1₁⋯𝐴^𝑛𝑘_𝑘)=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 integer 𝛼 whose conjugate roots 𝛼=𝛼₁,…,𝛼_𝑑 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 𝑃_𝑛(𝑥)∶=^𝑑_𝑖=1𝑥−𝛼^𝑛_𝑖=𝑥^𝑑+𝑎^(𝑛)_𝑑−1𝑥^𝑑−1+⋯+𝑎^(𝑛)_𝑘𝑥^𝑘+⋯+𝑎^(𝑛)₁𝑥+𝑎^(𝑛)₀.(4.14) In particular, since |||𝑎^(𝑛)_𝑘|||=|||||_𝑖1<_{⋯<𝑖𝑑−𝑘}𝛼^(𝑛)_𝑖1⋯𝛼^(𝑛)_{𝑖𝑑−𝑘}|||||≤𝐾∶=𝑑!,(4.15) we see that {𝑃_𝑛(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 {𝛼^𝑛∶𝑛≥0} shows that there exists 0≤𝑝<𝑞≤𝐾+1 such that 𝛼^𝑝=𝛼^𝑞, and thus 𝛼^𝑞−𝑝=1.

We can also prove the following variant.

Lemma 4.6. Given 𝑑≥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,…,𝛼_𝑑 cannot all be contained in the annulus 𝐴(𝜖)={𝑧∈ℂ∶1−𝜖≤|𝑧|≤1+𝜖}.(4.16)

Proof. Since the proof is elementary, we include it for convenience. Assume for a contradiction that for some 𝑑≥2 we can find an infinite sequence of monomials 𝑃_𝑛(𝑥)=𝑥^𝑑+𝑎^(𝑛)_𝑑−1𝑥^𝑑−1+⋯+𝑎^(𝑛)_𝑘𝑥^𝑘+⋯+𝑎^(𝑛)₁𝑥+𝑎^(𝑛)₀∈ℤ[𝑥]],for𝑛≥2,(4.17) whose roots 𝛼^(𝑛)₁,…,𝛼^(𝑛)_𝑑∈𝐴(1/𝑛) do not lie on the unit circle. In particular, since 𝑃_𝑛(𝑥)=∏^𝑑_𝑖=1(𝑥−𝛼^(𝑛)_𝑖), we see that |||𝑎^(𝑛)_𝑘|||=|||||_𝑖1<_{⋯<𝑖𝑑−𝑘}𝛼^(𝑛)_𝑖1⋯𝛼^(𝑛)_{𝑖𝑑−𝑘}|||||≤𝐾∶=1+1𝑛^𝑑!.(4.18) Since for each 𝑘, we have 𝑎^(𝑛)_𝑘∈ℤ∩[−𝐾,𝐾], for all 𝑛≥1, we can use the pigeonhole principle to choose an infinite subsequence with 𝑃(𝑥)∶=𝑃_𝑛1(𝑥)=𝑃_𝑛2(𝑥)=𝑃_𝑛3(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 if 𝛼 is not a root of unity, then |𝛼|≥1+1/4^2+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 on 𝕋⁶ defined by the matrices 𝐴₁=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝010000001000000100000010000001−125352⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝐴₂=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0−6−6−3−62−24407−22−6−6−2−103−389313−44−11−12−3−175−51414322−7⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(4.19) The number of fixed points 𝑁(𝑛₁,𝑛₂) for |𝑛₁|,|𝑛₂|≤4 is presented in Table 3.

Table 3 

The number of fixed points 𝑁(𝑛₁,𝑛₂) for |𝑛₁|,|𝑛₂|≤4. The columns correspond to 𝑛₁ and the rows correspond to 𝑛₂.

The matrix 𝐴₁ has eigenvalues 𝛼₁=3.68631…,𝛼₂=−1.32361…,𝛼₃=0.0607659…+0.998152𝑖…,𝛼₄=0.0607659…−0.998152𝑖…,𝛼₅=−0.75551…,𝛼₆=0.271274…,(4.20) and the matrix 𝐴₂ has corresponding eigenvalues 𝛽₁=−0.463258…,𝛽₂=−22.1542…,𝛽₃=0.910592…−0.413307𝑖…,𝛽₄=0.910592…+0.413307𝑖…,𝛽₅=−0.0451382…,𝛽₆=−2.15862…(4.21)

In this example, we see that 𝜆_{_}=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 𝜆𝜃₁,𝜃₂=sup_{𝜃1≤𝜃≤𝜃2}max_𝑖=1,2,3cos𝜃log||𝛼_𝑖||+sin𝜃log||𝛽_𝑖||,𝜆_{_}𝜃₁,𝜃₂=inf_{𝜃1≤𝜃≤𝜃2}max_𝑖=1,2,3cos𝜃log||𝛼_𝑖||+sin𝜃log||𝛽_𝑖||.(5.1)

We then have the following natural refinement of Theorem 1.3.

Theorem 5.2 (Sector Theorem). Let 𝐴₁,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)𝜆𝜃₁,𝜃₂∶=limsup_{‖‖𝑛1,𝑛2‖‖2→+∞,𝑛1,𝑛2∈𝒮𝜃1,𝜃2}1‖‖𝑛₁,𝑛₂‖‖₂log𝑁𝑛₁,𝑛₂,𝜆_{_}𝜃₁,𝜃₂∶=liminf_{‖‖𝑛1,𝑛2‖‖2→+∞,𝑛1,𝑛2∈𝒮𝜃1,𝜃2}1‖‖𝑛₁,𝑛₂‖‖₂log𝑁𝑛₁,𝑛₂(5.2) satisfy 0<𝜆_{_}(𝜃₁,𝜃₂)<𝜆(𝜃₁,𝜃₂)<+∞.

Proof. The proof follows easily by modifying the proof of Theorem 1.3. 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 𝑁𝑛₁,𝑛₂=||det𝐼−𝐴^𝑛1₁𝐴^𝑛2₂||=||1−𝛼^𝑛1₁𝛽^𝑛2₁+𝛼^𝑛1₂𝛽^𝑛2₂+𝛼^𝑛1₂𝛽^𝑛2₂+𝛼^−𝑛1₁𝛽^−𝑛1₁+𝛼^−𝑛2₂𝛽^−𝑛2₂+𝛼^−𝑛3₃𝛽^−𝑛3₃−1||.(5.3) We can again consider the vectors 𝑣₁,𝑣₂,𝑣₃, but this time we only need to consider unit vectors 𝑣_𝜃 with 𝜃₁≤𝜃≤𝜃₂. We can again write that ⟨𝑣_𝑖,𝑤_𝜃⟩=cos𝜃log𝛼_𝑖+sin𝜃log𝛽_𝑖 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 ||𝛼^𝑛1_𝑖𝛽^𝑛2_𝑖||=exp𝑅cos𝜃log||𝛼_𝑖||+sin𝜃log||𝛽_𝑖||.(5.4) 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 𝜆𝜃₁,𝜃₂=sup_{𝜃1≤𝜃≤𝜃2}max_𝑖=1,2,3⟨𝑣_𝑖,𝑤_𝜃⟩≥inf_{𝜃1≤𝜃≤𝜃2}max_𝑖=1,2,3⟨𝑣_𝑖,𝑤_𝜃⟩=𝜆_{_}𝜃₁,𝜃₂,(5.5) as required.

Definition 5.3. Let us denote 𝜆(𝜃)∶=max_𝑖=1,2,3cos𝜃log||𝛼_𝑖||+sin𝜃log||𝛽_𝑖||.(5.6)

We then have the following corollary.

Corollary 5.4 (Directional growth). Let 𝐴₁,𝐴₂∈SL(3,ℤ) be commuting independent hyperbolic matrices. Let 0≤𝜃<2𝜋. The following limits exist and agree: 𝜆(𝜃)∶=lim_𝜖→0limsup_{‖‖𝑛1,𝑛2‖‖2→+∞,𝑛1,𝑛2∈𝒮𝜃−𝜖,𝜃2+𝜖}1‖‖𝑛₁,𝑛₂‖‖₂log𝑁𝑛₁,𝑛₂,𝜆_{_}(𝜃)∶=lim_𝜖→0liminf_{‖‖𝑛1,𝑛2‖‖2→+∞,𝑛1,𝑛2∈𝒮(𝜃−𝜖,𝜃+𝜖)}1‖‖𝑛₁,𝑛₂‖‖₂log𝑁𝑛₁,𝑛₂,(5.7) and 𝜆(𝜃)=𝜆_{_}(𝜃)=𝜆(𝜃).

Proof. This follows immediately from Theorem 5.2 and continuity of 𝜆(𝜃).

Remark 5.5. We have that for each fixed choice (𝑛₁,𝑛₂)∈𝒮(𝜃₁,𝜃₂) that ℎ𝒜𝑛₁,𝑛₂,⋅=lim_𝑘→+∞1𝑘logCard𝑥∶𝐴𝑘𝑛₁,𝑘𝑛₂𝑥=𝑥.(5.8) We see that for any 𝜖>0 we have that 𝜆_{_}𝜃₁,𝜃₂−𝜖≤ℎ𝒜𝑛₁,𝑛₂‖‖𝑛₁,𝑛₂‖‖₂≤𝜆𝜃₁,𝜃₂+𝜖(5.9) providing ‖(𝑛₁,𝑛₂)‖₂ is sufficiently large. In particular, by continuity we see that we have the limit lim_𝑅→+∞ℎ(𝒜([𝑅cos𝜃,𝑅sin𝜃]))𝑅=𝜆(𝜃).(5.10)