Table of Contents
ISRN Geometry
Volume 2012, Article ID 165808, 15 pages
Research Article

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

Department of Mathematics, The University of Warwick, Coventry CV4 7AL, UK

Received 1 April 2012; Accepted 3 May 2012

Academic Editors: J. Keesling and S. Troubetzkoy

Copyright © 2012 Mark Pollicott. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 𝑇𝕋𝑑𝕋𝑑. In particular, if 𝐴SL(𝑑,) is the associated hyperbolic matrix, then the number of fixed points for 𝑇𝑛 is given by Card{𝑇𝑛𝑥=𝑥}=||det(𝐴𝑛1)||.(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 𝑇𝑛 grows exponentially fast in 𝑛. We then recall that the growth rate of the number of periodic points for 𝑇 is given by (𝑇)=lim𝑘+1𝑘logCard𝑥𝑇𝑘𝑥=𝑥>0,(1.2) where (𝑇) is the topological entropy of 𝑇𝕋𝑑𝕋𝑑 (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 𝑑3, and for simplicity of exposition we shall initially assume that 𝑑=3. Let us, therefore, consider a pair of commuting hyperbolic matrices 𝐴1,𝐴2SL(3,) (i.e., 𝐴1𝐴2=𝐴2𝐴1 and neither matrix has an eigenvalue of modulus one) and associate the natural 2-action on the three dimensional torus 𝕋3=3/3 defined by 𝒜2×𝕋3𝕋3givenby𝒜𝑛1,𝑛2,𝑥=𝐴𝑛11𝐴𝑛22𝑥+3.(1.3) We will also ask for this action to be nondegenerate, that is, if 𝑛1,𝑛2 satisfy 𝐴𝑛11𝐴𝑛22=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)2.

Definition 1.1. We denote the number of fixed points of by 𝐴𝑛11𝐴𝑛22 on 𝕋3 by 𝑁𝑛1,𝑛2=Card𝑥𝕋3𝒜𝑛1,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,)𝕋3𝕋3 in terms of (𝑛1,𝑛2)2. 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=sup0𝜃2𝜋max𝑖=1,2,3cos𝜃log||𝛼𝑖||+sin𝜃log||𝛽𝑖||,𝜆_=inf0𝜃2𝜋max𝑖=1,2,3cos𝜃log||𝛼𝑖||+sin𝜃log||𝛽𝑖||.(1.5)

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

Theorem 1.3. Let 𝐴1,𝐴2SL(3,) be commuting independent hyperbolic matrices. The growth rates of the fixed points 𝜆=limsup𝑛1,𝑛22+1𝑛1,𝑛22log𝑁𝑛1,𝑛2,𝜆_=liminf𝑛1,𝑛22+1𝑛1,𝑛22log𝑁𝑛1,𝑛2>0(1.6) satisfy 0<𝜆_<𝜆<+.

Related problems have been studied for 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 2-subshifts of finite type.

2. Examples

Let us consider some examples that illustrate Theorem 1.3.

Example 2.1. Consider the commuting matrices 𝐴1,𝐴2SL(3,) given by 𝐴1=110121012,𝐴2=201011112.(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=0.3279850.7369760.591009,𝑒2=0.5910090.3279850.736976,𝑒3=0.7369760.5910090.327985.(2.2) 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 0𝜃<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=0100011115,𝐴2=2100211115.(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 0.04406490.2075140.977239,0.9953050.09633370.00932395,0.1857540.4075320.894099.(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 0𝜃<2𝜋.

3. Proof of Theorem 1.3

We begin by fixing our notation. Let 𝐴1,𝐴2SL(3,) 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., 𝐴𝑛11𝐴𝑛22=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 v𝑖+3 is dense in 𝕋3); (3)each of the real numbers log|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 of 2-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,𝑚3 such that matrix 𝐴𝑛1𝐴𝑚2 either has |𝛼𝑛1𝛽𝑚1|>1>|𝛼𝑛2𝛽𝑚2|,|𝛼𝑛3𝛽𝑚3| or |𝛼𝑛1𝛽𝑚1|<1<|𝛼𝑛2𝛽𝑚2|,|𝛼𝑛3𝛽𝑚3|. In particular, 𝑇=𝒜(𝑛1,𝑛2,) corresponds to a linear hyperbolic toral automorphism 𝑇 for which 𝑣𝑖+3 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 log𝛼𝑖/log𝛽𝑖 is a rational 𝑝/𝑞, say, then by comparing the actions of 𝐴1 and 𝐴2 on the dense 𝑒1+3 set, we then see from the second part that 𝐴𝑝1𝐴𝑞2=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)2(0,0) we have that 𝐴𝑛11𝐴𝑛22 has no eigenvalues of modulus 1.

We recall the following standard result for the fixed points of the single transformation 𝒜(𝑛1,𝑛2,)𝕋3𝕋3.

Lemma 3.2. For each (𝑛1,𝑛2)2{(0,0)}, we can write 𝑁𝑛1,𝑛2=||det𝐼𝐴𝑛11𝐴𝑛22||.(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)2{(0,0)}, we can write 𝑁𝑛1,𝑛2=||1𝛼𝑛11𝛽𝑛21+𝛼𝑛12𝛽𝑛22+𝛼𝑛12𝛽𝑛22+𝛼𝑛11𝛽𝑛11+𝛼𝑛22𝛽𝑛22+𝛼𝑛33𝛽𝑛331||.(3.2)

Proof. The matrix 𝐴𝑛11𝐴𝑛22 has eigenvalues 𝛼𝑛11𝛽𝑛21, 𝛼𝑛12𝛽𝑛22, and 𝛼𝑛13𝛽𝑛23. Multiplying out this expression for 𝑁(𝑛1,𝑛2) gives 𝑁𝑛1,𝑛2=||det𝐼𝐴𝑛11𝐴𝑛22||=||1𝛼𝑛11𝛽𝑛211𝛼𝑛12𝛽𝑛221𝛼𝑛13𝛽𝑛23||=||1𝛼𝑛11𝛽𝑛21+𝛼𝑛12𝛽𝑛22+𝛼𝑛12𝛽𝑛22+𝛼1𝛼2𝑛1𝛽1𝛽2𝑛2+𝛼1𝛼3𝑛1𝛽1𝛽3𝑛2+𝛼2𝛼3𝑛1𝛽2𝛽3𝑛21||=||1𝛼𝑛11𝛽𝑛21+𝛼𝑛12𝛽𝑛22+𝛼𝑛12𝛽𝑛22+𝛼𝑛11𝛽𝑛21+𝛼𝑛12𝛽𝑛22+𝛼𝑛13𝛽𝑛231||,(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 2 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 𝑣𝑖=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 since 𝛼1𝛼2𝛼3=det𝐴1=1 and 𝛽1𝛽2𝛽3=det𝐴2=1 we immediately see that 𝑣1+𝑣2+𝑣2=0.

We now parameterize the unit vectors in 2 by 𝑤𝜃=cos𝜃sin𝜃,for0𝜃<2𝜋.(3.5) We can then write that 𝑣𝑖,𝑤𝜃=cos𝜃log𝛼𝑖+sin𝜃log𝛽𝑖,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 ||𝛼𝑛1𝑖𝛽𝑛2𝑖||=exp𝑅cos𝜃log||𝛼𝑖||+sin𝜃log||𝛽𝑖||.(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 0𝜃<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 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 𝛿0 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} will be dense on the real line . However, since +𝑣1+3 is dense in 𝕋3 this means that we can choose 𝑛𝑘,𝑚𝑘 such that 𝐴𝑛𝑘,𝐵𝑚𝑘𝐼 as 𝑘+, but with 𝐴𝑛𝑘,𝐵𝑚𝑘𝐼. However, this is clearly false in the lattice SL(3,). On the other hand, if 𝛿=𝑝/𝑞 were a rational then by again considering the action on the dense set +𝑤1+3 we see that 𝐴𝑝𝐵𝑞=𝐼, which contradicts the nondegeneracy hypothesis.

4. Generalizations to 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 2𝑘𝑑1.(1)We shall assume that 𝐴1,,𝐴𝑘SL(𝑑,) are commuting matrices, that is, 𝐴𝑖𝐴𝑗=𝐴𝑗𝐴𝑖 for 1𝑖,𝑗𝑘.(2)We shall assume that each matrix 𝐴𝑛11𝐴𝑛𝑘𝑘, (𝑛1,,𝑛𝑘)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,,𝑛𝑘 such that 𝐴𝑛11𝐴𝑛22𝐴𝑛𝑘𝑘=𝐼then𝑛1==𝑛𝑘=0.(4.1)(4)We shall assume that the action is irreducible, that is, no 𝒜(𝑛1,,𝑛𝑘)𝕋𝑑𝕋𝑑 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) and 𝐴𝑖 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=𝑑1. In particular, in this case 𝐴𝑖 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 𝒜𝑘×𝕋𝑑𝕋𝑑 be the action given by 𝒜(𝑛1,,𝑛𝑑,𝑥)=𝐴𝑛11𝐴𝑛𝑘𝑘𝑥+𝑑, then we denote 𝑁𝑛1,,𝑛𝑘=Card𝑥𝕋𝑑𝒜𝑛1,𝑛𝑘,𝑥=𝑥.(4.2)

Definition 4.2. We can define 𝜆_=inf𝑣2=1sup𝑤𝑣,𝑤,𝜆=sup𝑣2=1sup𝑤𝑣,𝑤,(4.3) where the supreme ranges over all unit vectors 𝑣 in 𝑘.

The natural generalization of Theorem 1.3 is the following.

Theorem 4.3. The growth rates of the number of fixed points 𝜆=limsup𝑛1,,𝑛𝑘2+1𝑛1,,𝑛𝑘2log𝑁𝑛1,,𝑛𝑘},𝜆_=liminf𝑛1,,𝑛𝑘2+1𝑛1,,𝑛𝑘2log𝑁𝑛1,,𝑛𝑘}>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,,𝑛𝑘)2{(0,,0)}, we can write 𝑁𝑛1,,𝑛𝑘=||det𝐼𝐴𝑛11𝐴𝑛𝑘𝑘||.(4.5)

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

In particular, we can use Lemma 4.4 to write 𝑁𝑛1,,𝑛𝑘=||det𝐼𝐴𝑛11𝐴𝑛𝑘𝑘||=𝑑𝑗=1|||||1𝑑1𝑖=1𝛼(𝑖)𝑗𝑛𝑖|||||.(4.6) It is convenient to use the parameterization (𝑛1,,𝑛𝑘)=(𝑝1𝑅,,𝑝𝑘𝑅), where (1)0𝑝1,,𝑝𝑘1 with 𝑝21++𝑝2k=1; (2)𝑅=(𝑛1,,𝑛𝑘)2.

We can now introduce the notation 𝑣𝑝=(𝑝1,,𝑝𝑘) and 𝑣𝑗=(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 𝑁𝑛1,,𝑛𝑘𝑗𝑣𝑝,𝑣𝑗>0exp𝑅𝑣𝑝,𝑣𝑗1𝑗𝑣𝑝,𝑣𝑗<01exp𝑅𝑣𝑝,𝑣𝑗(1𝛿)𝑗𝑣𝑝,𝑣𝑗0exp𝑅𝑣𝑝,𝑣𝑗(1𝛿)exp𝑅𝑣𝑝,𝑗𝑣𝑝,𝑣𝑗0𝑣𝑗,(4.7) for 𝑅𝑅0. In particular, we see that 𝑁𝑛1,,𝑛𝑘(1𝛿)exp𝜆_𝑛1,,𝑛𝑘2,(4.8) where 𝜆_=inf𝑝𝑣𝑝,𝑗𝑣𝑝,𝑣𝑗0𝑣𝑗.(4.9) Similarly, we see that for 𝑅𝑅0, 𝑁𝑛1,,𝑛𝑘(1+𝛿)exp𝜆𝑛1,,𝑛𝑘2,(4.10) where 𝜆=sup𝑝𝑣𝑝,𝑗𝑣𝑝,𝑣𝑗0𝑣𝑗.(4.11)

To see that 𝜆_>0, we need to know that 𝑣1,,𝑣𝑘 are not confined to a codimension one hyperplane in 𝑘 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=(𝑣(1)𝑝,,𝑣(𝑘)𝑝), then by Dirichlet's theorem of simultaneous diophantine approximation, for any 𝜖>0, we choose 1𝑞([1/𝜖]+1)𝑘 and (𝑛1,,𝑛𝑘)𝑘 with 𝑛1,,𝑛𝑘𝑞𝑣𝑝𝜖.(4.12) In particular, the eigenvalues (𝛼1𝑗)𝑛1(𝛼𝑘𝑗)𝑛𝑘, 𝑗=1,,𝑘, for the matrix 𝐴𝑛11𝐴𝑛𝑘𝑘SL(𝑑,) satisfy |||log|||𝛼(1)𝑗𝑛1𝛼(𝑘)𝑗𝑛𝑘||||||=|||||𝑘𝑙=1𝑛𝑙log|||𝛼(𝑙)𝑗||||||||𝑘𝜖+𝑞||||𝑘𝑙=1𝑣𝑙log|||𝛼(𝑙)𝑗|||||||=||𝑣𝑝,𝑣𝑖||=0.(4.13) In particular, the algebraic integers, and its conjugates, occurring as zeros of the characteristic polynomial det(𝑧𝐼𝐴𝑛11𝐴𝑛𝑘𝑘)=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,,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 𝑃𝑛(𝑥)=𝑑𝑖=1𝑥𝛼𝑛𝑖=𝑥𝑑+𝑎(𝑛)𝑑1𝑥𝑑1++𝑎(𝑛)𝑘𝑥𝑘++𝑎(𝑛)1𝑥+𝑎(𝑛)0.(4.14) In particular, since |||𝑎(𝑛)𝑘|||=|||||𝑖1<<𝑖𝑑𝑘𝛼(𝑛)𝑖1𝛼(𝑛)𝑖𝑑𝑘|||||𝐾=𝑑!,(4.15) we see that {𝑃𝑛(𝑥)𝑛1} is a finite set as is the set of roots 𝛼 of these polynomials. Thus for any such root, the pigeonhole principle applied to {𝛼𝑛𝑛0} shows that there exists 0𝑝<𝑞𝐾+1 such that 𝛼𝑝=𝛼𝑞, and thus 𝛼𝑞𝑝=1.

We can also prove the following variant.

Lemma 4.6. Given 𝑑2, 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 𝐴(𝜖)={𝑧1𝜖|𝑧|1+𝜖}.(4.16)

Proof. Since the proof is elementary, we include it for convenience. Assume for a contradiction that for some 𝑑2 we can find an infinite sequence of monomials 𝑃𝑛(𝑥)=𝑥𝑑+𝑎(𝑛)𝑑1𝑥𝑑1++𝑎(𝑛)𝑘𝑥𝑘++𝑎(𝑛)1𝑥+𝑎(𝑛)0[𝑥]],for𝑛2,(4.17) whose roots 𝛼(𝑛)1,,𝛼(𝑛)𝑑𝐴(1/𝑛) do not lie on the unit circle. In particular, since 𝑃𝑛(𝑥)=𝑑𝑖=1(𝑥𝛼(𝑛)𝑖), we see that |||𝑎(𝑛)𝑘|||=|||||𝑖1<<𝑖𝑑𝑘𝛼(𝑛)𝑖1𝛼(𝑛)𝑖𝑑𝑘|||||𝐾=1+1𝑛𝑑!.(4.18) Since for each 𝑘, we have 𝑎(𝑛)𝑘[𝐾,𝐾], for all 𝑛1, 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 if 𝛼 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 on 𝕋6 defined by the matrices 𝐴1=010000001000000100000010000001125352,𝐴2=06636224407226621033893134411123175514143227.(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=0.06076590.998152𝑖,𝛼5=0.75551,𝛼6=0.271274,(4.20) and the matrix 𝐴2 has corresponding eigenvalues 𝛽1=0.463258,𝛽2=22.1542,𝛽3=0.9105920.413307𝑖,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 0𝜃<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)2𝑛2tan(𝜃1)𝑛1𝑛2tan(𝜃2)}, for 0𝜃1<𝜃22𝜋.

Definition 5.1. We can denote 𝜆𝜃1,𝜃2=sup𝜃1𝜃𝜃2max𝑖=1,2,3cos𝜃log||𝛼𝑖||+sin𝜃log||𝛽𝑖||,𝜆_𝜃1,𝜃2=inf𝜃1𝜃𝜃2max𝑖=1,2,3cos𝜃log||𝛼𝑖||+sin𝜃log||𝛽𝑖||.(5.1)

We then have the following natural refinement of Theorem 1.3.

Theorem 5.2 (Sector Theorem). Let 𝐴1,𝐴2SL(3,) be commuting independent hyperbolic matrices. Let 0𝜃1<𝜃22𝜋. The growth rates of the fixed points in the sector 𝒮(𝜃1,𝜃2)𝜆𝜃1,𝜃2=limsup𝑛1,𝑛22+,𝑛1,𝑛2𝒮𝜃1,𝜃21𝑛1,𝑛22log𝑁𝑛1,𝑛2,𝜆_𝜃1,𝜃2=liminf𝑛1,𝑛22+,𝑛1,𝑛2𝒮𝜃1,𝜃21𝑛1,𝑛22log𝑁𝑛1,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,)𝕋3𝕋3, this time restricting to (𝑛1,𝑛2)𝒮, can be written as 𝑁𝑛1,𝑛2=||det𝐼𝐴𝑛11𝐴𝑛22||=||1𝛼𝑛11𝛽𝑛21+𝛼𝑛12𝛽𝑛22+𝛼𝑛12𝛽𝑛22+𝛼𝑛11𝛽𝑛11+𝛼𝑛22𝛽𝑛22+𝛼𝑛33𝛽𝑛331||.(5.3) We can again consider the vectors 𝑣1,𝑣2,𝑣3, but this time we only need to consider unit vectors 𝑣𝜃 with 𝜃1𝜃𝜃2. 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 ||𝛼𝑛1𝑖𝛽𝑛2𝑖||=exp𝑅cos𝜃log||𝛼𝑖||+sin𝜃log||𝛽𝑖||.(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 𝜆𝜃1,𝜃2=sup𝜃1𝜃𝜃2max𝑖=1,2,3𝑣𝑖,𝑤𝜃inf𝜃1𝜃𝜃2max𝑖=1,2,3𝑣𝑖,𝑤𝜃=𝜆_𝜃1,𝜃2,(5.5) as required.

Definition 5.3. Let us denote 𝜆(𝜃)=max𝑖=1,2,3cos𝜃log||𝛼𝑖||+sin𝜃log||𝛽𝑖||.(5.6)

We then have the following corollary.

Corollary 5.4 (Directional growth). Let 𝐴1,𝐴2SL(3,) be commuting independent hyperbolic matrices. Let 0𝜃<2𝜋. The following limits exist and agree: 𝜆(𝜃)=lim𝜖0limsup𝑛1,𝑛22+,𝑛1,𝑛2𝒮𝜃𝜖,𝜃2+𝜖1𝑛1,𝑛22log𝑁𝑛1,𝑛2,𝜆_(𝜃)=lim𝜖0liminf𝑛1,𝑛22+,𝑛1,𝑛2𝒮(𝜃𝜖,𝜃+𝜖)1𝑛1,𝑛22log𝑁𝑛1,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 𝒜𝑛1,𝑛2,=lim𝑘+1𝑘logCard𝑥𝐴𝑘𝑛1,𝑘𝑛2𝑥=𝑥.(5.8) We see that for any 𝜖>0 we have that 𝜆_𝜃1,𝜃2𝜖𝒜𝑛1,𝑛2𝑛1,𝑛22𝜆𝜃1,𝜃2+𝜖(5.9) providing (𝑛1,𝑛2)2 is sufficiently large. In particular, by continuity we see that we have the limit lim𝑅+(𝒜([𝑅cos𝜃,𝑅sin𝜃]))𝑅=𝜆(𝜃).(5.10)


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