ISRN Geometry
Volume 2012, Article ID 165808, 15 pages
http://dx.doi.org/10.5402/2012/165808
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.

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 . In particular, if is the associated hyperbolic matrix, then the number of fixed points for is given by Since the hyperbolicity of is equivalent to the fact that the eigenvalues for do not lie on the unit circle, it is easy to see that the number of fixed points for grows exponentially fast in . We then recall that the growth rate of the number of periodic points for is given by where is the topological entropy of (or, equivalently, the sum of the logarithms of the eigenvalue of the matrix 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 , and for simplicity of exposition we shall initially assume that . Let us, therefore, consider a pair of commuting hyperbolic matrices (i.e., and neither matrix has an eigenvalue of modulus one) and associate the natural -action on the three dimensional torus defined by We will also ask for this action to be nondegenerate, that is, if satisfy , then this necessarily implies that . We say that and are independent.

We can now consider the growth of the number of fixed points for the action associated to any element .

Definition 1.1. We denote the number of fixed points of by on by

We want to give uniform estimates on the rate of growth of the number of fixed points for the actions in terms of . In particular, we want to give a lower bound on the growth of the fixed points in terms as . In the present context, we can assume without loss of generality that the eigenvalues of and the eigenvalues of are real.

Definition 1.2. We denote

Our main result, in the particular case , is the following.

Theorem 1.3. Let be commuting independent hyperbolic matrices. The growth rates of the fixed points satisfy .

Related problems have been studied for -actions in algebraic and symbolic examples by Miles and Ward . Interestingly, whereas their analysis relies on deep results in diophantine approximation, in the present context the required analysis is completely elementary.

The quantity is related to the supremum of the sum of the Lyapunov exponents for the action. In particular, the bound can then be deduced from (, Lemma 4.3 (a)).

Remark 1.4. The values and 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 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 -subshifts of finite type.

2. Examples

Let us consider some examples that illustrate Theorem 1.3.

Example 2.1. Consider the commuting matrices given by The number of fixed points for is presented in Table 1.

Table 1: The number of fixed points for . The columns correspond to and the rows correspond to .

The eigenvalues of are , , and , and the eigenvalues of are , , and (which happen to be a permutation of those for ). Corresponding to these eigenvalues are the common eigenvectors Using these eigenvalues we can now plot the function (cf. Figure 1) and then read off the values of and as the maximum and minimum values, respectively.

Figure 1: A plot of as a function of .

In this example, we see that (occurring at ) and (occurring at ).

Example 2.2 (cf. ). We can let
The number of fixed points for is presented in Table 2.

Table 2: The number of fixed points for . The columns correspond to and the rows correspond to .

The eigenvalues for are , , and , and the eigenvalues for are , , and . These correspond to the eigenvectors

In this case, we can compute (occurring at ) and (occurring at ) (cf. Figure 2).

Figure 2: A plot of as a function of .

3. Proof of Theorem 1.3

We begin by fixing our notation. Let 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., implies ).

We next recall the following standard results.

Lemma 3.1. Under the above hypotheses, (1)the eigenvalues of are real, and the eigenvalues of are real; (2)each of the common eigenvectors for and has irrational slope (i.e., each is dense in ); (3)each of the real numbers , , 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 -actions .
For the second part, we can restrict to the case , with the other cases being similar. It is easy to see that we can make an appropriate choice of such that matrix either has or . In particular, corresponds to a linear hyperbolic toral automorphism for which 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 is a rational , say, then by comparing the actions of and on the dense set, we then see from the second part that . This contradicts the nondegeneracy condition, completing the proof.

In particular, we see from parts 2 and 3 of Lemma 3.1 that for all we have that has no eigenvalues of modulus 1.

We recall the following standard result for the fixed points of the single transformation .

Lemma 3.2. For each , we can write

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 , we can write

Proof. The matrix has eigenvalues , , and . Multiplying out this expression for gives where we have used the identities and for the last line.

We want to use this lemma to estimate the growth of . 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 defined by Each of these has irrational slope, by the final part of Lemma 3.1.

Lemma 3.4. All of the vectors , , and are nonzero and satisfy .

Proof. For the first part, we need only observe that if , say, then this would require , 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 and we immediately see that .

We now parameterize the unit vectors in by We can then write that In particular, if we write , say, where , then we can write

To prove Theorem 1.3, it suffices to show that the vectors , , are not collinear. Since and the vectors 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 there is some such that (se Figure 3). For typical , 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 onto one of the vectors , , must have a strictly positive component.

Assume for a contradiction that the vectors , , and are collinear. Then we can choose such that First we observe that cannot be irrational since otherwise will be dense on the real line . However, since is dense in this means that we can choose such that as , but with . However, this is clearly false in the lattice . On the other hand, if were a rational then by again considering the action on the dense set we see that , which contradicts the nondegeneracy hypothesis.

4. Generalizations to ℤ𝑘-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 .(1)We shall assume that are commuting matrices, that is, for .(2)We shall assume that each matrix , 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 such that (4)We shall assume that the action is irreducible, that is, no preserves a proper invariant toral subgroup of . (5)We shall additionally assume, mainly for convenience, that the matrices are semisimple (i.e., they diagonalize over the complex numbers) and has complex eigenvalues for .

The special case that bears closest comparison with the special case of and is when . In particular, in this case has real eigenvalues for .

We now generalize two definitions from the first section.

Definition 4.1. Let be the action given by , then we denote

Definition 4.2. We can define where the supreme ranges over all unit vectors in .

The natural generalization of Theorem 1.3 is the following.

Theorem 4.3. The growth rates of the number of fixed points satisfy .

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

Lemma 4.4. For each , we can write

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

In particular, we can use Lemma 4.4 to write It is convenient to use the parameterization , where (1) with ; (2).

We can now introduce the notation and , for . We can now easily see from (4.6) that for any , there exists such that for . In particular, we see that where Similarly, we see that for , where

To see that , we need to know that are not confined to a codimension one hyperplane in orthogonal to some . Assume for a contradiction that there is a unit vector such that for . Let , then by Dirichlet's theorem of simultaneous diophantine approximation, for any , we choose and with In particular, the eigenvalues , , for the matrix satisfy In particular, the algebraic integers, and its conjugates, occurring as zeros of the characteristic polynomial 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, ). Any algebraic integer whose conjugate roots 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 In particular, since we see that is a finite set as is the set of roots of these polynomials. Thus for any such root, the pigeonhole principle applied to shows that there exists such that , and thus .

We can also prove the following variant.

Lemma 4.6. Given , there exists such that if is an algebraic number of degree, which is not an algebraic integer, then the conjugate values cannot all be contained in the annulus

Proof. Since the proof is elementary, we include it for convenience. Assume for a contradiction that for some we can find an infinite sequence of monomials whose roots do not lie on the unit circle. In particular, since , we see that Since for each , we have , for all , we can use the pigeonhole principle to choose an infinite subsequence with 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 is not a root of unity, then . (cf. ).

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 .

Example 4.9 (cf. ). We can consider the action on defined by the matrices The number of fixed points for is presented in Table 3.

Table 3: The number of fixed points for . The columns correspond to and the rows correspond to .

The matrix has eigenvalues and the matrix has corresponding eigenvalues

In this example, we see that (occurring at ) and (occurring at ) (cf. Figure 4).

Figure 4: A plot of as a function of .

5. A Sector Theorem and Directional Growth

A natural refinement is to estimate the number of fixed points for lying in a sector of the form , for .

Definition 5.1. We can denote

We then have the following natural refinement of Theorem 1.3.

Theorem 5.2 (Sector Theorem). Let be commuting independent hyperbolic matrices. Let . The growth rates of the fixed points in the sector satisfy .

Proof. The proof follows easily by modifying the proof of Theorem 1.3. Recall that the number of fixed points of the single transformation , this time restricting to , can be written as We can again consider the vectors , but this time we only need to consider unit vectors with . We can again write that for . In particular, if we write , say, where , then we have that We now want to estimate in terms of the largest expression of the form (5.4) where . In particular, modifying the proof of Theorem 1.3, we observe that as required.

Definition 5.3. Let us denote

We then have the following corollary.

Corollary 5.4 (Directional growth). Let be commuting independent hyperbolic matrices. Let . The following limits exist and agree: and .

Proof. This follows immediately from Theorem 5.2 and continuity of .

Remark 5.5. We have that for each fixed choice that We see that for any we have that providing is sufficiently large. In particular, by continuity we see that we have the limit

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