Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.


Introduction
Toral automorphisms or cat maps, by which we mean the action of matrices M ∈ GL(d, Z) on the d-torus T d , are a widely used and versatile class of dynamical systems, see [45,28] for some classic results in the context of ergodic theory. Of particular interest are the hyperbolic and quasi-hyperbolic ones, which are characterised by having no root of unity among their eigenvalues. All periodic orbits of such automorphisms lie on the rational (or finite) invariant lattices L n = {x ∈ T d | nx = 0 mod 1}, which are also known as the n-division points. One can encode the possible periods of a toral automorphism M on T d via the dynamical zeta function in a systematic way, which is always a rational function [8,18]. The literature on classifying periodic orbits of toral automorphisms when d = 2 is vast (compare [21,25,37] and references therein). An extension beyond d = 2 is difficult due to the fact that the conjugacy problem between integer matrices is then much harder because (unlike d = 2) no complete set of conjugacy invariants mod n is known. Therefore, our focus will also be on d = 2, with occasional extensions to higher dimensions.
The larger ring Mat(d, Z) of toral endomorphisms (which includes integer matrices without integer inverses) has received far less attention [2,14,8], particularly those in the complement of GL(d, Z). Note that the resulting dynamics induced by M ∈ Mat(d, Z) \ GL(d, Z) on a finite lattice L n may or may not be invertible. In the latter case, beyond periodic orbits, there exist eventually periodic orbits which possess points that lead into a periodic orbit. We call these points and the periodic point to which they attach the 'pretails' to the periodic orbit (see Eq. (15) for a formal definition). The action of M induces a directed graph on L n (e.g. see our three figures below). Alternatively, the pretails can be combined to form a rooted tree which is a characteristic attribute to any pair (M, L n ).
As well as their interest from a mathematical viewpoint, toral automorphisms also have been well-studied from a physics perspective, in particular as quantum cat maps (see [30,19,32] and references therein). Here, the action of the integer matrix on a rational lattice L n , for some n, is all-important as quantum cat maps and their perturbations are built from (classical) cat maps and their perturbations restricted to a particular rational lattice (called the Wigner lattice in this instance). There has been recent interest in dealing with so-called pseudo-symmetries of quantum cat maps that are manifestations of local symmetries of cat maps restricted to some rational lattice [30,19,32]. Although, in the context of quantisation, matrices from the group Sp(2d, Z) play the key role, we prefer to work with the larger group of unimodular integer matrices and consider the former as a special case.
The main aims of this paper are twofold: (i) to elucidate the orbit structure of toral endomorphisms on rational lattices, equivalently the periodic orbits together with the related pretail tree structure; (ii) to further characterise the nature of symmetries or (time) reversing symmetries of toral automorphisms, these being automorphisms of the torus (or of a rational lattice) that commute with the cat map, respectively conjugate it into its inverse.
We expand a little on our results, where we refer to the actual formulation below in the paper. The results are readable without the surrounding notational details.
With respect to aim (i), Section 3 characterises the splitting of L n into periodic and eventually periodic points under a toral endomorphism M . Every periodic point has a pretail graph isomorphic to that of the fixed point 0 (Corollary 1), which is trivial if and only if M is invertible on L n . In general, the pretail tree codes important information on the action of M . One question in this context is whether all maximal pretails have the same length, for which we give a partial answer via a sufficient condition on ker(M ) in Proposition 3. Given M , the lattice L n can be decomposed into into 2 invariant submodules, one of which captures the invertible part of M and the other the nilpotent part. This way, we are able to characterise the dynamics that is induced by M on L n in the case of n = p r , p prime, in Corollary 3 and Lemma 3.
Our contribution towards aim (ii) continues the investigations from [9,10,12]. The key quantity for integer matrices of dimension 2 is the mgcd (see Eq. 13 below), and one consequence of [12,Thm. 2] is that M ∈ SL(2, Z) is always conjugate to its inverse on L n , for each n ∈ N. The conjugating element -called a reversing symmetry or reversor -is an integer matrix that has an integer matrix inverse on L n , which typically depends on n. In this way, any SL(2, Z) matrix that fails to be conjugate to its inverse on the torus (e.g. M = ( 4 9 7 16 ) from [9,Ex. 2]) is still conjugate to its inverse on every rational lattice. In [12], we did not consider the nature of the reversor on the lattice. Theorem 1 of Section 4 establishes that it is an orientation-reversing involution, what is called an anticanonical (time-reversal) symmetry in the language of [30]. Section 4.2 uses normal forms of GL(2, F p ) to characterise the symmetries and possible reversing symmetries of such matrices; the underlying structure of the conjugacy classes of GL(2, F p ) is summarised in Table 4.2. The symmetry structure has some extensions to higher dimensions (Section 4.3) and to general modulus n (Section 4.4). Section 4.5 presents some results for the case when M ∈ GL(d, F p ) has a root in the same group.
The structure of the paper is as follows. Section 2 summarises some properties of integer matrices that we use later in the paper, with some reformulations or slight generalisations that we find useful. In particular, throughout the paper, we formulate the results for arbitrary dimension whenever it is possible without extra complications, though this is not our main focus. As described, Section 3 addresses aim (i) above, while Section 4 deals with aim (ii). In the Appendix, we briefly discuss two classic examples of toral automorphisms for d = 2 and some aspects of their dynamics.

Preliminaries and powers of integer matrices
The purpose of this section is to summarise important properties of and around integer matrices that are needed later on, with focus on those that are not standard textbook material. At the same time, we introduce our notation. For general background on integer matrices and their connections to algebraic number theory, we refer to the classic text by Taussky [43].
2.1. Lattices, rings and groups. The most important lattices on the torus T d = R d /Z d , which is a compact Abelian group, consist of the n-division points Clearly, the L n are invariant under toral endomorphisms (with the action of the representing matrices taken mod 1). It is sometimes easier to replace L n by the set L n := {(k 1 , . . . , k d ) | 0 ≤ k i < n}, with the equivalent action of M defined mod n. This also applies to various theoretical arguments involving modular arithmetic. Consequently, we use L n (with action of M mod 1) andL n (with action mod n) in parallel.
Our discussion will revolve around the residue class ring Z/nZ with n ∈ N, which is a principal ideal ring, but not a domain, unless n = p is a prime. In the latter case, Z/pZ = F p is the finite field with p elements, while the ring has zero divisors otherwise. For general n, the unit group is an Abelian group (under multiplication) of order φ(n), where φ is Euler's totient function from elementary number theory [26]. In general, it is not a cyclic group. The integer matrices mod n form the finite ring Mat(d, Z/nZ) of order n d 2 . The invertible elements in it form the group GL(d, is well-known from the standard literature [33,34]. Formula (2) follows from the corresponding one for n = p r via the Chinese remainder theorem, while the simpler prime power case is a consequence of the observation that each element of a non-singular matrix M over Z/p s Z can be covered (independently of all other matrix elements) by p elements in Z/p s+1 Z without affecting its non-singularity. Let us finally mention that SL(n, Z/nZ), the subgroup of matrices with determinant 1, is a normal subgroup (it is the kernel of det : GL(n, Z/nZ) −→ (Z/nZ) × ). The factor group is GL(n, Z/nZ)/ SL(n, Z/nZ) ≃ (Z/nZ) × and thus has order φ(n).

2.2.
Orbit counts and generating functions. The orbit statistics of the action of a matrix M ∈ Mat(d, Z) on the lattice L n is encapsulated in the polynomial m denotes the number of m-cycles of M on L n . Recall that, if a m and c m denote the fixed point and orbit count numbers of M (dropping the upper index for a moment), they are related by where µ(k) is the Möbius function from elementary number theory [26]. Despite the way it is written, Z n is a finite product and defines a polynomial of degree at most n d . Note that the degree of Z n can be smaller than n d (as the matrix M need not be invertible on L n ), but Z n (t) is always divisible by (1−t), because 0 is a fixed point of every M . The polynomials Z n are closely related [12,36] to the zeta function of toral endomorphisms, which can be calculated systematically; compare [8] and references therein. Dynamical zeta functions give access to the distribution and various asymptotic properties of periodic orbits [18,40], and also relate to topological questions; compare [23] for a systematic exposition of the latter aspect in a more general setting. Further aspects on the asymptotic distribution of orbit lengths on prime lattices can be found in [29].
2.3. Matrix order on lattices and plateau phenomenon. Assume that M is invertible on L n (hence also onL n ). Then, its order is given by Clearly, ord(M, 1) = 1 in this setting. When M is not invertible on L n , the definition results in ord(M, n) = 0; otherwise, ord(M, n) is the smallest m ∈ N with M m = ½ mod n.
Let M ∈ GL(d, Z) be arbitrary, but fixed. To determine ord(M, n) for all n ≥ 2, it suffices to do so for n an arbitrary prime power, since the Chinese remainder theorem [26] gives (7) ord(M, n) = lcm ord(M, p r 1 1 ), . . . , ord(M, p r ℓ ℓ ) when n = p r 1 1 · · · p r ℓ ℓ is the prime decomposition of n. It is clear that ord(M, p r )| ord(M, p r+1 ) for all r ∈ N, see also [14,Lemma 5.2].
Let us now assume that M ∈ Mat(d, Z) is not of finite order, meaning that M k = ½ for all k ∈ N, which excludes the finite order elements of GL(d, Z). If p is a prime, we then obtain the unique representation (8) M ord(M,p) = ½ + p s B with s ∈ N and an integer matrix B ≡ 0 mod p. Starting from this representation, an application of the binomial theorem for powers of it, in conjunction with the properties of the binomial coefficients mod p, gives the following well-known result. In what follows, we will refer to the structure described in Proposition 1 as the plateau phenomenon. Such a plateau can be absent (p odd with s = 1, or the first case for p = 2), it can be at the beginning (p odd with s ≥ 2), or it can occur after one step (p = 2 when t ≥ 2 exists as described), but it cannot occur later on.
Proposition 1 is a reformulation of [14, Thms. 5.3 and 5.4], which are originally stated for M ∈ GL(2, Z). As one can easily check, the proofs do not depend on the dimension. Similar versions or special cases were also given in [13] and [41] (with focus on SL(2, Z)-matrices), in [37] (for the order of algebraic integers), in [44] (for the Fibonacci sequence), in [15] (for linear quadratic recursions) and in [22] and [46] (for general linear recursions). Let us also mention that, based on the generalised Riemann hypothesis, Kurlberg has determined a lower bound on the order of unimodular matrices mod N for a density 1 subset of integers N in [31].

2.4.
Powers of integer matrices. Consider a matrix M ∈ Mat(d, Z) with d ≥ 2 and characteristic polynomial P M (z) = det(z½ − M ), which (following [46]) we write as so that c d = (−1) d+1 det(M ). Let us define a recursion by u 0 = u 1 = . . . = u d−2 = 0 and u d−1 = 1 together with This results in an integer sequence. Moreover, when c d = 0, we also define for m ≤ −1. In particular, since d ≥ 2, one always has u −1 = 1/c d and u −2 = −c d−1 /c 2 d , while the explicit form of u m with m < −2 depends on d. Note that the coefficients with negative index are rational numbers in general, unless |c d | = 1.
The Cayley-Hamilton theorem together with (9) can be used to write down an explicit expansion of powers of the matrix M in terms of M k with 0 ≤ k ≤ d − 1, where the coefficients satisfy γ The coefficients are explicitly given as (12) γ where m ≥ d and the second expression follows from the first by (9). Formulas (10) and (12) can be proved by induction from for all m ≥ 0 in this formulation. When det(M ) = 0, the representation (12) also holds for m < d, as follows from checking the cases 0 ≤ m < d together with a separate induction argument for m < 0. In particular, one then has which is again an integer matrix when |c d | = 1.  Let n ∈ N and assume gcd(n, D) = 1. This allows us to introduce κ(n) := period of (u m ) m≥0 mod n which is well-defined, as the sequence mod n is then indeed periodic without 'pretail'. Recall that (u m ) m≥0 mod n must be periodic from a certain index on, as a result of Dirichlet's pigeon hole principle. Since D is a unit in Z/nZ, the recursion (14) can be reversed, and (u m ) m≥0 mod n must thus be periodic, with κ(n) being the smallest positive integer k such that u k = 0 and u k+1 = 1 mod n.
One can now relate κ(n) and ord(M, n) as follows, which provides an efficient way to calculate ord(M, n).
In the remaining case, N n = 1, the matrix satisfies M ≡ α½ mod n with α ∈ (Z/nZ) × , so that ord(M, n) is the order of α modulo n.
Proof. If M = a b c d , the iteration formula (14) implies that M m ≡ ½ mod n if and only if so that also u m (a − d) ≡ 0 mod n. Consequently, n divides u m b, u m c and u m (a − d). This implies that u m is divisible by n gcd(n,b) , n gcd(n,c) and n gcd(n,a−d) , hence also by the least common multiple of these three numbers, which is the integer .
which is the period of the sequence (u m ) m≥0 modulo N n . Since κ(N n ) clearly divides κ(n), one finds ord(M, n)| κ(n).

Remark 1.
Instead of the characteristic polynomial P M , any other monic polynomial that annihilates M can be employed to derive a recursive sequence whose period is a multiple of the matrix order modulo n. For n = p a prime, the unique minimal polynomial Q M of M suggests itself to be chosen. For d = 2, Q M has smaller degree than P M precisely when M = α½, whence mgcd(M ) = 0 and Q M (z) = z − α. Consequently, ord(M, p) is then always equal to the order of α modulo p. ♦

Orbit pretail structure of toral endomorphisms
In this section, we look at the action of M ∈ Mat(d, Z) on a lattice L n , with special emphasis on the structure of general endomorphisms. When M is not invertible, this manifests itself in the existence of non-trivial 'pretails' to periodic orbits, with rather characteristic properties. More precisely, given a periodic point y of M , a finite set of iterates (or suborbit) Consider an arbitrary x ∈L n and its iteration under M . Since |L n | = n d is finite, Dirichlet's pigeon hole principle implies that this orbit must return to one of its points. Consequently, every orbit is a cycle or turns into one after finitely many steps, i.e. it is eventually periodic. By elementary arguments, one then finds the following result.   In general, the projective summands need not be free. As a simple example, let us consider L 6 with d = 1 and M = 2. Here, per(M ) = {0, 2, 4} covers the fixed point 0 and a 2-cycle, while ker(M ) = {0, 3}. Both are modules (and also principal ideals, hence generated by a single element) over Z/6Z, but do not have a basis, hence are not free. Nevertheless, one has Z/6Z = per(M ) ⊕ ker(M ). We will return to this question below.
3.2. The pretail tree. Consider the equation M ℓ x = y, with ℓ ∈ N, for some arbitrary, but fixed y inL n . In general, this equation need not have any solution x ∈L n . On the other hand, when there is a solution x ∈L n , the set of all solutions is precisely x + ker(M ℓ ), which has cardinality |ker(M ℓ )|. If y is a periodic point, the first case can never occur, as there is then at least one predecessor of y. Due to the linearity of M , the structure of the set of pretails of a periodic point y must be the same for all y ∈ per(M ) (note that there is precisely one predecessor of y in the periodic orbit, which might be y itself, while all points of the pretail except y are from the complement of the periodic orbit).
Consequently, we can study the pretail structure for y = 0. Let us thus combine all pretails of the fixed point 0 into a (directed) graph, called the pretail graph from now on; see [47] for general background on graph theory. A single pretail is called maximal when it is not contained in any longer one. By construction, there can be no cycle in the pretail graph, while y = 0 plays a special role. Viewing each maximal pretail of 0 as an 'ancestral line', we see that this approach defines a rooted tree with root 0. Note that an isomorphic tree also 'sits' at every periodic point y.
Corollary 1. Every periodic point of M onL n has a directed pretail graph that is isomorphic to that of the fixed point 0. Up to graph isomorphism, it thus suffices to analyse the latter. By reversing the direction, it is a rooted tree with root 0. This tree is trivial if and only if M is invertible onL n .
Two illustrative examples are shown in Figures 1 and 2. Each M defines a unique (rooted) pretail tree on a given lattice. If v i denotes the number of nodes (or vertices) of this tree with graph distance i from the root, we have v 0 = 1 and where v i = 0 for all i larger than the maximal pretail length. Also, one has for j ≥ 1, where the second equality follows from the submodule property. Recall that terminal nodes of a rooted tree (excluding the root in the trivial tree) are called leaves. With this definition, the total number of leaves onL n is |L n \ ML n |. For i ∈ N, define w i to be the number of nodes with graph distance i from the root that fail to be leaves, and complete this with w 0 = 0 for the trivial tree and w 0 = 1 otherwise. It is clear that this leads to via the number of solutions to M x = 0 and the special role of the root, and inductively to whenever w 0 = 1, with both relations being valid for all i ≥ 0.
Lemma 2. If M acts onL n , its uniquely defined pretail tree of the fixed point 0 has height m ≥ 0, and the following properties are equivalent. In particular, m is the integer from Fact 1.
Proof. By Corollary 1, the pretail tree is trivial (hence w 0 = m = 0) if and only if M is invertible onL n . Since all claims are clear for this case, let us now assume that M is not invertible onL n . All maximal pretails have the same length if and only if all leaves of the pretail tree of 0 have the same graph distance from the root 0. Clearly, the latter must be the height m of the tree. When M is not invertible onL n , the tree is not the trivial one, so m ≥ 1. The equivalence of (i) and (ii) is then clear, since both conditions characterise the fact that all leaves have distance m from the root.
The implication (ii) ⇒ (iii) can be seen as follows. The first claim is trivial for i = 0, as ker for 1 ≤ i < m, which (inductively) reduces to the first condition of (iii), while the second is clear from the meaning of m.
Conversely, the second condition of (iii) means w m+j = 0 for all j ≥ 0, while the first condition, together with Eqns. (18) and (19), successively gives v i = w i for all 0 ≤ i < m.
On the latticeL p r , when |ker(M )| = p, one can say more. Proof. By assumption, M is not invertible, and the last claim is obvious from Lemma 2 in conjunction with Eq. (16). We thus need to prove the formula for the cardinality of ker(M i ) for arbitrary i ≥ 0.
Since 0 is a fixed point of M , we clearly have v 0 = 1 and v 1 = p − 1, together with the inequality 0 ≤ w 1 ≤ p − 1. If w 1 = 0, we have m = 1 and we are done. Otherwise, ker(M ) ker(M 2 ), hence |ker(M 2 )| = p j for some j ≥ 2, as the kernel is a subgroup of our lattice (which has cardinality p rd ). This forces w 1 = p − 1 and j = 2. More generally, when In general, the maximal pretails need not share the same length, which means that we still have to extend our point of view.   3.3. Decomposition onL p r . When the integers u, v are coprime, one has L uv ≃ L u ⊕ L v , wherefore the action on L n with n ∈ N is completely determined by that on L p r , for all p r ||n.
In particular, the pretail orbit structure on an arbitrary L n can be derived from that on the sublattices associated with the factors in the prime factorisation of n.
Define R r = Z/p r Z, which is a local ring, with unique maximal ideal (p) = pR r . The latter contains all zero divisors. By [33,Thm. X.4.4], we then know that the two projective modules per(M ) and ker(M k ) of Fact 2 are free, so each has a basis. Consequently, one knows that the linear map onL p r defined by M induces unique linear maps on Fix(M k(r) ) and ker(M m(r) ), and M is conjugate to the direct sum of these maps, compare [1,Prop. 4.3.28]. Each of the latter, in turn, admits a matrix representation with respect to any chosen basis of the corresponding submodule. Different choices of bases lead to conjugate matrices, by an application of [1,Prop. 4.3.23].
Furthermore, d ′ is independent of r. When comparing the above objects as modules over the ring R s for different s, one has rank 1 (per 1 (M )) = rank r (per r (M )) = d ′ and rank 1 (ker 1 (M m(1) )) = rank r (ker r (M m(r) )) = d − d ′ , where an index s at per, ker or rank refers to R s as the underlying ring. , respectively. Here, B s = 0 mod p r for some s ∈ N and gcd(det(A), p) = 1, because A is invertible mod p r and det(A) must be a unit in R r . But this means that the reduction of A mod p is also invertible over R 1 = Z/pZ, while the reduction of B mod p is still nilpotent. Consequently, these reductions provide the blocks for the direct sum overL p , and the claim is obvious.
Since two free modules of the same rank are isomorphic [33, Cor. III.4.3], we also have the following consequence.  At this point, it is reasonable to link the properties of M onL p r to its minimal polynomial over F p . When M is nilpotent, the claim is obvious, because 0 is then the only possible root of the minimal polynomial over F p , as all other elements of the splitting field of f are units.
In all remaining cases, M is similar to A ⊕ B with A invertible and B nilpotent, by Corollary 3. We thus know that µ M (x)|x s (x k −1) with s and k as claimed, since the latter Since B is nilpotent, its powers cannot return to a non-zero matrix, hence B s ≡ 0 mod p.
Similarly, A s+k ≡ A s mod p is equivalent with A k ≡ ½ mod p, as A is invertible. This shows that we must indeed have µ M (x) = x s f (x) with ord(f, p) = k.

3.4.
Classification onL p . When we consider n = p, we can go one step further, because F p is a field and one can classify nilpotent matrices via their Jordan normal form. This follows from the observation that 0 is the only possible eigenvalue. Recall that an elementary shift matrix is an upper triangular matrix with entries 1 on the upper super-diagonal and 0 everywhere else (this includes the 0-matrix in one dimension). An elementary shift matrix is nilpotent, with nil-degree equal to its dimension. The following result is now a standard consequence of the Jordan normal form over fields [27,33]. Some of this structure survives also for general n. For instance, the 0-matrix in dimension d ≥ 1 leads to the regular (n d −1)-star as its pretail tree onL n . When d ≥ 2, the d-dimensional elementary shift matrix, onL n , results in a semi-regular tree with w 0 = 1, w 1 = n − 1 and w i = n for 2 ≤ i ≤ d − 1, while w j = 0 for all j ≥ d. These trees have the property that all maximal pretails share the same length, which is the nil-degree of the matrix. One can now go through all possible block-diagonal combinations of such elementary shift matrices. This is a combinatorial problem and gives the possible pretail trees over F p .
As already suggested by Proposition 3, the structure of ker(M ) plays an important role for the structure of the pretail tree. Together with the linearity of M , it constrains the class of trees that are isomorphic to the pretail tree of some integer matrix. A more detailed analysis is contained in [36].

Symmetry and reversibility
Reversibility is an important concept in dynamics, compare [39] and references therein for background, and [20] for an early study in continuous dynamics. Here, we focus on discrete dynamics, as induced by toral auto-and endomorphisms.
A matrix M is called reversible, within a given or specified matrix group G, if it is conjugate to its inverse within G. Note that a nilpotent matrix M (or a matrix with nilpotent summand, as in Corollary 3) cannot be reversible in this sense. However, they can still possess interesting and revealing symmetry groups, although it is more natural to look at the ring of matrices that commute with M in this case.
One can now study the action of S(M ) on the pretail graph of Figure 3, which actually explains all its symmetries. ♦ In what follows, we derive certain general properties, where we focus on the reversing symmetry group, with invertible matrices M in mind.

4.1.
Reversibility of SL(2, Z)-matrices mod n. Recall the matrix mgcd from Eq. (13), which is a conjugation invariant. It can be used to solve the reversibility at hand as follows.
Theorem 1. Let M ∈ SL(2, Z) and n ∈ N be arbitrary. Then, the reduction of M mod n is conjugate to its inverse within the group GL(2, Z/nZ). The action mod 1 of any M ∈ SL(2, Z) on L n is thus reversible for all n ∈ N.
Proof. When M ∈ SL(2, Z), also its inverse is in SL(2, Z), and M and M −1 share the same determinant and trace. Moreover, they also have the same mgcd, so that the first claim follows from [12, Thm. 2] (or from Lemma 1). This immediately implies, for all n ∈ N, the reversibility of the action mod n of M on the latticeL n , so that the statement on the equivalent action of M mod 1 on L n is clear. as defined in the proof of [12,Prop. 6], and note that they are not inverses of each other. However, by [12,Prop. 5], there is some matrix P n ∈ GL(2, Z/nZ) with M = P n N (M )P −1 n , hence we also have M −1 = P n N (M ) −1 P −1 n . Observe next that  Z)). More interesting (beyond Theorem 1) is the question which matrices M ∈ Mat(2, Z), when considered mod n for some n ∈ N, are reversible in GL(2, Z/nZ). Let us begin with n = p being a prime, where Z/pZ ≃ F p is the finite field with p elements. GL(2, F p ). Let us consider the symmetry and reversing symmetry group of an element of GL(2, F p ) with p prime, the latter being a group of order

Reversibility in
compare Eq. (3). For our further discussion, it is better to distinguish p = 2 from the odd primes. For convenience, we summarise the findings also in Table 4.2.
Example 3. For p = 2, one has GL(2, F 2 ) = SL(2, F 2 ) ≃ D 3 , the latter denoting the dihedral group of order 6. There are now three conjugacy classes to consider, which may be represented by the matrices ½, the involution R = ( 0 1 1 0 ), and the matrix M = ( 1 1 1 0 ) of order 3. The corresponding cycle structure on L 2 is encapsulated in the generating polynomials Z 2 (t). They read respectively, and apply to entire conjugacy classes of matrices.
For the (reversing) symmetry groups, one clearly has R(½) = S(½) = GL(2,  Table 4.2. We summarise the reversibility and orbit structure here, but omit proofs whenever they emerge from straight-forward calculations. I. The first type of conjugacy class is represented by matrices M = a½ with a ∈ F × p ≃ C p−1 . The order of M coincides with the order of a mod p, ord(a, p), which divides p−1. One clearly has R(M ) = S(M ) = GL(2, F p ) in this case, either because a 2 = 1 (so that M = M −1 ) or because a 2 = 1 (so that no reversors are possible). The corresponding orbit structure on L p comprises one fixed point (x = 0) together with p 2 −1 ord(a,p) orbits of length ord(a, p). The non-trivial orbits starting from some x = 0 must all be of this form, as x gets multiplied by a under the action of M and returns to itself precisely when a k = 1, which first happens for k = ord(a, p).
II. The next type of conjugacy class is represented by matrices M = ( a 1 0 a ) with a ∈ F × p . Its symmetry group is given by , which is Abelian. As generators of the cyclic groups, one can choose ( 1 1 0 1 ), which has order p in GL(2, F p ), and γ½, with γ a generating element of F × p . The reversible cases are precisely the ones with a 2 = 1 in F p , hence with det(M ) = 1. Here, R = diag(1, −1) is a possible choice for the (involutory) reversor, so that A matrix M of type II (in its normal form as in Table 4.2) satisfies For a type III matrix, one has M k (x, y) t = (a k x, b k y) t , so each of the p − 1 non-zero points (x, 0) t is fixed by M ord(a,p) ; analogously, each of the p − 1 non-zero points (0, y) t is fixed by M ord(b,p) . The remaining points that are non-zero in both coordinates have period lcm (ord(a, p), ord(b, p)). In summary, this gives one fixed point, p−1 ord(a,p) orbits of length ord(a, p), p−1 ord(b,p) orbits of length ord(b, p), and (p−1) 2 lcm(ord(a,p),ord(b,p)) orbits of length lcm(ord(a, p), ord(b, p)).
IV. Finally, the last type of conjugacy class can be represented by companion matrices of the form 0 −D 1 T with the condition that the characteristic polynomial z 2 − T z + D is irreducible over F p . The determinant and the trace satisfy D = ηη ′ and T = η + η ′ , where η and η ′ are not in F p , but distinct elements of the splitting field of the polynomial, which can be identified with F p 2 . One consequence is that 1 + D ± T = (1 ± η)(1 ± η ′ ) = 0.
The symmetry group is S(M ) = {α½ + γM | α, γ ∈ F p , not both 0}, which is an Abelian group with p 2 − 1 elements. The order follows from the observation that det(α½ + γM ) = (α + γη)(α + γη ′ ) vanishes only for α = γ = 0 in this case. In fact, one has S(M ) ≃ C p 2 −1 , as any matrix 0 −ηη ′ 1 η+η ′ ∈ GL(2, F p ) with η ∈ F p 2 \ F p has order p 2 − 1 or possesses a root in GL(2, F p ) of that order. This relies on the facts that we can always write η = λ m , where λ is a generating element of F × p 2 ≃ C p 2 −1 , and that λλ ′ and λ + λ ′ are in F p . This is a special case of Fact 9 below and of a statement on the existence of roots in GL(d, Z); see Lemma 6 below.
The condition for reversibility, in view of the above restriction on D and T , can only be satisfied when D = 1, in which case R = ( 0 1 1 0 ) turns out to be an involutory reversor, so that again R(M ) = S(M ) ⋊ C 2 in this case.
Matrices with irreducible characteristic polynomial χ M produce orbits of one length r only, where r is the smallest integer such that χ M (z)|(z r − 1), or, equivalently, the order of its roots in the extension field F p 2 . Table 1. Summary of conjugacy structure for GL(2, F p ) via normal forms. Note that class III is absent for p = 2. The second possibility for R(M ) always applies when det(M ) = 1. Only non-trivial orbits are counted.
Putting these little exercises together gives the following result. Remark 2. Since F p is a field, we can use the following dichotomy to understand the structure of S(M ), independently of the chosen normal forms. A matrix M ∈ GL(2, F p ) is either a multiple of the identity (which then commutes with every element of Mat(2, F p )) or it possesses a cyclic vector (meaning an element v ∈ F 2 p such that v and M v form a basis of F 2 p ). In the latter case, M commutes precisely with the matrices of the ring 4.3. Some extensions to higher dimensions. In principle, a similar reasoning, based on a normal form approach, can be applied to arbitrary dimensions. Over the finite field F p , normal forms are given by the rational canonical form and the elementary divisor normal form ('First' and 'Second natural normal form' in the terminology of [24, §6]), which are block diagonal matrices with companion matrices on the diagonal. The advantage of dealing with companion matrices is that one can employ the theory of linear recursions: there is a one-to-one correspondence between the cycle lengths modulo n ∈ N of a certain initial condition u = (u 0 , . . . , u d−1 ) under the recursion induced by the polynomial f , and the period of the corresponding point u t under the matrix iteration of C f ; compare the final remark in [48], and Section 2.4.
Working with a block diagonal matrix of this shape, the analysis can be done block-wise; in particular, the symmetry groups are the direct product of the symmetry groups of the component matrices on the diagonal, augmented by all additional symmetries that emerge from equal blocks, which can be permuted.
Determining the period lengths associated with irreducible polynomials amounts to finding their orders in the sense of [34,Def. 3.3.2]. The periods and their multiplicities arising from the powers of irreducible polynomials that show up in the factorisation of the invariant factors (the elementary divisors) are then given by [48,Thm. 4].
Extending the analysis to matrices over the local rings Z/p r Z is more difficult. In general, it seems hard to write down an exhaustive system of normal forms for the similarity classes, and to decide whether given matrices are similar. However, a solution for a large subclass of square matrices over the p-adic integers Z p and the residue class rings Z/p r Z is presented in [17]. For a polynomial f ∈ Z p [x] whose reduction modulo p has no multiple factors, a complete system of d × d matrix representatives X with respect to similarity that satisfy f (X) ≡ 0 mod p r is given by all direct sums of companion matrices which are in agreement with the factorisation of f mod p. For instance, if the reduction of the common characteristic polynomial modulo p of two matrices does not have any quadratic factors, the matrices are conjugate mod p r if and only if they are conjugate mod p [17,Thm. 3 and Corollary].
An exhaustive treatment of conjugacy classes of 3 × 3 matrices over an arbitrary local principal ideal ring can be found in [4].
Remark 3. In [4], it is pointed out that 2 × 2 matrices over a local ring can be decomposed into a scalar and a cyclic part. Over Z/p r Z, this decomposition reads where p ℓ = gcd(mgcd(M ), p r ) = p vp(mgcd(M )) with v p denoting the standard p-adic valuation, unique d ∈ { ℓ−1 j=0 a j p j | p ∤ a j } and cyclic C ∈ Mat(2, Z/p r−ℓ Z), which is unique up to Proof. For p = 2, Fact 4 implies D 2 ≡ 1 mod p r . Since p cannot divide both D − 1 and D + 1, one has p r |(D − 1) or p r |(D + 1), which gives the first claim. When 2 r |(D − 1)(D + 1), 2 divides one of the factors and 2 r−1 the other one, so D ≡ 1 or D ≡ −1 mod 2 r−1 . If D ≡ −1 mod 2 r−1 , Fact 5 gives 2 tr(M ) ≡ 0 mod 2 r−1 and thus M 2 ≡ ½ mod 2 r−2 by Fact 6.
One immediate consequence for d = 2 is the following.
Corollary 6. If M ∈ GL(2, Z) with D = −1 is reversible for infinitely many n ∈ N, one has Fact 7. Let A be an integer matrix whose determinant is coprime with n ∈ N. The reduction of the inverse of A over Z/nZ, taken modulo k|n, is then the inverse of A over Z/kZ.

Remark 4.
To see that reversibility mod p for all primes p which divide n is not sufficient for reversibility mod n, one can consider a locally reversible matrix M with det M = 1: according to Fact 4, only finitely many n exist such that M is reversible mod n, so for each prime p there must be a maximum r for which M is reversible mod p r . Recalling an example from above, M = 0 −4 1 0 is reversible mod 3 but not mod 9 as can be verified by explicit calculation. It is an involution mod 5, hence also reversible mod 15, but not mod 45. ♦ Reversibility can be viewed as a structural property that reflects additional 'regularity' in the dynamics, in the sense that it typically reduces the spread in the period distribution. For 2 × 2-matrices, the normal form approach shows that reversibility implies the existence of only one non-trivial period length on L p ; compare our comments in Section 4.5.
4.5. Matrix order and symmetries over F p . Let us now discuss the order of a matrix M ∈ GL(d, F p ), with p a prime, in conjunction with the existence of roots of M in that group. We begin by recalling the following result from [ Fact 8. If f is an irreducible polynomial of degree d over F p , its splitting field is isomorphic with F p d . There, it has the d distinct roots α, α p , . . . , α p d−1 that are conjugates and share the same order in (F p d ) × .
In particular, two irreducible polynomials over F p of the same degree have isomorphic splitting fields.
From now on, we will identify isomorphic fields with each other. In particular, we write F p d for the splitting field of an irreducible polynomial of degree d over F p .
Next, let K be an arbitrary finite field, consider an irreducible, monic polynomial f ∈ K[x] of degree d, and let L be the splitting field of f . When λ 1 , λ 2 , . . . , λ d are the roots of f in L, one has the well-known factorisation (20) f where the e i denote the elementary symmetric polynomials, The elementary symmetric polynomials, when evaluated at the roots of f , are fixed under all Galois automorphisms of the field extension L/K, so that the following property is clear. Let M be a d × d integer matrix with irreducible characteristic polynomial χ M over F p . Let α be a root of χ M in F p d and λ a generating element of the unit group (F p d ) × . Clearly, there is an n ∈ N with α = λ n . By Fact 8, one has F p (α) = F p d = F p (λ), where the degree of the extension field over F p equals d. Consequently, the minimal polynomial of λ over F p is an irreducible monic polynomial of degree d over F p , and the conjugates of α are powers of the conjugates of λ. Let α 1 , . . . , α d and λ 1 , . . . , λ d denote the respective collections of conjugates. Thus, over F p d , one has the matrix conjugacy (20) and C(f ) denoting the companion matrix of f . Here, it was exploited that a d × d matrix whose characteristic polynomial f has d distinct roots is always similar to the companion matrix of f . Note that C(f ) ∈ GL(d, F p ) by Fact 9. Now, M and C(f ) are matrices over F p that are conjugate over F p d , so (by a standard result in algebra, see [1,Thm. 5.3.15]) they are also conjugate over F p , which means that we have the relation with some A ∈ GL(d, F p ). By similarity, ord(W ) = ord(C(f )) = ord(diag(λ 1 , . . . , λ d )) = p d − 1. This gives the following result.
Lemma 6. A matrix M ∈ GL(d, F p ) with irreducible characteristic polynomial has either the maximally possible order p d − 1, or admits an n-th root W ∈ GL(d, F p ) as in (21). Here, n can be chosen as n = p d −1 ord(M ) , so that the root has order p d − 1.
Fact 10. Let A be a matrix over F p with minimal polynomial of degree d. Then, the ring has precisely p d elements, which correspond to the different d-tuples (ξ 1 , . . . , ξ d ).
Proof. Two distinct d-tuples producing the same matrix would give rise to a non-trivial linear combination that vanishes, involving powers of A of degree d − 1 at most, which contradicts the minimal polynomial having degree d.
where W denotes the cyclic group generated by W .  [27] and the following Corollary. In such a case, S(M ) is a true group extension of F p [M ] × . The situation is thus particularly simple for matrices M ∈ Mat(2, F p ): Either they are of the form M = a½ (then with S(M ) = GL(2, F p )), or they are cyclic (then with S(M ) = F p [M ] × ).

Appendix: Two classic examples
If one reads through the literature, two matrices are omnipresent as examples, the Arnold and the Fibonacci cat map. Still, several aspects of them are unclear or conjectural, despite the effort of many. Let us sum up some aspects, with focus on properties in line with our above reasoning.
A.1. Arnold's cat map. Here, we collect some results for the matrix M A = ( 2 1 1 1 ) ∈ SL(2, Z) in an informal manner. This case was studied in [37,21,25] and appeared in many other articles as main example. It was introduced in [3, Example 1.16] as a paradigm of (discrete) hyperbolic dynamics.
The integer matrix M A is reversible within the group GL(2, Z), with a reversor of order 4, but none of order 2. One has S(M A ) ≃ C 2 × C ∞ , where C 2 = {±1} and the infinite cyclic group is generated by the unique square root of M A in GL(2, Z) (see below), while R(M A ) = S(M A ) ⋊ C 4 ; see [9] for more. In particular, M A inherits local reversibility in GL(2, Z/pZ) for all primes p from its 'global' reversibility within GL(2, Z).
It was shown in [25] that M A , except for the trivial fixed point 0, has orbits of only one period length on each prime lattice L p . In view of the normal forms, this is clear whenever the characteristic polynomial is irreducible. However, a matrix of type III from Table 4.2 has reducible characteristic polynomial and occurs for primes with 5 p = −1. Here, different orbit lengths would still be possible in general, but reversibility forces the two roots to be multiplicative inverses of one another and thus to have the same order modulo p.
The iteration numbers are p m = f 2m , where the f k are the Fibonacci numbers, defined by the recursion f k+1 = f k + f k−1 for k ∈ N with initial conditions f 0 = 0 and f 1 = 1. Since mgcd(M A ) = 1, Proposition 2 implies ord(M A , n) = κ A (n) = period (f 2m ) m≥0 mod n , where the periods for prime powers (with r ∈ N) are given by κ A (2 r ) = 3 · 2 max{0,r−2} and κ A (5 r ) = 10 · 5 r−1 together with κ A (p r ) = p r−1 κ A (p) for all remaining plateau-free primes. It has been conjectured that this covers all primes [44]. No exception is known to date; the conjecture was tested for all p < 10 8 in [5]. Note that each individual prime can be analysed on the basis of Proposition 1.
The periods mod p are κ A (2) = 3, κ A (5) = 10, together with κ A (p) = p − 5 p 2m p − 1 2 1 − 5 p for odd primes p = 5, where 5 p denotes the Legendre symbol and m p ∈ N is a characteristic integer that covers the possible order reduction. It is 1 in 'most' cases (in the sense of a density definition), but there are infinitely many cases with m p > 1; this integer is tabulated to some extent in [44,25].
Let us write down the generating polynomials for the distribution of cycles on the lattices L n . Once again, this is only necessary for n a prime power. We use a formulation with a factorisation that shows the structure of orbits on L p r \ L p r−1 . In the notation of [12], one finds Z 1 (t) = (1 − t) and with r ≥ 1 for the prime p = 2, as well as with r ≥ 1 for p = 5. As usual, we adopt the convention to treat an empty product as 1. The remaining polynomials read Z p r (t) = (1 − t) as long as the plateau phenomenon is absent (see above).
A.2. Fibonacci cat map. Closely related is the matrix M F = ( 1 1 1 0 ) ∈ GL(2, Z), which is the unique square root of the Arnold cat map M A in GL(2, Z). It appears in numerous applications; see [38,6,7,16] and references therein for some of them. Here, the iteration numbers are the Fibonacci numbers themselves, and the periods are the so-called Pisano periods; compare [42, A001175] and references given there, or [44].
The matrix M F is not reversible in GL(2, Z) (while its square M A is, see above), and has the same symmetry group as M A . In fact, ±M F are the only roots of M A in GL(2, Z). This situation implies that the orbit structure for M F must be such that the iteration of its square gives back the counts we saw in the previous example.
For prime powers p r , with r ∈ N, one finds κ F (5 r ) = 20 · 5 r−1 together with κ F (p r ) = p r−1 κ F (p) for all remaining primes, with the same proviso as for the Arnold cat map. The periods κ F (p) are given by κ F (2) = κ A (2) = 3 together with κ F (p) = 2 κ A (p) for all odd primes, which is not surprising in view of the relation between the two matrices M F and M A . The orbit distribution is more complicated in this case, as usually orbits of two possible lengths arise in each step. One finds for the primes 2 and 5 (with r ∈ N 0 as before), as well as for all remaining primes that are free of the plateau phenomenon (which possibly means all, see above). Here, n p ∈ N 0 is a characteristic integer which often takes the values 1 or 0, but does not seem to be bounded.