A Profinite Group Invariant for Hyperbolic Toral Automorphisms

For a hyperbolic toral automorphism, we construct a profinite completion of an isomorphic copy of the homoclinic group of its right action using isomorphic copies of the periodic data of its left action. The resulting profinite group has a natural module structure over a ring determined by the right action of the hyperbolic toral automorphism. This module is an invariant of conjugacy that provides means in which to characterize when two similar hyperbolic toral automorphisms are conjugate or not. In particular, this shows for two similar hyperbolic toral automorphisms with module isomorphic left action periodic data, that the homoclinic groups of their right actions play the key role in determining whether or not they are conjugate. This gives a complete set of dynamically significant invariants for the topological classification of hyperbolic toral automorphisms.


Introduction
The topological classification of hyperbolic, irreducible, toral automorphisms brings together the subjects of dynamical systems, algebra, and algebraic number theory. It is well-known [1] that two T n automorphisms induced by A, B ∈ GL(n, Z) are topologically conjugate if and only if A and B are conjugate in within the group GL(n, Z), i.e., there is C ∈ GL(n, Z) such that AC = CB. Furthermore, by a wellknown result of Latimer, MacDuffee and Taussky (see [14,17,18]), this happens if and only if A and B are associated to the same ideal class in the number ring determined by their common characteristic polynomial. In both these settings, algorithms have been developed that, in principle, determine when two automorphisms are conjugate [4,6]. For instance, the case of n = 2 uses classical results about continued fractions [2,11], while the case n ≥ 3 uses recently developed geometric continued fractions [9]. Yet there is not yet a clear understanding of the topological classification as would be given by a complete set of dynamically significant invariants.
The periodic data for a hyperbolic, irreducible, toral automorphism is insufficient to characterize its conjugacy class, and yet provides a dynamically significant invariant of conjugacy (see [13]). Let Per l (A) denote the finite group of l-periodic points for the left action of A in T n = R n /Z n , represented by columns vectors. It is possible for hyperbolic, irreducible, toral automorphisms A and B to have conjugate actions on the Per l level, for every l, without A and B being conjugate while being similar, i.e., there is C ∈ GL(n, Q) such that AC = CB. More generally, suppose that A and B have the same irreducible characteristic polynomial p(x), let R = Z[x]/(p(x)), and set Per g (A) = ker(g(A)) ⊂ T n , where g ∈ Z[x] with g(A) invertible. It is still possible that A and B are not conjugate on T n , and yet are strongly Bowen-Franks equivalent (also written strongly BF-equivalent), i.e., associated to every g ∈ Z[x] with g(A) invertible, is an R-module isomorphism φ g : Per g (A) → Per g (B) that conjugates A and B on the level of periodic points determined by g. It is then natural to consider what additional dynamically significant invariants are needed to determine when two strongly BF-equivalent similar hyperbolic, irreducible, toral automorphisms are conjugate or not, i.e., when the level conjugacies would imply the existence of a global conjugacy.
This paper explores this approach, from a dual point of view: instead of the direct limit of the R-modules Per g (A), the main object is a profinite limit of the Pontryagin dual R-modules Z n /Z n g(A), given by the right action of g(A) on Z n represented by row vectors. We review in Section 2 some basic definitions and facts about profinite groups. Then in Section 3, we present the construction of a profinite group G A associated to a hyperbolic A ∈ GL(n, Z). This G A is an R-module. It is a profinite completion of the right Z[A]-module Z n which is R-module isomorphic to the homoclinic group H ′ A of the right action of hyperbolic toral automorphism induced by A. As an invariant of conjugacy, the R-module H ′ A naturally embeds into G A and plays the key role in determining when two strongly BF-equivalent similar hyperbolic toral automorphisms are conjugate or not.
Obviously, strong BF-equivalence is a necessary condition for conjugacy of similar hyperbolic toral automorphisms. In Section 4, we describe conditions by which two similar hyperbolic toral automorphisms are strongly BF-equivalent in the dual sense, i.e., Z n /Z n g(A) is R-module isomorphic to Z n /Z n g(B) for every g ∈ Z[x] with g(A) invertible. Then in Section 5 we show that strong BF-equivalence of similar hyperbolic A, B ∈ GL(n, Z) implies the existence of a topological R-module isomorphism Ψ : G A → G B . In Section 6, we detail the exact manner by which embedded copies of H ′ A and H ′ B in G A and G B respectively, characterize when A and B are conjugate or not. Specifically, for strong BF-equivalent similar hyperbolic A and B, it is how the image of the embedded copy of H ′ A under any topological R-module isomorphism Ψ : G A → G B intersects the embedded copy of H ′ B in G B , that determines conjugacy or the lack thereof. This characterization of conjugacy applies when the characteristic polynomial of the similar A and B is irreducible or reducible. When the similar hyperbolic A and B have an irreducible characteristic polynomial, we show in Section 7 that the intersection of the image of the embedded copy of H ′ A in G A under any topological R-module isomorphism Ψ : G A → G B with the embedded copy of H ′ B in G B is either trivial or has finite index in the embedded copy of H ′ B in G B The characterization of conjugacy for strongly BF-equivalent similar hyperbolic A, B ∈ GL(n, Z) show that the embedded copies of H ′ A and H ′ B in G A and G B are the additional dynamically significant invariants that are needed. It is known [12] that the homoclinic groups H ′ A and H ′ B are complete invariants of conjugacy when A and B are Pisot, i.e., have one real eigenvalue larger than 1 in modulus with all the remaining eigenvalues smaller than 1 in modulus. Our characterization of conjugacy extends the role of the homoclinic groups as classifying invariants from the Pisot case to the general case. This characterization of conjugacy also resolves the problem of classification of quasiperiodic flows of Koch type according to the equivalence relation of projective conjugacy (see [3]), and gives a dynamical systems resolution to the ideal class problem in algebraic number theory.
From a computational point of view, our dynamical characterization of conjugacy for strongly BF-equivalent similar hyperbolic A, B ∈ GL(n, Z) is not completely satisfactory. We would like to better understand how to computationally detect when the conditions for characterization hold or are violated for the R-submodules H ′ A and H ′ B in G A and G B respectively. We are currently investigating some computational avenues for this detection that we hope to include in a subsequent paper.

Basic Theory of Profinite Groups
We review the construction of and set the notation for profinite groups built from the left and right actions of an A ∈ GL(n, Z) on the integer lattice Z n . We list below many of the basic properties of these profinite groups. The statements and proofs of these basic properties are from [15], where the proofs appropriately adapt, when needed, from the group-theoretic setting to the R-module setting without difficulty. In the next section, we construct a particular profinite group with an R-module structure for a hyperbolic toral automorphism.
The ring for the module structure on the profinite groups is is the ring of polynomials with integer coefficients. The abelian group G = Z n , represented by row vectors, is naturally a Z[A]-module with the right action of A on m ∈ Z n defined by m → mA.
We define a topology on G as follows. Let N be a collection of Z[A]-submodules of Z n of finite index that is filtered from below, i.e., for N, M ∈ N there is K ∈ N such that K ≤ N ∩ M . The elements of N , considered as a fundamental system of neighbourhoods of the identity element 0 of G, endows G with a profinite topology.
We A profinite completion of G with respect to its profinite topology defined by N is the inverse limit of the surjective inverse system {G/N, ϕ N M , N }. The inverse limit is the Z[A]-submodule,  The collection U includes the collection {ker(ϕ N ) : N ∈ N }. We make U into a directed poset by defining V U when U ≤ V . This gives a surjective inverse A subcollection of open Z[A]-submodules of G N may sometimes realize G N as well. A cofinal subsystem of {G N /U, ϕ UV , U} is a surjective inverse system {G N /K, ϕ KL , K} where K is a subset of U such that K is a directed poset with respect to the binary relation on U, and such that for

A Z[A]-Module For Hyperbolic A
We use the Pontryagin dual of periodic data for the left action of a hyperbolic toral automorphism to construct a profinite group associated to it. Let A ∈ GL(n, Z) be hyperbolic, and let T A be the hyperbolic toral automorphism that the left action of A induces on T n , i.e., T A π = πA where π : R n → T n is the canonical covering epimorphism with R n and T n represented by column vectors. Hyperbolicity of A implies that det(A r − I) = 0 for all r ∈ Z. Let T ′ A be the hyperbolic toral automorphism that the right action A on Z n , represented by row vectors, induces on T n , represented by row vectors. Let , the homoclinic group for the right action [10]). For each k ∈ Z, we also use the right action of A on Z n to define the abelian groups The finite abelian groups G k,A , having elements of the form [m] k,A = m + N k,A , are generalized Bowen-Franks groups which are isomorphic to the Pontryagin duals of Per k! (T A ), the group of periodic points of period k! for the left action T A on T n . Obviously, the collection Lemma 3.1. For any A ∈ GL(n, Z), the collection N A satisfies N k,A ≥ N k+1,A for all k ∈ N and is filtered from below. If, in addition, A is hyperbolic, then N A also satisfies where within the first set of braces on the right-hand side, the power of A eventually In the first set of braces on the right-hand side, the power of the A eventually becomes Since the term in the first set of braces on the right side of the factorization is an This means that A k! sends an element of Z n to another element of Z n that is a distance of m away from ξ k . (Here · is the usually Euclidean norm on R n .) Let E s (A) and E u (A) be the stable and unstable spaces of A acting on R n . The only element of Z n that belongs to E s (A) ∪ E u (A) is 0 because no lattice point can iterate under A or A −1 to 0. The hyperbolicity of A implies that ξA k! , for ξ ∈ Z n \ {0}, goes to ∞ with exponential speed as k → ∞ (see Proposition 1.2.8 on p. 28 in [7]). This means that for all sufficiently large k, each A k! sends the points in Z n a distance of m away from ξ k , to a distance much larger than m away from ξ k . This is a contradiction.
We associate to a hyperbolic A ∈ GL(n, Z) a Z[A]-module G A as follows, where, to set some notation in this context, we state some of the basic properties from Section 2 without explicit reference. By Lemma 3.1, the binary relation on the filtered from below collection N A , indexed by N, simplifies to l ≤ k for l, k ∈ N implies N l,A ≥ N k,A . We denote the elements of We could have constructed G A from the filtered from below directed poset N of all the finite Z[A]-modules BF g (A) for g ∈ Z[x] with g(A) invertible. However, because N A is cofinal in N , we would have constructed a Z[A]-module from N that would have been Z[A]-module isomorphic to G A . We prefer the cofinal subcollection because of its linear ordering. For is the ring endomorphism defined by ϑ D q(x) + (p(x)) = q(D); it does matter whether D = A or D = B because the similarity of A and B implies that ker(θ A ) = ker(θ B ).

Sufficient Conditions for Strong BF-Equivalence
For similar hyperbolic A, B ∈ GL(n, Z), the finite R-modules G k,A , G k,B for k ∈ N are invariants of the conjugacy classes of A and B respectively in terms of their R-module structure. More generally, for g ∈ Z[x] with g(A) (and hence g(B)) invertible, the finite R-modules Z n /Z n g(A) and Z n /Z n g(B) are invariants of the conjugacy classes of A and B in terms of their R-module structures. The generalized Bowen-Franks groups are BF g ( For two Rmodules G 1 and G 2 , we write G 1 ∼ =R G 2 when G 1 and G 2 are R-module isomorphic. Similar hyperbolic A, B ∈ GL(n, Z) are said to be strongly BF-equivalent when BF g (A) ∼ =R BF g (B) for all all g(x) ∈ Z[x] with g(A) invertible. When BF g (A) and BF g (B) are not R-module isomorphic for some g ∈ Z[x] with g(A) invertible, then the similar A and B fail to be conjugate. So a necessary condition for similar hyperbolic A and B to be conjugate is that A and B are strongly BF-equivalent. is said to be equivalent to I if there is a nonzero z ∈ K such J = zI. The Latimer-MacDuffee-Taussky Theorem asserts that there is a one-to-one correspondence between the conjugacy classes of matrices in GL(n, Z) with characteristic polynomial p(x) and the equivalence classes of the fractional ideals of Z[β]. With this one-toone correspondence, we have that G k,  Consider the map from I to J given by multiplication by γ, i.e., x ∈ I gets maps to γx ∈ J. It obviously maps αI into αJ. On the other hand, if γx ∈ αJ, say γx = αz for z ∈ J, then x = aαx + bγx = α(ax + bz).
Using Z-bases for I and J we get an invertible integer-entry matrix X g defined by γv = wX g that satisfies X g A = BX g .
This semi-conjugacy induces an R-module isomorphism between the groups BF g (A) and BF g (B). Since g ∈ Z[x] with g(A) invertible is arbitrary, we have strong BFequivalence of A with B.
Theorem 4.2 extends a result from [13], which states that for hyperbolic A, B ∈ GL(n, Z) with the same irreducible characteristic polynomial p(x), whose associated fractional ideals I and J of Z[β] satisfy O(I) = O(J) with I and J invertible in O(I), the matrices A and B are strongly BF-equivalent. Invertible fractional ideals of Z[β] are always weakly equivalent. One way to guarantee the invertibility of nonzero fractional ideals of Z[β] is when the discriminant of p(x) is square-free; then Z[β] = o K (see [4]), a Dedekind domain in which every nonzero fractional ideal is invertible (see [16]). Example 4.3. Consider the irreducible polynomial p(x) = x 3 − 2x 2 − 8x − 1 whose discriminant of 1957 is square-free (see Table B4 in [4]). The two matrices 2 and the comments that followed its proof, the matrices A and B are strongly BF-equivalent. But, I and J are inequivalent fractional ideals (see [8]), and so A and B are not conjugate.

Strongly BF-Equivalent Hyperbolic Toral Automorphisms
Unfortunately, for similar hyperbolic A, B ∈ GL(n, Z), strong BF-equivalence of A and B does not imply the conjugacy of A and B (see [13]). As illustrated in Example 4.3, an algebraic obstruction for this is the existence of inequivalent invertible ideals with the same ring of coefficients. However, strong BF-equivalence does relate the profinite R-modules G A and G B . The first claim is that G A and G B have the same finite quotients through their respective open R-submodules: where we understand equality in terms of R-module isomorphisms. Let H = G A /U for U ∈ U A . Since {ker(ϕ k,A ) : k ∈ N} is a fundamental set of open R-submodule neighbourhoods of 0, there is a k ∈ N such that ker(ϕ k,A ) is an R-submodule of U . Thus U/ker(ϕ k,A ) is an R-submodule of G A /ker(ϕ k,A ) and (G A /ker(ϕ k,A ))/(U/ ker(ϕ k,A ) By the assumption of strong BF-equivalence of A and B, we have that G k,A ∼ =R G k,B ∼ =R G B /ker(ϕ k,B ). Then there is an open R-submodule V of G B containing ker(ϕ k,B ) such that U/ker(ϕ k,A ) ∼ =R V /ker(ϕ k,B ). This implies that H = G A /U ∼ =R G B /V for V ∈ U B . Reversing the roles of A and B in this argument gives the other inclusion, and thus the first claim.
From this point on, we adapt the argument from ( [15], Theorem 3.2.7 on pp. 88-89). For each k ∈ N set Since G A and G B are both finitely generated, there are for each k ∈ N only finitely many U ∈ U A and V ∈ U B such that [G A : U ] ≤ k and [G B : V ] ≤ k. Thus U k ∈ U A and V k ∈ U B for all k ∈ N. By the first claim, for each k ∈ N there is a K ∈ U A such that The second claim is that this K is an R-submodule of U k . We show this by setting up an association between certain open R-submodules of G A and G B . Take

This implies that G
These two arguments account for all U ∈ U A with K an R-submodule of U such that [G A : U ] ≤ k. Thus we obtain that K is an R-submodule of U k , giving the second claim.
The third claim is that

Reversing the roles gives by a similar argument that |G
This gives the third claim.
For each k ∈ N, let Z k be the set of all R-module isomorphisms from G A /U k to G B /V k . By the third claim, Z k = ∅ for all k ∈ N. Let σ k denote an element of Z k .
The fourth claim is that each R-module isomorphism σ k+1 ∈ Z k+1 induces an R-module isomorphism σ k ∈ Z k . Since U k+1 is an R-submodule of U k and V k+1 is an R-submodule of V k , we have that U k /U k+1 is an R-submodule of G A /U k+1 and and so σ k+1 induces an R-module isomorphism σ k between G A /U k and G B /V k . It remains to check that By the argument used in the second claim, we have that L is an Hence |G A /U k | = |G A /L|, and so giving the fourth claim. Denote by ξ k+1,k : Z k+1 → Z k the map defined by σ k+1 → σ k as given by the fourth claim. Then {Z k , ξ k+1,k } is an inverse system of nonempty sets. Hence (by Proposition 1.1.4 in [15]), there is an element (σ k ) ∞ k=1 in the inverse limit

The inverse systems {G
where each of the R-module isomorphisms is a topological R-module isomorphism. Here, the first and last profinite R-modules are topologically R-module isomorphic to G A and G B respectively. Therefore, there is a topological R-module isomorphism between G A and G B .

Characterization of Conjugacy
For similar A, B ∈ GL(n, Z), the profinite R-modules G A and G B are invariants of conjugacy. Furthermore, they provide means to characterize when A and B are conjugate or not in terms of the embedded copies of the R-modules H ′ A and H ′ B in G A and G B respectively. Specifically, for a topological R-module isomorphism Ψ : Lemma 6.1. Suppose A, B ∈ GL(n, Z n ) are similar and hyperbolic. If there are R-module isomorphisms Ψ k : G k,A → G k,B such that for all l ≤ k, then the collection of R-module isomorphisms Ψ k induces a topological R-module isomorphism Ψ : Proof. The isomorphisms Ψ k are homeomorphisms because of the discrete topologies on G k,A and on G k,B . Assuming that ϕ k,l,B Ψ k = Ψ l ϕ k,l,A for all l ≤ k implies that the Ψ k are components of a morphism from the inverse system {G k,A , ϕ k,l,A , N} to the inverse system {G k,B , ϕ k,l,B , N}. The maps Ψ k ϕ k,A : G A → G k,B form a collection of compatible maps: for all l ≤ k, there holds The compatible maps Ψ k ϕ k,A induce a continuous homomorphism Ψ : G A → G B that satisfies ϕ k,B Ψ = Ψ k ϕ k,A for all k ∈ N. The map Ψ is given by and is a topological isomorphism because each Ψ k is a topological isomorphism and because G A is compact and G B is Hausdorff. Since each Ψ k is an R-module isomorphism, we have Ψ k A k = B k Ψ k for all k ∈ N. From this we get Thus, Ψ is a topological R-module isomorphism.
Theorem 6.2. For similar hyperbolic A, B ∈ GL(n, Z), the following are equivalent: (a) A is conjugate to B, Proof. (a)⇒(b) Suppose A and B are conjugate, i.e., there is C ∈ GL(n, Z) such that AC = CB. For k ∈ N define maps Ψ k : We show that the maps Ψ k are well-defined.
. , and so each Ψ k is surjective. Since Ψ k is a continuous bijection, G k,A is compact, and G k,B is Hausdorff, each Ψ k is a homeomorphism. The maps Ψ k satisfy We show that the Φ k induce a topological R-module isomorphism from G A to G B . For l ≤ k, the topological R-module isomorphisms Ψ k satisfy By Lemma 6.1, the maps Ψ k therefore induce a topological R-module isomorphism Ψ : We show that this topological R-module isomorphism Ψ satisfies Ψ ι A (Z n ) = ι B (Z n ). For m ∈ Z n , we have The invertibility of C implies the opposite inclusion, so that Ψ(ι A (Z n ) = ι B (Z n ).
(b)⇒(c) This follows because ι B is an R-module monomorphism.
(c)⇒(a). Let ∆ Ψ = Ψ ι A (Z n ) ∩ ι B (Z n ), and suppose there is a R-module isomorphism h : Z n → ∆ Ψ , i.e., hB = Γ B h. Then h −1 : ∆ Ψ → Z n is also a Rmodule isomorphism, i.e., Bh −1 = h −1 Γ B , and so the map h −1 Ψι A is an R-module isomorphism from the right Z[A]-module Z n to the right Z[B]-module Z n . This implies that h −1 Ψι A is an automorphism of Z n , and so there is C ∈ GL(n, Z) such that C = h −1 Ψι A . Thus for all m ∈ Z n , we have Since mAC = mCB holds for all m ∈ Z n , we have that AC = CB, i.e., that A and B are conjugate.
We detail another proof of part (b) implying part (a) of Theorem 6.2 that highlights the role that hyperbolicity and the profinite topology play. Suppose that there is a topological R-module isomorphism Ψ : G A → G B such that Ψ ι A (Z n ) = ι B (Z n ). Since ι B : Z n → G B is a monomorphism, there is an isomorphism ι −1 B : ι B (Z n ) → Z n , and so Ψ ι A (Z n ) = ι B (Z n ) implies that ι −1 B Ψι A : Z n → Z n is an automorphism. Since the automorphism group of Z n is GL(n, Z), there is C ∈ GL(n, Z) such that For all m ∈ Z n , it follows that Since Ψ is an R-module isomorphism, we have Γ B Ψ = ΨΓ A , and so for all m ∈ Z n , This means for all k ∈ N and for all m ∈ Z n that m(CB − AC) = mCB − mAC ∈ N k,B = Z n (B k! − I).
For the standard basis of row vectors e 1 , . . . , e n of Z n , we have e j (CB −AC) ∈ N k,B for all k ∈ N and for all j = 1, . . . , n. By Lemma 3.1, the hyperbolicity of B implies that Thus e j (CB − AC) = 0 for j = 1, . . . , n. Therefore AC = CB, and so A and B are conjugate. This completes this alternative proof of (b) implies (a) in Theorem 6.2. When similar hyperbolic A, B ∈ GL(n, Z) are nonconjugate but strongly BFequivalent, it is not the lack of a topological R-module isomorphism between G A and G B that is responsible for lack of conjugacy between A and B. By Theorem 5.1 there exists at least one topological R-module isomorphism Ψ : G A → G B . By part (b) of Theorem 6.2, the failure of Ψ(ι A (Z n )) = ι B (Z n ) for every topological R-module isomorphism Ψ : G A → G B is responsible for the lack of conjugacy between A and B. For instance, there is no way to take any topological R-module isomorphism from G A to G B and, by adjusting G A and G B by topological R-module automorphisms, achieve a topological R-module isomorphism Ψ : G A → G B such that Ψ(ι A (Z n )) = ι B (Z n ). By part (c) of Theorem 6.2, similar statements hold regarding the failure of Ψ ι A (Z n ) ∩ ι B (Z n ) to be R-module isomorphic to the right Z[B]-module Z n .
The R-submodule ∆ Ψ of ι B (Z n ) defined in the proof of part (c) of Theorem 6.2 plays the key role in the characterization of conjugacy for strongly BF-equivalent similar hyperbolic A, B ∈ GL(n, Z). Since Ψ(ι A (Z n )) = Ψ(ι A (θ −1 A (H ′ A ))) and ι B (Z n ) = ι B (θ −1 B (H ′ B )) as R-modules, then ∆ Ψ = Ψ(ι A (θ −1 A (H ′ A ))) ∩ ι B (θ −1 B (H ′ B )). The characterizations of conjugacy given Theorem 6.2 imply that strongly BFequivalent similar hyperbolic A, B ∈ GL(n, Z) are conjugate if and only if there exists a topological R-module isomorphism Ψ : G A → G B such that either Thus the embedded copies of H ′ A and H ′ B in G A and G B respectively, determine the conjugacy of A with B or the lack thereof.

Irreducible Hyperbolic Toral Automorphisms
No assumption is made about the irreducibility of the characteristic polynomial of the similar hyperbolic GL(n, Z) matrices in Theorem 5.1 or Theorem 6.2. A hyperbolic toral automorphism A ∈ GL(n, Z) is irreducible if its characteristic polynomial is irreducible. For a topological R-module isomorphism Ψ : G A → G B , more can be deduced about the nature of the intersection of Ψ(ι A (Z n ) with ι B (Z n ) when A and B have the same irreducible characteristic polynomial.
Proof. Suppose Ψ : G A → G B is a topological R-module isomorphism such that Since ι A (Z n ) is an R-submodule of G A and Ψ is an R-module isomorphism, it follows that Ψ ι A (Z n ) = Ψ Γ A ι A (Z n ) = Γ B Ψ ι A (Z n ) , and so Ψ ι A (Z n ) is an R-submodule of G B . Since ι B (Z n ) is an R-submodule of G B , it follows that ∆ Ψ is an R-submodule of G B . The inclusion of R-submodules ∆ Ψ < ι B (Z n ) and the conjugacy Γ B ι B = ι B B imply that ι −1 B (∆ Ψ ) is a B-invariant subgroup of Z n : . Let S be the nontrivial subspace R n spanned by a set of generators of ι −1 B (∆ Ψ ). This subspace is B-invariant, i.e., B(S) = S, and it projects to a right action T Binvariant subtorus of T n . The irreducibility of the characteristic polynomial of B implies that the only T B -invariant subtori of T n are {0} and T n (see Proposition 3.1 on p. 726 in [8]). Since S is nontrivial, it follows that ι −1 B (∆ Ψ ) contains a basis for R n . Thus ι −1 B (∆ Ψ ) has finite index in Z n , and hence ∆ Ψ has finite index in ι B (Z n ).
When ∆ Ψ = {0}, then by Theorem 7.1, ∆ Ψ has finite index within ι B (Z n ) and so ∆ Ψ is group-theoretically isomorphic to Z n . By part (c) of Theorem 6.2, only when ∆ Ψ is R-module isomorphic to the right Z[B]-module Z n can we conclude that A and B are conjugate. For any irreducible similar hyperbolic strongly BFequivalent A and B that are not conjugate, such as in Example 4.3, any topological R-module isomorphism Ψ : G A → G B has either ∆ Ψ = {0} or it has finite index in ι B (Z n ), but there is no topological R-module isomorphism Ψ : G A → G B with ∆ Ψ = Ψ(ι A (θ −1 A (H ′ A ))) ∩ ι B (θ −1 B (H ′ B )) being R-module isomorphic H ′ B , and thus the embedded copies of H ′ A and H ′ B in G A and G B respectively, distinguish A and B dynamically.