@ Appl. Gen. Topol. 22, no. 2 (2021), 321-330doi:10.4995/agt.2021.14589 © AGT, UPV, 2021 Periodic points of solenoidal automorphisms in terms of inverse limits Sharan Gopal and Faiz Imam Department of Mathematics, BITS-Pilani, Hyderabad Campus, India. (sharanraghu@gmail.com, mefaizy@gmail.com) Communicated by F. Balibrea Abstract In this paper, we describe the periodic points of automorphisms of a one dimensional solenoid, considering it as the inverse limit, lim ← k (S1, γk) of a sequence (γk) of maps on the circle S 1 . The periodic points are discussed for a class of automorphisms on some higher dimensional solenoids also. 2010 MSC: 54H20; 37C25; 11B50; 37P35; 22C05. Keywords: solenoid; periodic points; inverse limits; Pontryagin dual. 1. Introduction We consider a dynamical system of the form (G,f), where G is a compact group and f is an automorphism of G. The study of dynamics involves the eventual behaviour of trajectories of its points i.e., the sequences (fn(x))∞n=0, where x ∈ G and fn = f ◦ f ◦ ... ◦ f (n times for n ∈ N) and f0 is the identity map on G. A point x ∈ G is said to be periodic with a period n if there is an n ∈ N such that fn(x) = x. The sets of periods and periodic points of a family of dynamical systems are well studied in literature. See for instance, [3], [6], [7], [8], [12], [17] and [18]. In this paper, we give a description of the sets of periodic points of automorphisms of a one dimensional solenoid i.e., we describe the set P(f) = {x ∈ Σ : x is a periodic point of f}, where Σ is a one Received 07 November 2020 – Accepted 29 May 2021 http://dx.doi.org/10.4995/agt.2021.14589 S. Gopal and F. Imam dimensional solenoid and f is an automorphism of Σ. It is then extended to some automorphisms of certain higher dimensional solenoids. Solenoids are extensively studied in literature. Some of the papers consider solenoids as dual groups of subgroups of Qn,n ∈ N, while others consider them as inverse limits of certain maps on the n-dimensional torus, Tn. In [19], it is shown that an ergodic automorphism of a solenoid is measure theoretically isomorphic to a Bernoulli shift. The papers [1], [5] and [14] discuss about the structure of a solenoid, whereas [13] describes the structure of group of automorphisms of a solenoid. The articles [2] and [11] calculate the entropy and the zeta function respectively, for an automorphism of a solenoid. The papers [9] and [10] consider the flows on higher dimensional solenoids. We use results from [9] to describe the sets of periodic points of some automorphisms on certain higher dimensional solenoids. There are articles on counting the number of periodic points of a dynamical system; this forms a crucial part in defining the zeta function. The number of periodic points of any given period for some continuous homomorphisms of a one dimensional solenoid was discussed in [15]. We show that our description of periodic points of one dimensional solenoidal automorphisms is in accordance with this result. A characterization of sets of periodic points for automorphisms of a one dimensional solenoid was given in [12], where it was described in terms of adeles, a number theoretic concept. It was based on a description of subgroups of Q. It may however be noted that this characterization for one dimensional solenoids may not be extended to higher dimensions with similar ideas, as there is no neat description of subgroups of Qn for n > 1. We now describe the sets of periodic points, again for automorphisms of one dimensional solenoids in a different manner, namely in terms of inverse limits and then use it to describe the periodic points of some automorphisms on certain higher dimensional solenoids that are inverse limits of sequences of maps on Tn,n > 1. In all these cases, we show that the set of periodic points of a given period is the inverse limit of the same maps (that define the solenoid) restricted to a subgroup of Tn. This may help in giving a characterization of periodic points for automorphisms of other higher dimensional solenoids also. 1.1. Definitions and Notations. A solenoid is a compact connected finite dimensional abelian group. An equivalent interpretation is that, a topological group Σ is an n−dimensional solenoid if and only if its Pontryagin dual Σ̂ is (isomorphic to) a subgroup of the discrete additive group Qn and contains Zn (see [16]). Thus, a one dimensional solenoid is a topological group whose dual is a subgroup of Q and contains Z. Definition 1.2 gives an equivalent description of a one dimensional solenoid Σ as the inverse limit of a sequence of maps on the circle S1, where Z $ Σ̂ ⊆ Q. In section 3, we consider n- dimensional solenoids, for n > 1, which are inverse limits of sequences of maps on Tn. We use the notations Z, N, N0 and P to denote the sets of integers, positive integers, non-negative integers and the prime numbers respectively. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 322 Periodic points of solenoidal automorphisms in terms of inverse limits For a,b ∈ Z, we use the notations a | b, if a divides b; else, we write a 6 | b. To represent a sequence (a1,a2, · · · ) of positive integers, besides the customary notations (ak) ∞ k=1 and (ak), we also use a single capital letter. For instance, we write A = (ak) = (ak) ∞ k=1 = (a1,a2, · · · ). We write diag [m1,m2, · · · ,mn] for an n × n diagonal matrix with m1,m2, · · · ,mn on the principal diagonal. Definition 1.1. Let Xk be a topological space for each k ∈ N0 and fk : Xk → Xk−1 be a continuous map for each k ∈ N. Then the subspace of ∞∏ k=0 Xk defined as lim ← k (Xk,fk) = {(xk) ∈ ∞∏ k=0 Xk : xk−1 = fk(xk),∀k ∈ N} is called the inverse limit of the sequence of maps (fk). Definition 1.2. Let A = (a1,a2, · · · ) be a sequence of integers such that ak ≥ 2 for every k ∈ N. The solenoid corresponding to the sequence A, denoted by ΣA, is defined as ΣA = {(xk) ∈ (S 1)(N0) : xk−1 = akxk (mod1) for every k ∈ N}. In other words, the one dimensional solenoid ΣA is the inverse limit, lim ← k (S1,γk), where γk : S 1 → S1 is defined as γk(x) = akx(mod1). 2. One Dimensional Solenoids The descriptions of a one dimensional solenoid as an inverse limit and as the dual group of a subgroup of Q are very closely related. The dual of a one dimensional solenoid ΣA, where A = (ak) is isomorphic to the subgroup of Q generated by { 1 a1a2···ak : k ∈ N}. Now, a subgroup of Q is characterized by a sequence, called the height sequence, indexed by prime numbers and with values in N0 ∪ {∞}. We will now discuss about this sequence and establish a relation between the terms of this sequence and the integers ak’s. One may refer to [4] for more details about the structure of subgroups of Q. Let S ⊆ Q and x ∈ S. For a p ∈ P , the p−height of x with respect to S, denoted by h (S) p (x) is defined as the largest non-negative integer n, if it exists, such that x pn ∈ S; otherwise, define h (S) p (x) = ∞. Thus, we have a sequence (h (S) p (x)), p ranging over prime numbers in the usual order, with values in N0 ∪ {∞}. We call such sequences as height sequences. If (up) and (vp) are two height sequences such that up = vp for all but finitely many primes and up = ∞ ⇔ vp = ∞, then they are said to be equivalent. If S is a subgroup of Q, then there is a unique height sequence (up to equivalence) associated to all non-zero elements of S. Also, two subgroups of Q are isomorphic if and only if their associated height sequences are equivalent. Given a subgroup S of Q, for every p ∈ P , we assign an element n(S)p of N0 ∪ {∞} as follows. Let Qp and Zp denote the field of p−adic numbers and the ring of p−adic integers respectively and |u|p denote the p−adic norm of u ∈ Qp. Then define n (S) p = sup{h (S) p (x) : x ∈ S ∩ Z∗p}, where Z ∗ p is the multiplicative group {x ∈ Zp : |x|p = 1}. Now, the information whether n (S) p © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 323 S. Gopal and F. Imam is finite or not, for a given p, is going to play a crucial role in our discussion. So, we define D (S) ∞ = {p ∈ P : n (S) p = ∞}. We will use the notations n (S) p and D (S) ∞ , as defined here, throughout this paper. We now have the following relation between the sequences (n (S) p ) and A, where S is the dual of ΣA. Proposition 2.1. Let ΣA be a one dimensional solenoid and S = Σ̂A, where A = (ak). Let p ∈ P and np = n (S) p . Then, (1) p ∈ D (S) ∞ if and only if for every j ∈ N, there exists a k ∈ N such that pj|a1a2 · · ·ak. (2) If p /∈ D (S) ∞ , then np is the largest non-negative integer such that pnp|a1a2 · · ·ak for some k. Proof. (1) Suppose p ∈ D (S) ∞ . Since np = ∞, for any j ∈ N, there exists an x ∈ S ∩ Z∗p with h (S) p (x) > j. Now, x ∈ Z∗p implies that x = a b , where a,b ∈ Z and (a,p) = (b,p) = 1. Also, h(S)p (x) > j implies that xpj = a pj b ∈ S. But, S = { i a1a2···ak : i ∈ Z,k ∈ N } . Thus, a pj b = i a1a2···ak for some i ∈ Z and k ∈ N. Then, we have aa1a2 · · ·ak = ipjb implying that pj|a1a2 · · ·ak. For the converse, let j ∈ N. Then, there exists a k ∈ N such that a1a2 · · ·ak = p ji for some i ∈ N. This implies that 1 pj = i a1a2···ak ∈ S and thus h (S) p (1) ≥ j. Since j is chosen arbitrarily and 1 ∈ S ∩ Z∗p, we get, np = ∞ i.e., p ∈ D (S) ∞ . (2) Suppose p /∈ D (S) ∞ . Then, np = max{h (S) p (x) : x ∈ S ∩ Z∗p}. Say np = h (S) p (x0) for some x0 ∈ S ∩ Z∗p i.e., x0 pnp ∈ S. Let x0 = u0 v0 , for some u0,v0 ∈ Z. Then (u0,p) = (v0,p) = 1. Now, x0 pnp ∈ S implies that u0 pnp v0 = i a1a2···ak for some i ∈ Z and k ∈ N i.e, u0a1a2 · · ·ak = ipnpv0 and hence pnp|a1a2 · · ·ak. If possible, let l > np such that p l|a1a2 · · ·aj for some j. But then, a1a2 · · ·aj = p li′ for some i′ ∈ N implying that 1 pl = i ′ a1a2···aj ∈ S and thus h (S) p (1) ≥ l > np which is a contradiction. Therefore, np is the largest integer such that pnp|a1a2 · · ·ak for some k. � The following corollary follows from the above proposition. Corollary 2.2. Let ΣA, S and D (S) ∞ be defined as above. Then, for a p ∈ P, p ∈ D (S) ∞ if and only if p divides infinitely many ak’s. If f is an automorphism of a one dimensional solenoid Σ, then its dual is an automorphism of a subgroup of Q and thus, it is multiplication by a non-zero rational number, say α β and for any (xk) ∈ Σ, f((xk)) = ( α β xk(mod1)). We say that f is induced by α β . It is known that f is ergodic if and only if α β 6= ±1. Further, we can assume that A = (βbk), where each bk is a positive integer © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 324 Periodic points of solenoidal automorphisms in terms of inverse limits coprime to β. In this case, we can write f((xk)) = (αb1x1,αb2x2, ...) for each (xk) ∈ Σ(βbk). See [19] for all these details about automorphisms. We now state and prove our main results, namely the description of periodic points (Theorem 2.5) and the number of periodic points (Theorem 2.7) of an automorphism of a one dimensional solenoid. Before that, the following proposition describes the elements of a one dimensional solenoid with rational coordinates, in terms of the prime factors of ak’s and the succeeding proposition shows that a periodic point should have only rational coordinates. Proposition 2.3. Let ΣA be a one dimensional solenoid where A = (ak) and (xk) = ( uk vk ) ∈ ΣA ∩ QN0, where uk,vk ∈ Z such that (uk,vk) = 1. For a p ∈ P, denote |vk|p = 1 pck , for every k ≥ 0 and let |ak|p = 1 pdk , for every k ≥ 1. If h is the least integer such that ch > 0, then ck = ch + dh+1 + dh+2 + · · · + dk, for every k > h. Proof. It follows from the definition of a one dimensional solenoid that uh vh = ah+1ah+2 · · ·ak uk vk + j for some j ∈ Z. Since ch > 0, it follows that (uh,p) = (uk,p) = 1. Then, we can find positive integers a ′ h+1,a ′ h+2, · · · ,a ′ k,v ′ k and v ′ h, each of which is coprime to p, such that uh pch v ′ h = p dh+1+dh+2+···dk a ′ h+1a ′ h+2···a ′ kuk pck v ′ k + j ⇒ pckv ′ kuh = p ch+dh+1+···dkv ′ ha ′ h+1 · · ·a ′ kuk + jp ck+chv ′ kv ′ h ⇒ pck ( v ′ kuh − p chjv ′ kv ′ h ) = pch+dh+1+···dkv ′ ha ′ h+1 · · ·a ′ kuk Now, since ch > 0, p does not divide ( v ′ kuh − p chjv ′ kv ′ h ) . Thus, ck = ch + dh+1 + · · ·dk for every k > h. � Proposition 2.4. Let ΣA be a one dimensional solenoid and S = Σ̂A, where A = (ak). If (xk) is periodic in (ΣA,φ), where φ is an automorphism of ΣA induced by α β , then xk ∈ Q for every k ∈ N0. Further, for any p ∈ D (S) ∞ , we have |xk|p ≤ 1 for every k ∈ N0. Proof. Say φl ((xk)) = (xk) for some l ∈ N. Then, for any k ∈ N0, αl βl xk = xk + jk for some jk ∈ Z and thus xk ∈ Q. Let xk = uk vk , where uk, vk ∈ Z and (uk,vk) = 1. Then, (α l − βl)uk = β lvkjk for every k ≥ 0. For a prime number p, let us now denote |vk|p = 1 pck , for every k ≥ 0 and |ak|p = 1 pdk , for every k ≥ 1. Let p ∈ D (S) ∞ . Then, by Corollary 2.2, p|ak for infinitely many k and thus dk > 0 for infinitely many k. Suppose there exists an r ∈ N0 such that p|vr. Then, cr > 0 and (α l − βl)ur = β lvrjr implies that p cr|(αl − βl). Now from Proposition 2.3, cr+k = ch+dh+1+· · ·+dr+dr+1+· · ·+dr+k, where h is the least integer such that ch > 0. Then, h ≤ r and cr+k = cr +dr+1 +dr+2 +· · ·+dr+k. Again, since (αl−βl)ur+k = α lvr+kjr+k for every k ≥ 0, we get p cr+dr+1+···+dr+k| © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 325 S. Gopal and F. Imam (αl − βl). This is a contradiction, as infinitely many of dr+1,dr+2, · · · are non- zero. Hence, p 6 | vk for any k. Therefore, |xk|p ≤ 1 for every k ≥ 0. � In the following theorem about the set of periodic points of the dynamical system(Σ(ak), α β ), we assume that ak = βbk, where each bk is a positive integer coprime to β. As noted already, there is no loss of generality in assuming this (see [19]). Theorem 2.5. Let φ be an automorphism of a one dimensional solenoid ΣA induced by α β , where A = (βbk), each bk being co-prime to β. For each l ∈ N, define Ul = ⋂ p∈P ( 1 p ep,l Zp ∩ Q ∩ S1 ) , where pep,l = 1 |αl−βl|p . If γk,l : Ul → Ul is the map defined as γk,l(x) = βbkx(mod 1) for each k ∈ N and l ∈ N, then P(φ) = ∞⋃ l=1 lim ← k (Ul,γk,l). Proof. Let (xk) be a periodic point with a period l. Then, xk ∈ Q for every k ≥ 0; say xk = uk vk , where uk, vk ∈ Z such that (uk,vk) = 1. Again, for every prime p, let |vk|p = 1 pck , for every k ≥ 0. Now, φl ((xk)) = (xk) implies that (αl − βl)uk = β lvkjk for some jk ∈ Z. Since pck|vk, it follows that pck|(αl − βl) and thus ck ≤ ep,l. We can now write xk = 1 p ep,l .p ep,l−ck .uk v ′ k , for some v ′ k ∈ Z such that (v ′ k,p) = 1. It then follows that xk ∈ 1 p ep,l Zp, because |p ep,l−ck .uk v ′ k |p ≤ 1 p ep,l−ck ≤ 1. Since p was chosen arbitrarily, we conclude that xk ∈ Ul, for every k ≥ 0. On the other hand, let (xk) ∈ lim ← k (Ul,γk,l) for some l ∈ N. Say xk = uk vk , where uk, vk ∈ Z such that (uk,vk) = 1. Write vk = ∏ p|vk pcp, for some cp ∈ N. Then, for any p|vk, |xk|p = p cp. Also, |xk|p ≤ p ep,l, for any p ∈ P . Thus, cp ≤ ep,l and hence vk|(α l − βl). Therefore, α l−βl vk ∈ Z, for every k. Then, φl((xk)) − (xk) = ( (αl − βl)bk+1bk+2...bk+lxk+l ) = (0) implying that (xk) is periodic. � Remark 2.6. The set of periodic points of period l is equal to lim ← k (Ul,γk,l). Here Ul is a subgroup of S 1 and the map γk,l is the restriction of γk to Ul, where γk is a map on S 1 such that Σ(βbk) = lim← k (S1,γk). The following theorem about the number of periodic points, which follows from the above description, is in accordance with a similar result in [15]. Theorem 2.7. Let φ be an automorphism of a one dimensional solenoid ΣA induced by α β and for every l ∈ N, let ep,l = 1 |αl−βl|p . Then the number of periodic points of φ with a period l is ∏ p/∈D (S) ∞ pep,l. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 326 Periodic points of solenoidal automorphisms in terms of inverse limits Proof. Since αl − βl ∈ Z, ep,l is positive only for finitely many primes. Thus, there is a finite subset F of P \ D (S) ∞ such that for a p /∈ D (S) ∞ , ep,l 6= 0 if and only if p ∈ F . Therefore ∏ p/∈D (S) ∞ pep,l = ∏ p∈F pep,l. We first claim that (xk) is periodic with a period l if and only if for every k ∈ N0, xk = uk vk , where uk, vk ∈ Z, 0 ≤ uk < vk and vk = ∏ p∈F pfp,k with 0 ≤ fp,k ≤ ep,l. If φl ((xk)) = (xk), then for every k ∈ N0, xk ∈ 1 p ep,l Zp ∩ Q, for every p ∈ P . Let xk = uk vk for some uk, vk ∈ Z such that (uk,vk) = 1. Now, xk ∈ 1 p ep,l Zp implies that |xk|p ≤ p ep,l, for every p. From Proposition 2.4, if p ∈ D (S) ∞ , then p 6 | vk. Also, for a prime p not in F , ep,l = 0 implies that p 6 | vk. Thus, the prime factorisation of vk = ∏ p∈F pfp,k for some 0 ≤ fp,k ≤ ep,l. Since xk ∈ [0,1), we conclude that 0 ≤ uk < vk. Conversely, if xk = uk vk , where uk and vk satisfy the given conditions, then |xk|p ≤ 1, for p /∈ F and |xk|p ≤ p fp,k for p ∈ F . In any case |xk|p ≤ p ep,l and thus xk ∈ Ul. Hence the claim follows. For a p ∈ F , let |ak|p = 1 pdk , for every k ∈ N. As this dk depends on p we will denote dk = d (p) k . Again, there are at most finitely many k ∈ N for which d (p) k > 0, as F ⊆ P \ D (S) ∞ ; let these positive integers be denoted by d (p) k1 ,d (p) k2 , ...,d (p) kα(p) , where α(p) ∈ N0. Further, assume that k1 < k2 < ... < kα(p). Let K = max{kα(p) : p ∈ F}, if kα(p) > 0 for at least some p ∈ F ; otherwise, define K = 0. Then, d (p) k = 0 for every k > K and for every p ∈ F . Let (xk) ∈ ΣA be periodic; say xk = uk vk , where uk, vk ∈ Z such that (uk,vk) = 1. We have xK = uK vK , where 0 ≤ uK < vK and vK = ∏ p∈F pfp,K with 0 ≤ fp,K ≤ ep,l. For any k < K, the value of xk is uniquely determined by xK, as xk = ak+1ak+2...aKxK (mod 1). Now, let k > K. It follows from Proposition 2.3 that vk = vK. Also, xK = aK+1...akxk (mod 1) i.e., uK vK = aK+1...ak uk vk + j for some j ∈ Z. By denoting aK+1...ak = gk and using the fact that vk = vK, we have uK vK = gk uk vK +j. Since d (p) k = 0 for any k > K and every p ∈ F , it follows that p 6 | gk for any p ∈ F . Having defined uK vK , the distinct possible values for uk vK are uk vK = uK gkvK − j gk , where j ∈ {0,1, ...,gk −1}. Consider two such values, say u (1) k vK = uK gkvK − j1 gk and u (2) k vK = uK gkvK − j2 gk for some j1, j2 ∈ {0,1, ...gk − 1}. Then, u (1) k −u (2) k vK = j2−j1 gk and thus gk ( u (1) k − u (2) k ) = vK (j2 − j1). Now, if j1 6= j2, then |j2 − j1| < gk and thus gk 6 | (j2 − j1). But then, there will be a prime p such that p | gk and p | vK. On one hand, p | gk implies that p /∈ F . On the other hand, p | vK implies that p | αl − βl and also p /∈ D (S) ∞ , which means that p ∈ F leading © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 327 S. Gopal and F. Imam to a contradiction. Hence, j1 = j2 i.e., u (1) k = u (2) k . Thus, there is only one possible value for xk. Thus, a periodic point (xk) is uniquely determined by the coordinate xK. Now, since 0 ≤ fp,K ≤ ep,l, the possible values of xK are i∏ p∈F p ep,l , where 0 ≤ i < ∏ p∈F pep,l. Thus, the theorem follows. � 3. n-dimensional solenoids We now extend our result about periodic points to some automorphisms of certain higher dimensional solenoids. Though this seems to be a small class, the reason for considering it is that the result follows immediately from what we have shown for one dimensional case. The higher dimensional solenoids that we are going to consider are isomorphic to products of one dimensional solenoids, as described in [9]. We mention here some notations, definitions and results from this paper that are needed to discuss our result. For a positive integer n > 1, let πn : Rn → Tn be the homomorphism defined as πn((x1,x2, ...,xn)) = (x1 (mod 1),x2 (mod 1), ...,xn (mod 1)). Let M = (Mk) ∞ k=1 = (M1,M2, ...) be a sequence of n × n matrices with integer entries and non-zero determinant. Then, the n−dimensional solenoid ∑ M is defined as ∑ M = {(xk) ∈ (T n)N0 : πn(Mkxk) = xk−1 for every k ∈ N}. In other words, ∑ M = lim← k (Tn,δk), where δk : Tn → Tn is defined as δk(x) = πn(Mkx) If φ is an automorphism of ∑ M, then there is a matrix L ∈ GL(n,Q) such that φ((xk)) = (π n(Lxk)). We say that φ is induced by the matrix L. Now, consider n sequences of positive integers A1 = (a 1 1,a 1 2, ...), A2 = (a 2 1,a 2 2, ...), ...... An = (a n 1 ,a n 2 , ...). Then define the sequence M = (Mk) of matrices as Mk = diag[a 1 k,a 2 k, ...,a n k]. These sequences of positive integers and matrices give rise to n one-dimensional solenoids and an n−dimensional solenoid. The following lemma from [9] gives a connection between these. Lemma 3.1. The map η : ∏n i=1 ∑ Ai → ∑ M given by η((x 1 k) ∞ k=1,(x 2 k) ∞ k=1, . . . . . . ,(xnk ) ∞ k=1) = ((x 1 1,x 2 1, . . . ,x n 1 ),(x 1 2,x 2 2, . . . ,x n 2 ), . . . ,(x 1 k,x 2 k, . . . ,x n k), . . .) is a topological isomorphism. We reserve these symbols Ai, i = 1,2, ...,n for the sequences of posi- tive integers and Mk, k ∈ N for the corresponding diagonal matrices as de- scribed above. Now, let φ be an automorphism of ∑ M induced by a di- agonal matrix, say D = diag[α1 β1 , α2 β2 , ..., αn βn ]. Then for each i, αi βi induces an automorphism of the one dimensional solenoid ∑ Ai , say ψi. Again, by following [19], we assume that Ai = (βib i k) for some suitable sequence (b i k) of positive integers. Then, the map ψ : ((x1k) ∞ k=1.(x 2 k) ∞ k=1, . . . ,(x n k ) ∞ k=1) 7→ (ψ1((x 1 k) ∞ k=1),ψ2((x 2 k) ∞ k=1), . . . ,ψn((x n k ) ∞ k=1)) is an automorphism of ∏n i=1 ∑ Ai . It is easy to see that η ◦ ψ = φ ◦ η. Thus, we have the following proposition. Proposition 3.2. ( ∏n i=1 ∑ Ai ,ψ) is conjugate to ( ∑ M,φ). We now state and prove a theorem regarding the periodic points. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 2 328 Periodic points of solenoidal automorphisms in terms of inverse limits Theorem 3.3. For each l ∈ N, define Vl = ∏n i=1 ( ⋂ p∈P ( 1 p ep,l,i Zp ∩ Q ∩ S1 )) , where pep,l,i = 1 |αl i −βl i |p . If δk,l : Vl → Vl is the map defined as δk,l(x) = πn(Mkx) for each k ∈ N and l ∈ N, then P(φ) = ∞⋃ l=1 lim ← k (Vl,δk,l). Proof. Let Pl(φ) and Pl(ψ) be the sets of periodic points of φ and ψ respec- tively, with a period l ∈ N. Since η is a conjugacy from ( ∏n i=1 ∑ Ai ,ψ) to ( ∑ M,φ), it follows that Pl(φ) = η (Pl(ψ)). But Pl(ψ) = ∏n i=1 Pl(ψi), where ψi is the automorphism of ∑ Ai induced by αi βi . Thus by Theorem 2.5, Pl(ψ) = ∏n i=1 { (xik) ∞ k=1 ∈ ∑ Ai : xik ∈ Q and |x i k|p ≤ 1 p ep,l,i for every p ∈ P } . Then, Pl(φ) = {( (x1k,x 2 k, · · ·x n k ) )∞ k=1 ∈ ∑ M : x i k ∈ Q and |x i k|p ≤ 1 p ep,l,i for every p ∈ P } = lim ← k (Vl,δk,l). Thus, P(φ) = ∞⋃ l=1 lim ← k (Vl,δk,l). � Remark 3.4. The set of periodic points of φ with a period l is equal to lim ← k (Vl,δk,l). Here, Vl is a subgroup of Tn and δk,l is the restriction of δk to Vl, where each δk is a map on Tn such that ∑ M = lim← k (Tn,δk). 4. Conclusion The periodic points of an automorphism of a one dimensional solenoid are described here. There are papers that discuss the number of periodic points or in general the zeta function of such automorphisms, whereas this paper gives an explicit description of these points. The paper [12], on the other hand, describes the sets of periodic points using adeles, but these ideas may not be useful for higher dimensional solenoids. Here, we have extended this result to certain automorphisms of higher dimensional solenoids also. Hence, the present description in terms of inverse limits may be helpful in more general cases. Acknowledgements. 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