Periodic points of solenoidal automorphisms in terms of adeles

Abstract

This article is about the periodic points characterization of automorphisms of some solenoids, whose duals are subgroups of algebraic number fields. Here, we use the theory of adeles for describing a solenoid and the periodic points of its automorphisms. This is in line with the earlier characterizations which considered other classes of solenoids.

1 Introduction

This article is third in the series of articles by the second author on the study of periodic points of solenoidal automorphisms. The first article [10] characterizes the sets of periodic points of one dimensional solenoids and full solenoids of arbitrary dimension. The second one [9] was about certain higher dimensional solenoids and the characterization was done in terms of inverse limits. In the present article, we describe certain higher dimensional solenoids and the sets of periodic points of automorphisms on them in terms of adeles.

The study of periodicity in a dynamical system, particularly the characterizations of the sets of periods and periodic points have been well studied problems. The paper [15] is one of the earliest articles in this direction. Later, there are several papers like [3,4,5,6, 16] that characterize the sets of periods or sets of periodic points. Periodic points also lead to the notion of zeta function, which actually counts the number of periodic points of a given period and this is also an invariant of dynamical systems.

On the other hand, solenoids are also considered by several people in literature like [1, 2, 7, 8, 12]. Given the rich topological as well as algebraic attributes possessed by a solenoid, it is an important object to study. The author in [18] describes the automorphisms of a solenoid. Solenoids find a significant place in the book [14] on algebraic dynamical systems. Richard Miles [13] found the zeta function for solenoidal automorphisms.

A solenoid being a compact connected finite dimensional abelian group, can be studied using both the topological and as well as algebraic techniques. In this article, we use a number theoretic concept namely adeles to study solenoids. We depend on [11] and [17] for terminology and several results about these adeles.

In the next section, we define several terms required for this article and also state the results of earlier characterizations of certain solenoids and the periodic points of their automorphisms. Then we have a section stating and proving the main results. The first theorem describes a solenoid in terms of adeles and the second gives the set of periodic points of some automorphisms on it. These automorphisms of a solenoid are induced by matrices from $GL_n(\mathbb {Q})$ which also belong to $GL_n(p^b \mathbb {Z}_p)$ for some particular primes p that depend on the given solenoid.

2 Preliminaries

A dynamical system is a pair (X, f), where X is a topological group and f is an automorphism of X. A point $x \in X$ is said to be periodic with a period $n \in \mathbb {N}$, if $f^n(x) = x$, where $f^n = f \circ f \circ ... \circ f$ (n times). We denote by P(f), the set of all periodic points of (X, f).

A solenoid is defined as a compact connected finite dimensional abelian group. Equivalently, an $n-$dimensional solenoid $\Sigma$ is an abelian group whose Pontryagin dual $\widehat{\Sigma }$ is an (additive) subgroup of $\mathbb {Q}^n$ and contains $\mathbb {Z}^n$. If $\widehat{\Sigma }=\mathbb {Z}^n$, then $\Sigma$ is the $n-$dimensional torus and on the other hand when $\widehat{\Sigma }=\mathbb {Q}^n$, we call it a full solenoid. A solenoid can also be described as an inverse limit. For an integer $n>1$, let $\kappa : \mathbb {R}^n \rightarrow \mathbb {T}^n$ be the homomorphism defined as $\kappa ((x_1,x_2,...,x_n)) = (x_1~(mod~1), x_2~(mod~1),..., x_n~(mod~1))$, where $\mathbb {T}=\mathbb {R}/\mathbb {Z}$ is the unit circle. Let $\overline{M} = (M_k)_{k=1}^{\infty } = (M_1, M_2,... )$ be a sequence of $n \times n$ matrices with integer entries and non-zero determinant. Then, the $n-$dimensional solenoid $\sum _{\overline{M}}$ is defined as $\sum _{\overline{M}} = \{(\hbox {{\textbf {x}}}_k) \in (\mathbb {T}^n)^{\mathbb {N}_0}: \kappa (M_k \hbox {{\textbf {x}}}_k) = \hbox {{\textbf {x}}}_{k-1}$ for every $k \in \mathbb {N}\}$.

We also refer to [14] for another suitable description of higher dimensional solenoids in terms of an algebraic notion. Dynamical systems defined with the help of $R-$modules, where $R=\mathbb {Z}[u^{\pm 1}]$, are discussed in this book. It is shown that the dual of the $R-$module, R/ (f), where $R=\mathbb {Z}[u^{\pm 1}]$ and (f) is a prime ideal in R, is isomorphic to a $k-$dimensional solenoid, provided that there is an $n \in \mathbb {Z}$ such that $u^n f(u) = c_0 + c_1u +... +c_ku^k$ with $|c_0c_k|>1$. In fact, if $\alpha$ is an automorphism of a compact group X, then the dynamics of $\alpha$ viewed as a $\mathbb {Z}$-action on X gives an R-module. If this module is $R/ \mathfrak {P}$, where $\mathfrak {P}$ is a prime ideal, then X is a solenoid if and only if $\mathfrak {P}=\{f \in R| f(c)=0 \}$, for some $c \in \mathbb {Q}-\{0\}.$ Moreover, the $\mathbb {Z}$-action on X becomes the shift action on a subgroup of $\mathbb {T}^{\mathbb {Z}}$.

We will now describe the concept of adeles and we refer to [17] for these. A finite extension of the field of rational numbers $\mathbb {Q}$ is defined as an algebraic number field $\mathbb {K}$. We denote by $P^{\mathbb {K}}$, the set of all places of $\mathbb {K}$, i.e, the equivalence classes of valuations of $\mathbb {K}$ (where two valuations $\phi _1$ and $\phi _2$ are said to be equivalent if there is an $s>0$ such that $\phi _1(r) = \phi _2(r)^s$ for every $r \in \mathbb {K}$). A place is called finite if it contains a non-archimedian valuation and infinite otherwise. The collection of finite places will be denoted by $P_{f}^{\mathbb {K}}$ whereas $P_{\infty }^{\mathbb {K}}$ denotes the set of infinite places. It may be noted that $P_{\infty }^{\mathbb {K}}$ is a finite set. For each $v \in P^{\mathbb {K}}$, $\mathbb {K}_{v}$ denotes the completion of $\mathbb {K}$ with respect to v and $\Re _{v} = \{x \in \mathbb {K}_{v}$: $|x|_{v}\le 1\}$. $\Re _v$ is always a compact subset of $\mathbb {K}_v$ and when $v \in P_{f}^{\mathbb {K}}$, $\Re _{v}$ is an open, unique maximal compact subring of $\mathbb {K}_v$. We also consider $\Re _{v}^{*}:=\left\{ x \in \Re _{v}: \,\, |x|_{v}=1 \right\}$ in our discussion. The adele ring of $\mathbb {K}$, denoted by $\mathbb {K}_{\mathbb {A}}$ is then defined as $\mathbb {K}_{\mathbb {A}}=\left\{ (x_{v})\in \prod _{v \in \mathbb {P}^{\mathbb {K}}}\mathbb {K}_{v}: \,\, x_{v} \in \Re _{v} \text { for all but finitely many } v \in P_{f}^{\mathbb {K}} \ \right\}$.

For the field $\mathbb {Q}$, every finite valuation is equivalent to a $p-$adic valuation where p is a rational prime and an infinite valuation is equivalent to the usual absolute value. Thus we can view $P^{\mathbb {Q}}= \{p \,: \,\, p \text { is a rational prime} \} \cup \{\infty \}$, where $|\,\,\,|_{\infty }$ is the usual absolute value. Now, if $\mathbb {K}$ is an algebraic number field, then for each $p \in P^{\mathbb {Q}}$, there exists finitely many $v \in P^{\mathbb {K}}$ such that v lies above p (denoted as $v|_p$).

If $\Sigma$ is a one-dimensional solenoid, then we have $\mathbb {Z}\subsetneqq \widehat{\Sigma } \subseteq \mathbb {Q}$. Let P denote the set of all rational primes. Let $M \subseteq \mathbb {Q}$ and $x \in M$. For any prime $p \in P$, we define the $p-$height of x as the largest non-negative integer n, if it exists, such that $\frac{x}{p^n} \in M$; otherwise, define $h_p(x) = \infty$. This gives us a sequence $(h_p(x))$, where p varies over P in the natural increasing order and the height sequence takes values from $\mathbb {N}\cup \{0\} \cup \{\infty \}$. There exists an equivalence relation among two height sequences $(u_p)$ and $(v_p)$ defined as $u_p = v_p$ for all but finitely many primes and $u_p = \infty$ if and only if $v_p = \infty$. For any subgroup of rationals, there is a unique sequence (up to the equivalence relation) corresponding to all the non-zero elements of the subgroup. Also, two corresponding height sequences are equivalent if and only if there exists an isomorphism between these two subgroups.

In [10],¹ the solenoid $\Sigma$ and the set of periodic points of an automorphism are described using $\mathbb {Q}_{\mathbb {A}}$ in the following way:

Theorem 1

(Sharan, Raja; 2017) Let $\Sigma$ be a one-dimensional solenoid. Let $n_p:=sup\{h_p(x): x \in \widehat{\Sigma } \cap \mathbb {Z}_p^*\}$ and $D_{\infty } = \{p \in P: n_p = \infty \}$.

Then $\Sigma = \frac{\mathbb {Q}_{\mathbb {A}}}{i(\mathbb {Q})+L}$, where $L = \prod _{p \le \infty } U_p$ and $U_p = \left\{ \begin{array}{ll} (0) &{} \hbox { if }p \in D_{\infty } \cup \{\infty \} \\ p^{n_p} \mathbb {Z}_p &{} \hbox { if }p \notin D_{\infty } \cup \{\infty \} \\ \end{array}\right.$,

where $i: \mathbb {Q}\rightarrow \mathbb {Q}_{\mathbb {A}}$ is the diagonal inclusion given by a constant adele sequence.

Theorem 2

(Sharan, Raja; 2017) Let $\Sigma$, L and $D_{\infty }$ be defined as above. If $\alpha$ is an ergodic automorphism of $\Sigma$, then $P(\alpha ) = \frac{i(\mathbb {Q})+\prod ' \mathbb {Q}_p}{i(\mathbb {Q})+L}$, where $\prod ' \mathbb {Q}_p:=\{x \in \mathbb {Q}_{\mathbb {A}}: x_p=0$ for every $p \in D_{\infty } \cup \{\infty \}$ and $x_p \in p^{n_p}\mathbb {Z}_p$ for all but finitely many p in $P \setminus D_{\infty }\}$.

In this article, we consider a solenoid of an arbitrary dimension n whose dual is an additive subgroup of an algebraic number field $\mathbb {K}$, with $[\mathbb {K}:\mathbb {Q}]=n$. In the rest of this article, we denote a solenoid by $\Sigma$ and an algebraic number field by $\mathbb {K}$. Now consider $\mathbb {K}_{\mathbb {A}}$, the ring of adeles of $\mathbb {K}$. For any $p \in P^{\mathbb {Q}}$, $\mathbb {Z}_p$ can be considered as a subring of $\mathbb {Q}_{\mathbb {A}}$ by identifying $c \in \mathbb {Z}_p$ with $x \in \mathbb {Q}_{\mathbb {A}}$, where $x_p=c$ and $x_q = 0$ for $q \ne p$. Similarly, $\prod _{v|_{p}} \mathbb {K}_{v}$ can be considered as a subring of $\mathbb {K}_{\mathbb {A}}$ by identifying $\prod _{v|_{p}} a_{v} \in \prod _{v|_{p}} \mathbb {K}_{v}$ with $b \in \mathbb {K}_{\mathbb {A}}$, where $b_v=a_v$ for ${v|_{p}}$ and $b_w= 0$ otherwise. From Lemma 6.101 of [11], it follows that there is an isomorphism (of topological groups) $\alpha : \mathbb {K}_{\mathbb {A}} \; \rightarrow (\mathbb {Q}_{\mathbb {A}})^{n}$ such that $\alpha \left( \prod _{v|_{p}} \Re _{v} \right)$ is equal to $\left( \mathbb {Z}_p \right) ^{n}$ for almost all finite p. We further assume that $\alpha \left( \prod _{v|_{p}} \Re _{v} \right) =\left( \mathbb {Z}_p \right) ^{n}$ for all the finite places.

We write $\alpha (x)=\left( x^{(1)},x^{(2)},\cdots ,x^{(n)} \right) \in \left( \mathbb {Q}_{\mathbb {A}} \right) ^{n}$, for each $x \in \mathbb {K}_{\mathbb {A}}$ and write $x^{(j)} = \left( x^{(j)}_{p} \right) _{p \in P^{\mathbb {Q}}}$, for each $x^{(j)}\in \mathbb {Q}_{\mathbb {A}}$. For every $r \in \mathbb {K}$, we write $\beta (r)=\left( r^{(1)},r^{(2)},\cdots ,r^{(n)} \right) \in \mathbb {Q}^{n}$ where $r=\sum _{i=1}^{n} r^{(i)} \alpha _i$ and $\left\{ \alpha _1,\alpha _2, \cdots , \alpha _n \right\}$ is a $\mathbb {Q}$-basis for $\mathbb {K}$. Then, $\beta$ is an isomorphism from $\mathbb {K}$ to $\mathbb {Q}^n$. We further assume that $\beta (\widehat{\Sigma })$ is a $\mathbb {Z}^n$-module and also $\mathbb {Z}^{n} \subseteq \beta (\widehat{\Sigma })$; in particular $\alpha _i \in \beta (\widehat{\Sigma })$ for each $1 \le i \le n$.

For $a=\left( a_v \right) _{v \in P^{\mathbb {K}}} \in \mathbb {K}_{\mathbb {A}}$, let $\overline{a}_p=\prod _{v|_{p}}a_{v} \in \prod _{v|_{p}}\mathbb {K}_{v}, \text { for every } p \in P^{\mathbb {Q}}$. We know that $\prod _{v|_{p}}\mathbb {K}_{v}$ is a vector space over $\mathbb {Q}_p$. It follows from Lemma 6.69 and 6.101 of [11] that the $\mathbb {Q}_p$-coordinates of $\overline{a}_p$ are same as $\left( a^{(1)}_p,a^{(2)}_p,\cdots ,a^{(n)}_p \right)$, where $\left( a^{(1)},a^{(2)},\cdots ,a^{(n)} \right) =\alpha (a)$ and $a^{(j)}=\left( a^{(j)}_q \right) _{q \in P^{\mathbb {Q}}}$.

3 Main results

Consider the map $\eta :\mathbb {Q}_{\mathbb {A}}\rightarrow \widehat{\mathbb {Q}}$ given by $\eta (x)=\eta _x$, where $\eta _x:\mathbb {Q}\rightarrow S^1$ is defined as $\eta _x(r)=e^{-2\pi i x_{\infty }r} \cdot \prod \nolimits _{p<\infty }e^{2\pi i \{x_{p}r\}_p}$ and $x=(x_p)_{p\in P^\mathbb {Q}}$. It is known that this map $\eta$ is a surjective homomorphism. Now, consider the map $\xi :(\mathbb {Q}_{\mathbb {A}})^n\rightarrow \widehat{\mathbb {Q}^n}$ given by $\xi (\bar{x})=\xi _{\bar{x}}$, where $\xi _{\bar{x}}:\mathbb {Q}^n\rightarrow S^1$ is defined as $\xi _{\bar{x}}(\bar{r})=\eta _{x^{(1)}}(r^{(1)}) \cdot \eta _{x^{(2)}}(r^{(2)}) \cdots \eta _{x^{(n)}}(r^{(n)})$, where $\bar{x}=(x^{(1)},x^{(2)},\cdots ,x^{(n)})\in (\mathbb {Q}_{\mathbb {A}})^n$ and $\bar{r}=(r^{(1)},r^{(2)},\cdots ,r^{(n)})\in \mathbb {Q}^n$. Observe that $\xi$ is a homomorphism. Note that $\xi _{\bar{x}}(\bar{r})=e^{-2\pi i\sum \nolimits _{j=1}^{n}x_{\infty }^{(j)}r^{(j)}}\cdot \prod \nolimits _{p<\infty }e^{2\pi i\sum \nolimits _{j=1}^{n}\{x_{p}^{(j)}r^{(j)}\}_p}$. Now, define $\omega : \mathbb {K}_{\mathbb {A}}\rightarrow \widehat{\mathbb {Q}^n}$ as $\omega (a)=\omega _a$, where $\omega _a=\xi \circ \alpha (a)$; in other words, if $a\in \mathbb {K}_{\mathbb {A}}$ and $\alpha (a)=(a^{(1)},a^{(2)},\cdots ,a^{(n)})$, then $\omega _a(\bar{r})=e^{-2\pi i\sum \nolimits _{j=1}^{n}a_{\infty }^{(j)}r^{(j)}}\cdot \prod \nolimits _{p<\infty }e^{2\pi i\sum \nolimits _{j=1}^{n}\{a_{p}^{(j)}r^{(j)}\}_p}$. Since $\xi$ and $\alpha$ are homomorphisms, $\omega$ is also a homomorphism. Finally, define $\psi :\mathbb {K}_{\mathbb {A}}\rightarrow \widehat{\mathbb {K}}$ as $\psi (a)=\psi _{a}, \text { for every } a \in \mathbb {K}_{\mathbb {A}}$, where $\psi _{a}:\mathbb {K}\rightarrow S^1$ is given by $\psi _{a}(r)=\omega _a\circ \beta (r),$ for every $r\in \mathbb {K}$. Note that if $\alpha (a)=(a^{(1)},a^{(2)},\cdots ,a^{(n)})\in (\mathbb {Q}_{\mathbb {A}})^n$ and $\beta (r)=(r^{(1)},r^{(2)},\cdots ,r^{(n)}) \in \mathbb {Q}^n$ then $\psi _{a}(r)= \omega _a\circ \beta (r)= \omega _a (r^{(1)},r^{(2)},\ldots ,r^{(n)})= \xi _{\alpha (a)} (r^{(1)},r^{(2)},\ldots ,r^{(n)})= \xi _{(a^{(1)},a^{(2)},\ldots ,a^{(n)})} (r^{(1)},r^{(2)},\ldots ,r^{(n)})= e^{-2\pi i\sum \nolimits _{j=1}^{n}a_{\infty }^{(j)}r^{(j)}}\cdot \prod \nolimits _{p<\infty }e^{2\pi i\sum \nolimits _{j=1}^{n}\{a_{p}^{(j)}r^{(j)}\}_p}$. Note that $\psi$ is a homomorphism.

Proposition 3

$\psi$ is a surjective homomorphism that is trivial on $i(\mathbb {K})$.

Proof

Let $\chi \in \widehat{\mathbb {K}}$. For each $\, 1\le j\le n$, define $\eta ^{(j)}:\mathbb {Q}\rightarrow S^1$ as $\eta ^{(j)}(b)=\chi \Big (\beta ^{-1}(0,\cdots ,b,\cdots ,0)\Big )$ where b is in ${j}^{th}$ position. Then $\eta ^{(j)}\in \widehat{\mathbb {Q}}$ and hence there exists $x^{(j)}=(x^{(j)}_p) \in \mathbb {Q}_{\mathbb {A}} \text {, such that } \eta ^{(j)}=\eta _{x^{(j)}}$.

$$\begin{aligned}&\text {Then,} \text { for every } r \in \mathbb {K},\\ {}&\psi \Big (\alpha ^{-1}\big (x^{(1)},x^{(2)},\cdots ,x^{(n)}\big )\Big )\Big ( r\Big )\\&\quad = \xi _{(x^{(1)},x^{(2)},\cdots ,x^{(n)})}\Big (r^{(1)},r^{(2)},\cdots ,r^{(n)}\Big )\\&\quad =\eta _{x^{(1)}}\Big (r^{(1)}\Big ).\eta _{x^{(2)}}\Big (r^{(2)}\Big )\cdots \eta _{x^{(n)}}\Big (r^{(n)}\Big )\\&\quad =\chi \bigg (\beta ^{-1}\Big (r^{(1)},0,0,\cdots ,0\Big )\bigg ).\chi \bigg (\beta ^{-1}\Big (0,r^{(2)},0,0,\cdots ,0\Big )\bigg )\\&\qquad \cdots \chi \bigg (\beta ^{-1}\Big (0,0,\cdots ,0,r^{(n)}\Big )\bigg )\\&\quad =\chi \bigg (\beta ^{-1}\Big (r^{(1)},0,0,\cdots ,0\Big )+\beta ^{-1}\Big (0,r^{(2)},0,0,\cdots ,0\Big )\\&\qquad +\cdots +\beta ^{-1}\Big (0,0,\cdots ,0,r^{(n)}\Big )\bigg )\\&\quad = \chi \bigg (\beta ^{-1}\Big (r^{(1)},r^{(2)},\cdots ,r^{(n)}\Big )\bigg )\\&\quad = \chi \Big (r\Big ). \quad \text { Hence, } \psi \text { is surjective.} \end{aligned}$$

$\square$

We now claim that $\psi$ is trivial on $i(\mathbb {K})$. If $a =(a_v )\in i(\mathbb {K})$, then there exists an $x \in \mathbb {K}$ such that $a_v=x \text { for every } v\in P^{\mathbb {K}}$. Now, $\bar{a}_p=(x,x,\cdots ,x) \in \prod _{v|_{p}}\mathbb {K}_{v}$ and the $\mathbb {Q}_{p}-$coordinates of $\bar{a}_p$ are $a^{(1)}_p,a^{(2)}_p, \cdots , a^{(n)}_p$. Since $\bar{a}_p$ is same for all values of p, each sequence $(a^{(j)}_p)_{p\in P^{\mathbb {Q}} }$ is a constant sequence and thus $a^{(j)}\in i(\mathbb {Q})$. Say $(a^{(1)}_p)=(\delta ^{(1)}),\quad (a^{(2)}_p)=(\delta ^{(2)})\quad ,\cdots ,\quad (a^{(n)}_p)=(\delta ^{(n)})$, where $\delta ^{(j)}\in \mathbb {Q}$. Then, for any $r \in \mathbb {K}$ with $\beta (r)=(r^{(1)},r^{(2)},\cdots ,r^{(n)})\in \mathbb {Q}^{n}$, we have $\psi _{(a)}(r)=\xi _{((\delta ^{(1)}),(\delta ^{(2)}),\cdots ,(\delta ^{(n)}))}(r^{(1)},r^{(2)}, \cdots ,r^{(n)})=\prod \nolimits _{j=1}^{n}\eta _{(\delta ^{(j)})}(r^{(j)})$. But, $\eta _{(\delta ^{(j)})}(r^{(j)})=e^{-2\pi i \delta ^{(j)} r^{(j)}}.\prod \nolimits _{p<\infty }e^{2\pi i\{\delta ^{(j)} r^{(j)}\}_p}=1, \text { for every } j$. This is because, for any $s \in \mathbb {Q}$, $s - \sum \nolimits _{{p < \infty }} {\{ s\} _{p} \in \mathbb {Z}}$ and thus $e^{-2\pi i s}.\prod \nolimits _{p<\infty }e^{2\pi i \{s\}_p}=1$. Therefore, $\psi _{(a)}=1, \text { for every } a\in i (\mathbb {K})$. $\square$

Since $\widehat{\Sigma }$ is a subgroup of $\mathbb {K}$, we have $\widehat{\widehat{{\Sigma }}}=\widehat{\mathbb {K}}/ann(\widehat{\Sigma })$ and thus, $\Sigma =\widehat{\mathbb {K}}/ann(\widehat{\Sigma })$. Define $\psi ':\mathbb {K}_{\mathbb {A}}\rightarrow \Sigma$ as $\psi '=\pi \circ \psi$, where $\pi :\widehat{\mathbb {K}}\rightarrow \Sigma$ is the quotient map. Since $\pi$ and $\psi$ are surjective, $\psi '$ is surjective. We will now find Ker $\psi ^{'}$ and thus obtain $\Sigma$ as a quotient of $\mathbb {K}_{\mathbb {A}}$.

For every $p \in P_f^{\mathbb {Q}}$ and $1\le j\le n$, define $m_p^{(j)}=\sup \{|r^{(j)}|_p:\, r \in \widehat{\Sigma } \}$, where $\beta (r)=\left( r^{(1)},r^{(2)},\cdots ,r^{(n)}\right)$. Since $\mathbb {Z}^n \subset \beta (\widehat{\Sigma })$, we have $r=\beta ^{-1}\left( 0,\cdots ,p,\cdots ,0 \right) \in \widehat{\Sigma } \text { and thus } |r^{(j)}|_{p}=|p|_{p}=\frac{1}{p}\ne 0$ concluding that $m^{(j)}_{p}\ne 0.$ Let $n_p^{(j)} = \left\{ \begin{array}{ll} \frac{1}{m_p^{(j)}} &{} \hbox { if } m_p^{(j)}<\infty \\ 0 &{} \hbox { if } m_p^{(j)}=\infty \end{array}\right.$ and

$D=\{p\in P_f^{\mathbb {Q}}: m_p^{(j)}=\infty \text { for every } 1\le j\le n\}$. Now, define a subgroup $U_p$ of $\prod _{v|_{p}}\mathbb {K}_{v}$ for every $p \in P^{\mathbb {Q}}$ as $U_p = \left\{ \begin{array}{ll} (0) &{} \hbox { for } p\in D\cup \{\infty \} \\ \{x \ \in \prod _{v|_{p}}\mathbb {K}_{v} : |x^{(j)}|_{p} \le n_p^{(j)}\text { for every } j\} &{} \hbox { for } p\notin D\cup \{\infty \} \end{array}\right.$, where $x^{(1)},x^{(2)},\cdots ,x^{(n)}$ are $\mathbb {Q}_{p}-$coordinates of x. Finally, define $V=i(\mathbb {K})+\prod \nolimits _{p\in P^{\mathbb {Q}}} U_p$. Hereafter, we just write $\prod U_p$ instead of $\prod \nolimits _{p\in P^{\mathbb {Q}}} U_p$ to make the notation look simpler. Also, note that D is similar to the set $D_{\infty }$ of [10].

Theorem 4

$\Sigma$ is isomorphic to $\mathbb {K}_{\mathbb {A}}/V$.

Proof

We prove that Ker $\psi '=V$ so that the required result follows. If $a \in V$, then $\psi '(a)= \pi (\psi (a))$; so we need to prove that $\psi (a)\in ann(\widehat{\Sigma })$, that is $\psi (a)(r)=1 \text { for every } r \in \widehat{\Sigma }.$ Now, $a \in V \text { implies that } a= (\delta )+l \text { where } (\delta ) \in i(\mathbb {K}) \text { and } l \in \prod U_p.$ So, we have the equality $\psi (a)=\psi (\delta ).\psi (l)=\psi (l)$. Further, if $p \in D \cup \{\infty \}$, then $\bar{l}_p=0$, where $\bar{l}_p=\prod _{v|_{p}}l_{v} \in \prod _{v|_{p}}\mathbb {K}_{v}$ which implies that $l^{(j)}_p=0 \text { for all } 1\le j\le n$. So, $\psi (l)(r)=\prod \nolimits _{p\notin D\cup \{\infty \}}e^{2\pi i\sum _{j=1}^{n}\{l^{(j)}_p r^{(j)}\}_{p}}, \text { for all } r \in \widehat{\Sigma }.$ $\square$

$$\begin{aligned}&\text {Even for a } p\notin D\cup \{\infty \} ,\,\, \bar{l}_p \in U_p\\&\quad \Rightarrow | l^{(j)}_p|_p \le n^{(j)}_p=\frac{1}{m^{(j)}_{p}} \,\, or \,\, |l^{(j)}_p|_p=0, \,\, \text { for every } 1 \le j \le n \\&\quad \Rightarrow |l^{(j)}_pr^{(j)}|_p \le |l^{(j)}_p||r^{(j)}|_p \le \frac{1}{m^{(j)}_{p}}.m^{(j)}_{p} =1 \,\, or \,\, |l^{(j)}_pr^{(j)}|_p =0, \\&\quad \qquad \text { for every } r \in \widehat{\Sigma } \,\, and \,\, 1 \le j\le n.\\&\quad \Rightarrow |l^{(j)}_pr^{(j)}|_p\le 1, \text { for all } r \in \widehat{\Sigma } \,\, and \,\, 1\le j\le n.\\&\quad \Rightarrow \{l^{(j)}_pr^{(j)}\}_p =0, \text { for all } r \in \widehat{\Sigma } \,\, and \,\, 1\le j\le n.\\&\quad \Rightarrow e^{-2\pi i\sum _{j=1}^{n}\{l^{(j)}_p r^{(j)}\}_p}=1 \end{aligned}$$

Hence, $\psi (a)(r)=\psi (l)(r)=1, \, \text {for every } r \in \widehat{\Sigma }$. Therefore, $V \subset Ker(\psi ')$.

For the converse, let $a \in Ker(\psi ')$. Following Proposition 6 of Chapter V in [17], we can write $a=(\delta )+l$, where $(\delta ) \in i(\mathbb {K})$, $l_v \in \Re _v$ for every $v \in P^{\mathbb {K}}$ and $l_{\infty }^{(j)} \in [0,1) \text { for all } 1\le j\le n$. So, it is enough to prove that $l \in \prod U_p$, or equivalently $\bar{l_p} \in U_p$, where $\bar{l}_p=\prod \nolimits _{v|_{p}}l_v \in \prod \nolimits _{v|_{p}}\mathbb {K}_v$. Again, $\psi (a)=\psi (l)$. Note that for any $p < \infty ,$ $\bar{l}_p \in \prod \nolimits _{v|_{p}}\Re _{v} \text { implies that } l^{(j)}_p \in \mathbb {Z}_p \text { for all } 1\le j\le n \text { and thus } \{l^{(j)}_p\}_p=0 \text { for all } 1\le j\le n.$ Choose $r \in \mathbb {K}$ for each $1\le j\le n$, such that $r^{(j)}=1$ and $r^{(i)}=0$ for $i\ne j$. Since $\mathbb {Z}^n \subset \beta (\widehat{\Sigma })$, it follows that $r \in \widehat{\Sigma }\text { and } \psi (l)(r)=e^{-2 \pi i l^{(j)}_{\infty }}$. Now, since $\psi (a)\in ann(\widehat{\Sigma })$, we have $e^{-2 \pi i l^{(j)}_{\infty }}=1 \text { implying that } l^{(j)}_{\infty }=0$. Hence, $\psi (l)(r)=\prod \nolimits _{p<\infty }e^{-2\pi i\sum _{j=1}^{n}\{l^{(j)}_p r^{(j)}\}_p},\text { for all } 1 \le j \le n \text { and for every } r \in \widehat{\Sigma }.$

Fix a $j \in \{1,2, \cdots ,n\}.$ Suppose $m^{(j)}_p=\infty$. Therefore, for arbitrary $k \in \mathbb {N}$, we can choose $r\in \widehat{\Sigma }$ such that $|r^{(j)}|_p > p^k$. Choose $\delta \in \mathbb {K}$ such that $\delta =(\delta ^{(1)},\delta ^{(2)},\cdots ,\delta ^{(n)}) \in \mathbb {Z}^n,$ where $|\delta ^{(j)}|_p=1$, $|\delta ^{(j)} r^{(j)}|_q \le 1 \,\, \text { for all } q \ne p$ and $\delta ^{(m)}r^{(m)}\in \mathbb {Z}\,\, \text { for all } m\ne j$. Since $\beta (\widehat{\Sigma })\text { is a }\mathbb {Z}^{n}-$module, the element $t=\beta ^{-1}(r^{(1)}\delta ^{(1)},r^{(2)}\delta ^{(2)},\cdots ,r^{(n)}\delta ^{(n)}) \in \widehat{\Sigma }$. Say $\beta (t)=(t^{(1)},t^{(2)},\cdots ,t^{(n)})$. Then, $|t^{(j)}|_p =|r^{(j)}|_p > p^k,~ |t^{(j)}|_q \le 1 \,\, \text { for every } q \ne p \text { and } |t^{(m)}|_q=|r^{(m)}\delta ^{(m)}|_q \le 1 \,\, \text { for every } q \text { and for every } m \ne j.$ Therefore, $\psi (l)(t)= e^{-2\pi i \{l^{(j)}_p r^{(j)}\}_p}$. So, $\psi (l)(t)=1$, implies $\{l^{(j)}_p r^{(j)}\}_p =0$, giving us the inequality $|l^{(j)}_p|_p \le \frac{1}{|r^{(j)}|_p} < \frac{1}{p^k}.$ Since k is arbitrary, it follows that $l^{(j)}_p=0.$

Now, consider the case $m^{(j)}_p < \infty$. If possible, suppose $|l^{(j)}_p|_p > n^{(j)}_p =\frac{1}{m^{(j)}_p}$. So we have $m^{(j)}_p>\frac{1}{|l^{(j)}_p|_p}$ and thus there exists $r \in \widehat{\Sigma } \text { such that } \frac{1}{|l^{(j)}_p|_p}< |r^{(j)}|_p \le m^{(j)}_p$ which gives $|l^{(j)}r^{(j)}|_p >1$. As done above, choose $\delta \in \mathbb {K}$ such that $|\delta ^{(j)}r^{(j)}|_p = |r^{(j)}|_{p}, \,\, \text { also } |\delta ^{(j)}r^{(j)}|_q \le 1 \text { for every } q \ne p$ and $\delta ^{(m)}r^{(m)} \in \mathbb {Z}\text { for every } m \ne j$. Then, $t=\beta ^{-1}(\delta ^{(1)}r^{(1)},\delta ^{(2)}r^{(2)},\cdots \cdots ,\delta ^{(n)}r^{(n)})\in \widehat{\Sigma }$ with $|l^{(j)}_{p}t^{(j)}|_p=|l^{(j)}_{p} \delta ^{(j)}r^{(j)}|_p=|l^{(j)}_{p}r^{(j)}|_p >1$, $|l^{(j)}_{q}t^{(j)}|_q \le 1 \text { for every } q \ne p\text { and } |l^{(m)}_{q}t^{(m)}|_q=|l^{(m)}_{q} \delta ^{(m)}r^{(m)}|_q \le 1 \text { for every } q \text { and for every } m \ne j$. This implies that $\psi (l)(t)= e^{-2\pi i \{l^{(j)}_p t^{(j)}\}_p} \ne 1$ giving us a contradiction. Hence $|l^{(j)}_p|_p \le n^{(j)}_p$.

Finally, for any $p \in D$, $m^{(j)}_p=\infty$ for every j implies that $l^{(j)}_p=0$ and thus $\bar{l}_p \in U_p$. Even for $p \notin D$, either $l^{(j)}_p=0$ or $|l^{(j)}_p|_p \le n^{(j)}_p$ and again $\bar{l}_p \in U_p$. So, for every prime p, $\bar{l}_p \in U_p$ and thus $a \in V.$ $\square$

We now describe the periodic points of some automorphisms of $\Sigma$. To start with, let $n_p = min\{n_p^{(j)}: 1 \le j \le n\}$ and $N_p = max\{n_p^{(j)}: 1 \le j \le n\}$ for every $p \in P_f^{\mathbb {Q}} \setminus D$. Choose a $b \in \mathbb {N}$ such that $\frac{1}{p^{b}} \le \frac{n_p}{N_p}$ and a matrix $E \in GL_n(\mathbb {Q})$ such that $E^k-I$ is invertible for every $k \in \mathbb {N}$ and $E \in GL_n(p^{b} \mathbb {Z}_p)$ for each $p \notin D \cup \{\infty \}$. The invertibility of $E^k-I$ ensures that E has no eigenvalue that is a root of unity and hence the dual action of E on the full n-dimensional solenoid is ergodic. Since E is invertible over $\mathbb {Q}$, the multiplication by E, which we denote by $m_E$ is an isomorphism of $(\mathbb {Q}_{\mathbb {A}})^n$. This gives an automorphism $M_E:\mathbb {K}_{\mathbb {A}} \rightarrow \mathbb {K}_{\mathbb {A}}$ as $M_E=\alpha ^{-1}\circ m_E \circ \alpha$. The assumption that $E \in GL_n(p^{b} \mathbb {Z}_p)$ for each $p \notin D \cup \{\infty \}$ helps in ensuring that $M_E(V)=V$ and thus obtaining an automorphism $\overline{M_E}$ on the solenoid $\Sigma$.

Proposition 5

$M_E(V)=V$.

Proof

Proof: Let $(\delta )+l \in V, \text { where } (\delta ) \in i(\mathbb {K}) \text { and } l=\left( \bar{l_p} \right) \in \prod U_p$. It is easy to see that $E \alpha \big ((\delta )\big ) \in \big (i(\mathbb {Q})\big )^n$ and hence $M_E\big ((\delta )\big ) \in i (\mathbb {K})$.

We will henceforth use the usual notation $E = [e_{ij}]_{1 \le i,j \le n}$ for the entries of E (and a similar notation for any other matrix in the sequel), where $[e_{i1}~e_{i2}~... ~e_{in}]$ is the $i^{th}$ row of E. Now, $M_E(l) = \alpha ^{-1}\big (\sum \nolimits _{j=1}^ne_{1j}l^{(j)},...,\sum \nolimits _{j=1}^ne_{nj}l^{(j)}\big )$. For $p \in D \cup \{\infty \}$, we have $l^{(j)}_p=0$ for every j, implying that $\sum \nolimits _{j=1}^ne_{ij}l^{(j)}_p = 0$ for any such p and any $1 \le i \le n$. We will now show that $|\sum \nolimits _{j=1}^ne_{ij}l^{(j)}_p|_p \le n_p^{(i)}$ for every $1 \le i \le n$ and $p \notin D \cup \{\infty \}$. On one hand, $|\sum \nolimits _{j=1}^ne_{ij}l^{(j)}_p|_p \le max\{|e_{ij}l^{(j)}_p|_p: 1 \le j \le n\}$ and on the other hand, owing to the fact that $E \in GL_n(p^{b} \mathbb {Z}_p)$, we get $|e_{ij}|_p \le \frac{1}{p^{b}} \le \frac{n_p}{N_p}$. Thus, for any $1 \le j \le n$, we have $|e_{ij} l_p^{(j)}|_p \le \frac{n_p}{N_p} n_p^{j} \le n_p \le n_p^{(i)}$. This shows that $M_E(l) \in \prod U_p$ and hence $M_E(V) \subseteq V$. $\square$

Conversely, for any $a=(\delta )+l \in V$, we have $M_E^{-1}(a)=M_{E^{-1}}\big ((\delta )\big ) + M_{E^{-1}}(l)$. Again, $M_{E^{-1}}\big ((\delta )\big ) \in i(\mathbb {K})$ and $M_E^{-1}(l) \in \prod U_p$ follow from the same observations as above, as $E^{-1} \in GL_n(p^b\mathbb {Z}_p)$ for every $p \notin D \cup \{\infty \}$. Hence, $M_E^{-1}(V) \subseteq V$. $\square$

Since $M_E$ is an automorphism of $\mathbb {K}_{\mathbb {A}}$ and V is an $M_E-$invariant subgroup of $\mathbb {K}_{\mathbb {A}}$, $M_E$ induces an automorphism of $\Sigma$, say $\overline{M_E}$.

Theorem 6

The set of periodic points of $\overline{M_{E}}$ is given by $P(\overline{M_{E}})= \frac{i(\mathbb {K})+\prod ^{'}\mathbb {K}_{v}}{V}$, where $\prod ^{'}\mathbb {K}_{v} = \left\{ x \in \mathbb {K}_{\mathbb {A}} \,: for \,\, every \,1 \le j \le n, x^{(j)}_{p}=0 \right. \, whenever \, \left. p\in D\cup \{\infty \}\right.$ and $\left. |x^{(j)}_{p}|_{p} \le N_p \hbox { for all but finitely many } p \notin D\cup \{\infty \} \right\}$.

Proof

Let $\bar{a}=a+V$ be periodic in $\Sigma$ with a period k. Then, $E^k \alpha (a) - \alpha (a) \in \alpha (V)$; say $E^k \alpha (a) - \alpha (a) = \alpha \big ((\delta )+l\big )$ for some $(\delta ) \in i(\mathbb {K})$ and $l \in \prod U_p$. Since $(E^k-I)^{-1}$ is a rational matrix, it follows that $\alpha (a) = (E^k-I)^{-1}\alpha \big ((\delta )\big ) + (E^k-I)^{-1}\alpha (l)$ and $(E^k-I)^{-1}\alpha \big ((\delta )\big ) \in \big (i(\mathbb {Q})\big )^n$. So, it remains to show that $\big (\sum \nolimits _{j=1}^nd_{1j}l^{(j)},...,\sum \nolimits _{j=1}^nd_{nj}l^{(j)}\big ) \in \alpha \big (\prod ^{'}\mathbb {K}_v)$, where $(E^k-I)^{-1}=[d_{ij}]_{1 \le i,j, \le n}$. Again for $p \in D \cup \{\infty \}$, we have $l^{(j)}_p = 0$ for every j and so, $\sum \nolimits _{j=1}^nd_{ij}l^{(j)}_p = 0$ for any $1 \le i \le n$. On the other hand, for all but finitely many $p \in P_f^{\mathbb {Q}}$, we have $|d_{ij}|_p=1$ for every $1 \le i,j \le n$, as each $d_{ij} \in \mathbb {Q}$. Therefore, $|\sum \nolimits _{j=1}^nd_{ij}l^{(j)}_p|_p \le max \{|d_{ij}l^{(j)}_p|_p: 1 \le j \le n\} = max \{|l^{(j)}_p|_p: 1 \le j \le n\} \le N_p$ for all but finitely many $p \notin D \cup \{\infty \}$.

For the converse, let $\bar{a}=a+V \in \frac{i(\mathbb {K})+\prod ^{'}\mathbb {K}_{v}}{V}$; say $a = (\delta )+l$, where $(\delta ) \in i(\mathbb {K})$ and $l \in \prod ^{'}\mathbb {K}_{v}$. We now have to find a $k \in \mathbb {N}$ such that $\overline{M_E}^k(\bar{a})=\bar{a}$ i.e., $E^k \alpha (a)-\alpha (a) \in \alpha (V)$. Note that, for any $k \in \mathbb {N}$ and for any $1 \le j \le n$, $E^k \alpha (a) - \alpha (a) = (E^k-I) \alpha \big ((\delta )\big )+ (E^k-I) (l)$. By similar observations as above, we get $(E^k-I) \alpha \big ((\delta )\big ) \in i(\mathbb {K})$. Now, say $(E^k-I)=[f_{ij}]_{1 \le i,j \le n}$. Again, for any $p \in D \cup \{\infty \}$, $\sum \nolimits _{j=1}^nf_{ij}l^{(j)}_p = 0$ for any $1 \le i \le n$.

Now, let $G = \{ p\notin D\cup \{\infty \}: \quad |l^{(j)}_p|_p > n^{(j)}_p$ for some $1\le j \le n \}$ and consider a $p \notin G \cup D \cup \{\infty \}$. Also, say $E^k=[h_{ij}]_{1 \le i,j \le n}$. Since $p \notin D \cup \{\infty \}$, we have $|h_{ij}|_p \le \frac{n_p}{N_p}$. Moreover, $f_{ij} = h_{ij}$ whenever $i \ne j$ and $f_{ii}=h_{ii}-1$ imply that $|f_{ij}|_p \le \frac{n_p}{N_p}$ for $i \ne j$ and $|f_{ii}| \le 1$. So, if $max \{|f_{ij}l^{(j)}_p|_p: 1 \le j \le n\} = |f_{ij}l^{j}_p|_p$ for some $i\ne j$, then we have $|f_{ij}l^{j}_p|_p=|f_{ij}|_p|l^{j}_p|_p \le \frac{n_p}{N_p}n_p^{(j)} \le n_p \le n_p^{(i)}$; otherwise, $max \{|f_{ij}l^{(j)}_p|_p: 1 \le j \le n\} = |f_{ii}l^{(i)}_p|_p \le n_p^{(i)}$. Hence, $|\sum \nolimits _{j=1}^nf_{ij}l^{(j)}_p|_p \le n_p^{(i)}$.

Finally, it remains to find a $k \in \mathbb {N}$ such that $|\sum \nolimits _{j=1}^nf_{ij}l^{(j)}_p|_p \le n^{(i)}_p$ for every i and every $p \in G$. Let us say $G = \{p_1,p_2,... p_m\}$ and for each $\gamma \in \{1,2,...,m\}$, ket $d_{\gamma }$ denote the usual metric on $GL_n(\mathbb {Z}_{p_{\gamma }})$ given by $d_{\gamma }(M,N) = max\{|M_{ij}-N_{ij}|_{p_{\gamma }}: 1 \le i,j \le n\}$. Then, $d\big ((M^{(1)},M^{(2)},...,M^{(m)}), (N^{(1)},N^{(2)},...,N^{(m)})\big ) = max \{d_{\gamma }(M^{(\gamma )}, N^{(\gamma )}): 1 \le \gamma \le m\}$ gives the product topology on $\prod \nolimits _{\gamma =1}^m GL_n(\mathbb {Z}_{p_{\gamma }})$. Since $E \in GL_n(\mathbb {Z}_{p_{\gamma }})$ for every $\gamma$, we have $(E,E,...,E) \in \prod \nolimits _{\gamma =1}^m GL_n(\mathbb {Z}_{p_{\gamma }})$; in fact, $\{(E^{\beta },E^{\beta },..., E^{\beta }): \beta \in \mathbb {N}\}$ is a semigroup of $\prod \nolimits _{\gamma =1}^m GL_n(\mathbb {Z}_{p_{\gamma }})$. Since $\prod \nolimits _{\gamma =1}^m GL_n(\mathbb {Z}_{p_{\gamma }})$ is compact, the closure of this semigroup is a subgroup and thus contains a sequence converging to (I, I, ..., I). Thus, there exists a $k \in \mathbb {N}$ such that $d((E^{k},E^{k},..., E^{k}),(I,I,...,I)) < \epsilon$, where $\epsilon = min\{\frac{n_{p_{\gamma }}}{n'_{p_{\gamma }}}: 1 \le \gamma \le m\}$ and $n'_{p_{\gamma }}=max\{|l^{(j)}_{p_{\gamma }}|_{p_{\gamma }}: 1 \le j \le n\}$. Then, $d_{\gamma }(E^k,I) < \epsilon$ implying that $|f_{ij}|_{p_{\gamma }} < \epsilon$ for every $1 \le i,j \le n$ and $1 \le \gamma \le m$. Hence, for any $p \in G$ and any $1 \le i \le n$, we get $|f_{ij}l^{(j)}_p|_p < \frac{n_p}{n'_p} n'_p = n_p \le n_p^{(i)}$ and finally, $|\sum \nolimits _{j=1}^nf_{ij}l^{(j)}_p|_p \le max \{|f_{ij}l^{(j)}_p|_p: 1 \le j \le n\} < n^{(i)}_p$. $\square$

The results in this article for higher dimensional solenoids can be intricately compared with the previously obtained results for one-dimensional solenoids ($\mathbb {Z} \subseteq \widehat{\Sigma } \subseteq \mathbb {Q}$) in [10], the first of the series of articles as mentioned in the introduction. In the previous work, solenoids whose Pontryagin duals are subgroups of $\mathbb {Q}$ are considered but in the current work we consider all such solenoids where $\widehat{\Sigma } \subseteq \mathbb {K}$. The corresponding adelic structures are also similar i.e. $\mathbb {K}_ \mathbb {A}$ in comparsion to $\mathbb {Q}_ \mathbb {A}$. In both these works, the map $\psi$ is defined from their adelic structures to their Pontryagin duals as $\psi : \mathbb {Q}_ \mathbb {A} \rightarrow \widehat{\mathbb {Q}}$ and $\psi : \mathbb {K}_ \mathbb {A} \rightarrow \widehat{\mathbb {K}}$. As a result of these parallel assumptions and identical maps, we obtain similar isomorphisms between their duals: $\widehat{\mathbb {Q}} \cong \frac{\mathbb {Q}_{\mathbb {A}}}{i(\mathbb {Q})}$ and $\widehat{\mathbb {K}}\cong \frac{\mathbb {K}_{\mathbb {A}}}{i(\mathbb {K})}$. The solenoid $\Sigma$ in both these works is obtained as a quotient space of its corresponding adelic structures. In the former case, we obtain $\Sigma = \frac{\mathbb {Q}_{\mathbb {A}}}{i(\mathbb {Q})+L}$, whereas in latter we get $\Sigma = \frac{\mathbb {K}_{\mathbb {A}}}{V}$ with $V=i(\mathbb {K})+L$. We can also observe the striking similarity between the set of periodic points of $P(\alpha ) = \frac{i(\mathbb {Q})+\prod ' \mathbb {Q}_p}{i(\mathbb {Q})+L}$ and $P(\overline{M_{d}})= \frac{i(\mathbb {K})+\prod ^{'}\mathbb {K}_{v}}{V}$.

4 Conclusion

This article describes the periodic points of automorphisms on n-dimensional solenoids whose duals are subgroups of algebraic number fields. Using the concept of adeles, it provides the description of such solenoids as well as the characterization of the periodic points of their automorphisms. This work is a development of the previous work done in [9] and [10]. Hence, the description in terms of adeles (or inverse limits) may pave further ways for the more general case, which still remains as an open problem.