2.1 Adeles, local and global fields

We review some basic facts and definitions regarding local fields, global fields, and the adeles. A general reference to these topics is Weil’s classical book [Reference Weil38, Chs. I–IV]; note that Weil calls what is now commonly referred to as global fields $\mathbb A$ -fields. Throughout this paper, the term local field will denote a locally compact field of characteristic zero; these include $\mathbb {R}$ and $\mathbb {C}$ as well as finite extensions of the field of p-adic numbers $\mathbb Q_p$ . (The terminology of global and local fields was introduced to incorporate both the positive and zero characteristic cases on an equal footing, but dynamically there are rather fundamental differences (see e.g. [Reference Einsiedler3, Reference Kitchens and Schmidt24]) and we restrict ourselves in this paper to the zero characteristic case.) Let $\mathbb {K}$ be a local field and let be the Haar measure on $\mathbb {K}$ . We define $\delta (\mathbb {K})$ as the degree of the field extension $\mathbb {K}$ over the closure of $\mathbb Q$ in $\mathbb {K}$ , which can be isomorphic to either $\mathbb {R}$ if $\mathbb {K}$ is Archimedean or $\mathbb Q_p$ for some prime p otherwise (to make the notation more consistent, we will also write $\mathbb Q_\infty $ for $\mathbb {R}$ ). Local fields come equipped with an absolute value $|\cdot |_{\mathbb {K}}$ , which we will always normalize to coincide with the usual absolute value on $\mathbb {R}$ or $\mathbb Q_p$ . We note that in any of these cases we have

(2.1)

for any measurable set $C \subseteq \mathbb {K}$ .

We recall that a global field $\mathbb {K}$ is a finite field extension of $\mathbb Q$ . We will denote the completions of $\mathbb {K}$ by $\mathbb {K}_\sigma $ , where $\sigma $ stands for the (Archimedean or non-Archimedean) place—that is, an equivalence class of absolute values. We choose the representative to coincide with either ${ | {\cdot } |}_\infty $ of ${ | {\cdot } |}_p$ on $\mathbb Q$ . We recall that $\mathbb {K}_\sigma $ is a local field and will use the abbreviation ${ | {\cdot } |}_\sigma ={ | {\cdot } |}_{\mathbb {K}_\sigma }$ for the norms satisfying (2.1) on $\mathbb {K}_\sigma $ . If ${ | {\cdot } |}_\sigma $ coincides with ${ | {\cdot } |}_p$ on $\mathbb Q$ , then we say that $\sigma $ lies over p; if ${ | {\cdot } |}_\sigma $ is Archimedean, we say that $ \sigma $ is an infinite place of $\mathbb {K}$ .

For a global field $\mathbb {K}$ , the ring of adeles $\mathbb A_{\mathbb {K}}$ over $\mathbb {K}$ is defined as the restricted direct product of all completions of $\mathbb {K}$ with respect to the maximal compact subrings for all non-Archimedean completions. In other words, $(t_\sigma )_\sigma \in \mathbb A_{\mathbb {K}}$ if $t_\sigma \in \mathbb {K}_\sigma $ for all places $\sigma $ of $\mathbb {K}$ and, except for finitely many $\sigma $ (an exceptional set that is assumed to include all infinite places), we have that in fact $t_\sigma $ lies in the maximal compact subring $\mathcal {O}_{\mathbb {K},\sigma } < \mathbb {K}_\sigma $ . In the special case $\mathbb {K}=\mathbb Q$ , this takes the form

$$ \begin{align*} \mathbb A=\textstyle{\mathbb{R}\times\prod_p^{\prime}\mathbb Q_p}=\mathbb{R}\times\bigcup_S(\prod_{p\in S}\mathbb Q_p\times\prod_{p\notin S}\mathbb{Z}_p), \end{align*} $$

where the union runs over all finite subsets S of the primes. The general case of the ring of adeles $\mathbb A_{\mathbb {K}}$ over a global field $\mathbb {K}$ is defined similarly, but can also be obtained via

(2.2) $$ \begin{align} \mathbb A_{\mathbb{K}}=\mathbb A\otimes_{\mathbb Q}\mathbb{K}. \end{align} $$

We shall identify $\mathbb {K}_\sigma $ with the corresponding subring in $\mathbb A_{\mathbb {K}}$ . Using a basis of $\mathbb {K}$ over $\mathbb Q$ , we obtain an additive group isomorphism (indeed, an isomorphism of vector spaces over $\mathbb Q$ )

(2.3) $$ \begin{align} \mathbb A_{\mathbb{K}}=\mathbb A\otimes_{\mathbb Q}\mathbb{K}\cong \mathbb A^{[\mathbb{K}:\mathbb Q]}. \end{align} $$

We recall moreover that $\mathbb Q$ diagonally embedded into $\mathbb A$ is discrete and cocompact and that the Pontryagin dual $\widehat {\mathbb A}$ of $\mathbb A$ can be identified with $\mathbb A$ itself. Finally the isomorphism between $\widehat {\mathbb A}$ and $\mathbb A$ can be chosen so that the annihilator of $\mathbb Q$ is $\mathbb Q$ itself, which implies that the Pontryagin dual of $\mathbb Q$ can be identified with $\mathbb A/\mathbb Q$ . This extends similarly to global fields, see e.g. [Reference Weil38, pp. 64–69].

2.2 Adelic actions

For us, the adelic setup gives a concrete language to discuss actions on general solenoids. We note however that for automorphisms on tori, it suffices to consider all Archimedean places of $\mathbb {K}$ and for irreducible actions, it would suffice to consider only finitely many places (see also [Reference Einsiedler and Lindenstrauss7] for the latter).

Indeed, let us fix a dimension $m\geq 1$ , a rank $d\geq 1$ , and d commuting matrices $A_1,\ldots ,A_d\in \operatorname {GL}_m(\mathbb Q)$ . We use them to define a linear representation $\widetilde {\alpha }$ of $\mathbb {Z}^d$ on $\mathbb Q^m$ . Using the matrices in the same way as within vector spaces, this extends to an action of $\mathbb {Z}^d$ by group automorphisms on $\mathbb A^m$ , which we will also denote by $\widetilde {\alpha }$ . Finally, we take the quotient by the discrete cocompact invariant subgroup $\mathbb Q^m$ and obtain an action $\alpha $ of $\mathbb {Z}^d$ by automorphisms on the solenoid

$$ \begin{align*} X_m=\mathbb A^m/\mathbb Q^m. \end{align*} $$

We will refer to $X_m$ as an adelic solenoid and to this action as the adelic action on $X_m$ defined by the matrices (or equivalently the linear maps) $A_1,\ldots ,A_d$ .

Since every group automorphism of $\mathbb Q^m$ is in fact $\mathbb Q$ -linear and defined by an invertible matrix in $\operatorname {GL}_m(\mathbb Q)$ , it follows from Pontryagin duality that every action of $\mathbb {Z}^d$ by automorphisms on $X_m$ can be defined this way. We will explain this step in a more general form in §4.1.

We say that a closed subgroup $Y<X_m$ of an adelic solenoid $X_m=\widehat {\mathbb Q}^m$ is adelic if it is a linear subspace over $\mathbb Q$ (that is, $\mathbb Q Y\subseteq Y$ ). Since this notion will be useful for us, we wish to study it briefly in the following lemma.

Proof The equivalence of (1) and (2) follows from Pontryagin duality. Indeed $aY=Y$ for $a\in \mathbb {Z}\setminus \{0\}$ (and then also $a\in \mathbb Q\setminus \{0\}$ ) is equivalent to $a(Y^\perp )=Y^\perp $ (since $(aY)^\perp =a^{-1}Y^\perp $ ).

Suppose now $V<\mathbb Q^m$ is a linear subspace as in (3). Then $\mathbb A\otimes _{\mathbb Q} V$ is clearly invariant under $\mathbb Q$ and hence defines modulo $\mathbb Q^m$ , an adelic subgroup.

Finally assume that Y is adelic as in (1) (and equivalently (2)). Let $W=Y^\perp <\mathbb Q^m$ so that W is a linear subspace and $Y=W^\perp $ by Pontryagin duality. By [Reference Weil38, Ch. IV], there exists a character $\chi _0\in \widehat {\mathbb A}$ so that the isomorphism $\widehat {\mathbb A}\cong \mathbb A$ is induced by the definition $\langle a,b\rangle =\chi _0(ab)$ for all $a,b\in \mathbb A$ and with this isomorphism, we have $\mathbb Q^\perp =\mathbb Q$ . Moreover, this also gives $\widehat {\mathbb A^m}\cong \mathbb A^m$ using the pairing

$$ \begin{align*} \langle (a_1,\ldots,a_m),(b_1,\ldots,b_m)\rangle = \chi_0(a_1b_1+\cdots+a_mb_m) \end{align*} $$

for all $(a_1,\ldots ,a_m),(b_1,\ldots ,b_m)\in \mathbb A^m$ . Since $W<\mathbb Q^m$ is a linear subspace, we may apply a linear isomorphism $A\in \operatorname {GL}_m(\mathbb Q)$ so that $W_1=A(W)$ is precisely the span of the first k standard basis vectors. Applying the inverse of the dual (transpose) linear automorphism to Y, this shows that $Y_1=(A^t)^{-1}(Y)$ satisfies that $Y_1^\perp =W_1$ . Now let $(a_1,\ldots ,a_m)\in Y_1$ . Hence we have $\chi _0(a_jb)=1$ for all $b\in \mathbb Q$ and $j=1,\ldots ,k$ . However, this gives by the properties of $\chi _0$ that $a_j\in \mathbb Q$ for $j=1,\ldots ,k$ . It follows that $Y_1=\mathbb Q^m+\mathbb A\otimes _{\mathbb Q} V_1$ , where $V_1$ is the linear hull of the last $m-k$ basis vectors. Applying $A^t$ to this claim gives the description of Y as in (3).

2.3 Irreducible adelic actions

We say that an adelic action on $X_m$ is $\mathbb A$ -irreducible if the associated linear representation of $\mathbb {Z}^d$ on $\mathbb Q^m$ is irreducible over $\mathbb Q$ , that is, if there does not exist a rational nontrivial proper invariant subspace. Note however that $\mathbb A$ -irreduciblity does not coincide with the notion of irreducibility defined on p. 5. In fact, an adelic action is never irreducible but it will be convenient to study $\mathbb A$ -irreducible adelic actions as basic building blocks of other adelic actions.

We note that given a global field $\mathbb {K}$ and $d\geq 1$ elements $\zeta _1,\ldots ,\zeta _d\in \mathbb {K}$ , we may consider multiplication by these elements as a $\mathbb Q$ -linear map on the vector space $\mathbb {K}$ over $\mathbb Q$ to define an adelic action of $\mathbb {Z}^d$ on $\mathbb A_{\mathbb {K}}/\mathbb {K}$ . Using a fixed basis of $\mathbb {K}$ over $\mathbb Q$ , we may identify $\mathbb {K}$ with $\mathbb Q^m$ and multiplication by $\zeta _1,\ldots ,\zeta _d$ with certain matrices $A_1,\ldots ,A_d$ . In this way, our discussions of §2.2 also apply to the multiplication maps by $\zeta _1,\ldots ,\zeta _d$ on $\mathbb {K}$ . The point of the following proposition is that every $\mathbb A$ -irreducible action of $\mathbb {Z}^d$ arises in this way from a global number field and d of its elements.

Proposition 2.2. (Diagonalization of $\mathbb A$ -irreducible action)

Let $m,d\geq 1$ and let $\alpha $ be an $\mathbb A$ -irreducible adelic action of $\mathbb {Z}^d$ on $X_m=(\mathbb A/\mathbb Q)^m$ . Then there exists a global field $\mathbb {K}$ of degree m over $\mathbb Q$ and d non-zero elements $\zeta _1,\ldots ,\zeta _d\in \mathbb {K}^\times $ so that $\alpha $ is isomorphic to the action on $\mathbb A_{\mathbb {K}}/\mathbb {K}$ generated by the maps $a\in \mathbb {K}\mapsto \zeta _j a\in \mathbb {K}$ for $j=1,\ldots ,d$ . More explicitly, this action on $\mathbb A_{\mathbb {K}}/\mathbb {K}$ (which as an additive topological group is isomorphic to $X_m$ ) can be given as follows:

$$ \begin{align*} \widetilde{\alpha}^{\mathbf{n}}: (v_\sigma)_\sigma\in\mathbb A_{\mathbb{K}}\mapsto (\underbrace{\zeta_{1,\sigma}^{n_1}\ldots \zeta_{d,\sigma}^{n_d}}_{=\zeta_{\mathbf{n},\sigma}}v_\sigma)_\sigma, \end{align*} $$

where $\zeta _{1,\sigma },\ldots ,\zeta _{d,\sigma }\in \mathbb {K}_\sigma $ and $\zeta _{\mathbf {n},\sigma }$ denote the image of $\zeta _1,\ldots ,\zeta _d$ respectively of $\zeta _{\mathbf {n}}=\zeta _1^{n_1}\ldots \zeta _d^{n_d}$ in the completion $\mathbb {K}_\sigma $ .

Proof Let $\zeta _j=\widetilde {\alpha }^{\mathbf {e}_j}\in \operatorname {GL}_m(\mathbb Q)$ for $j=1,\ldots ,d$ be the matrices that define the action $\widetilde {\alpha }$ on $\mathbb Q^m$ and $\mathbb A^m$ associated to $\alpha $ .

We define $\mathbb {K}=\mathbb Q[\zeta _1,\ldots ,\zeta _d]\subseteq \operatorname {GL}_m(\mathbb Q)$ to be the ring of polynomial expressions f in the matrices $\zeta _1,\ldots ,\zeta _d$ and with rational coefficients. We note that Lemma 2.1 implies that $\mathbb Q^m$ has no proper rational subspaces invariant under $\mathbb {K}$ . Since $\zeta _1,\ldots ,\zeta _d$ commute, it follows that any such polynomial expression $f\in \mathbb {K}$ is either zero or is invertible (as an element of $\operatorname {GL}_m(\mathbb Q)$ ). In particular, we have that $\mathbb {K}$ is an integral domain. As it is also a finite dimensional algebra over $\mathbb Q$ , it follows that $\mathbb {K}$ is field extension of $\mathbb Q$ . Once more because $\mathbb Q^m$ has no proper invariant subspaces, it also follows that $\varphi : a\in \mathbb {K}\mapsto a(\mathbf {e}_1)\in \mathbb Q^m$ must be surjective. By definition the kernel $\ker (\varphi )$ is an ideal, which implies that $\varphi $ is injective since $\mathbb {K}$ is a field. It follows that $\varphi $ is a linear isomorphism.

To summarize, we have found a global field $\mathbb {K}$ and elements $\zeta _1,\ldots ,\zeta _d\in \mathbb {K}^\times $ so that up to a linear isomorphism our linear representation $\widetilde {\alpha }^{\mathbf {n}}$ on $\mathbb Q^m$ is defined for every $\mathbf {n}\in \mathbb {Z}^{d}$ by multiplication by $\zeta _{\mathbf {n}}=\zeta _1^{n_1}\ldots \zeta _d^{n_d}$ on the vector space $\mathbb {K}$ .

To obtain the adelic action, we tensorize with $\mathbb A$ . On one hand, for the action on $\mathbb Q^m$ this gives the action of $\mathbb {Z}^d$ on $\mathbb A^m$ we started with. On the other hand, we may tensorize the linear isomorphism between $\mathbb Q^m$ and $\mathbb {K}$ with $\mathbb A$ to obtain the group isomorphism

$$ \begin{align*} \mathbb A^m=\mathbb Q^m\otimes_{\mathbb Q}\mathbb A\cong\mathbb{K}\otimes_{\mathbb Q}\mathbb A\cong (\mathbb{K}\otimes_{\mathbb Q}\mathbb{R})\times\textstyle{\prod^{\prime}_p(\mathbb{K}\otimes_{\mathbb Q}\mathbb Q_p)}. \end{align*} $$

Now notice that we can identify $\mathbb {K}$ with the quotient $\mathbb Q[x]/(p(x))$ for some irreducible polynomial $p(x)\in \mathbb Q[x]$ , which implies that $\mathbb {K}\otimes \mathbb {R}$ is isomorphic $\mathbb {R}[x]/(p(x))$ . Since the irreducible factors of $p(x)\in \mathbb {R}[x]$ correspond precisely to the roots of $p(x)$ (all appearing with multiplicity one) and hence also to the Galois embeddings of $\mathbb {K}$ into $\mathbb {C}$ , it follows that $\mathbb {K}\otimes \mathbb {R}$ is as a ring isomorphic to $\prod _{\sigma |\infty }\mathbb {K}_\sigma $ , where the product runs over all places of $\mathbb {K}$ lying above $\infty $ , that is, over all Archimedean completions of $\mathbb {K}$ .

This argument applies similarly for the tensor product with $\mathbb Q_p$ so that $\mathbb {K}\otimes _{\mathbb Q}\mathbb Q_p$ is isomorphic as a ring to the product $\prod _{\sigma |p}\mathbb {K}_\sigma $ and $\sigma $ denotes here all places of $\mathbb {K}$ above p, see also [Reference Weil38, p. 56]. Applying this argument at all places of $\mathbb Q$ , we obtain that $\mathbb A^m$ is isomorphic to $\mathbb A_{\mathbb {K}}$ .

Application of $\widetilde {\alpha }^{\mathbf {e}_j}$ corresponds under this isomorphism from $\mathbb A^m$ to $\mathbb A_{\mathbb {K}}=\prod _{\sigma }^{\prime }\mathbb {K}_\sigma $ to multiplication by the image of $\zeta _j$ in the factor $\mathbb {K}_\sigma $ for every place $\sigma $ of $\mathbb {K}$ . This gives the proposition.

Let us write $\delta (\sigma )=\delta (\mathbb {K}_\sigma )\in \mathbb {N}$ for any place $\sigma $ of $\mathbb {K}$ . The following product formula is a crucial ingredient in our proof.

Proposition 2.3. (Product formula)

Let $\alpha $ be an $\mathbb A$ -irreducible adelic $\mathbb {Z}^{d}$ -action as in Proposition 2.2. Then we have

(2.4) $$ \begin{align} \prod_{\sigma}|a|_{\sigma}^{\delta(\sigma)}= 1\quad\mbox{for every }a \in \mathbb{K}\setminus\{0\} \end{align} $$

and this applies in particular to $a=\zeta _{\mathbf {n}}$ for every $\mathbf {n}\in \mathbb {Z}^d$ .

We note that one way to obtain this result is precisely to interpret the product as the modular character for the automorphism defined by multiplication by a on the compact group $\widehat {\mathbb {K}}=\mathbb A_{\mathbb {K}}/\mathbb {K}$ (cf. (2.1)). We refer to [Reference Weil38, p. 75] for a proof along these lines.

2.4 A filtration by $\mathbb A$ -irreducible adelic actions

The following lemma reveals an advantage of adelic actions by connecting structural questions concerning $\alpha $ to linear algebra on the dual.

Lemma 2.4. (Decomposition into $\mathbb A$ -irreducible factors)

Let $m,d\geq 1$ and let $\alpha $ be an adelic $\mathbb {Z}^d$ -action on $X_m$ . Then there exist closed $\alpha $ -invariant adelic subgroups

$$ \begin{align*} Y_0=\{0\}<Y_1<\cdots< Y_r=X_m, \end{align*} $$

such that the action induced by $\alpha $ on $Y_{j}/Y_{j-1}$ is an $\mathbb A$ -irreducible adelic $\mathbb {Z}^d$ -action for all $j=1,\ldots ,r$ .

We will refer to the $\mathbb A$ -irreducible adelic actions appearing in Lemma 2.4 as the $\mathbb A$ -irreducible factors associated to $\alpha $ .

Proof By Pontryagin duality, we may consider instead of $\alpha $ the linear representation $\widehat {\alpha }$ on $\mathbb Q^m$ . Let $V_1\subseteq \mathbb Q^m$ be a non-trivial subspace that is invariant under $\widehat {\alpha }$ and of minimal dimension. Note that this implies that the restriction of $\widehat {\alpha }$ to $V_1$ is irreducible over $\mathbb Q$ . If $V_1\neq \mathbb Q^m$ , we let $V_2$ be a subspace that is invariant under $\widehat {\alpha }$ , strictly contains $V_1$ and is among these of minimal dimension. Once more this implies that $V_2/V_1$ is irreducible over $\mathbb Q$ (for the representation induced by $\widehat {\alpha }$ ).

Continuing like this, we obtain a partial flag

(2.5) $$ \begin{align} V_0=\{0\}<V_1<V_2<\cdots<V_r=\mathbb Q^m \end{align} $$

consisting of $\widehat {\alpha }$ -invariant subspaces so that $V_{j}/V_{j-1}$ is irreducible over $\mathbb Q$ . Applying Pontryagin duality (and reversing the indexing), this gives the lemma.

3.1 Leafwise measures

Given a quotient $X=G/\Gamma $ of a locally compact abelian group G by a lattice $\Gamma <G$ and a closed subgroup $V<G$ with $V\cap \Gamma =\{0\}$ , we consider the foliation of X into V-orbits. Let $\pi _X$ denote the natural projection $G\to X$ . As we will reduce our main theorem to the adelic case (Theorem 4.1), we consider the case that G is $\mathbb A^m$ (though everything we say below is equally valid for $G=\mathbb {R}^m$ , or a finite product of local fields). We note that the metric on $\mathbb A^m$ is chosen so that the balls $B_r^V=B_r(0)\cap V$ have compact closures for all $r>0$ .

For our purposes, it will be important to work with an extension of X—a product $\widetilde {X}=X\times \Omega $ of X with an arbitrary compact metric space $\Omega $ . We let V act on $\widetilde {X}$ by translation on the first coordinate and trivially on the second coordinate, obtaining in this way a foliation of $\widetilde {X}$ into V orbits. This foliation of $\widetilde X$ into V-orbits does not admit in general a Borel cross-section, and typically one cannot find a countably generated $\sigma $ -algebra on $\widetilde X$ whose atoms coincide with almost every (a.e.) V-orbits. Given a probability measure $\mu $ on $\widetilde X$ , the foliation into V-orbits gives rise to a system of leafwise measures on $\widetilde X$ : a Borel measurable map $ x \mapsto \mu _ x ^ V$ from a subset of full measure $\widetilde X ^{\prime } \subset \widetilde X$ to locally finite (possibly infinite) measures on V. We say that a leafwise measure $\mu _x^V$ is trivial if it is a multiple of the Dirac measure at the identity; we say that the system of leafwise measures is trivial if it is trivial at a.e. point. We also note that almost surely $0$ belongs to the support of $\mu _x^V$ .

The system of leafwise measure satisfies the following compatibility condition: for any $v\in V$ and $x \in \widetilde {X}^{\prime }$ so that $x+v$ is also in ${\widetilde {X}}^{\prime }$

(3.1) $$ \begin{align} (\mu_{x+v}^V+v)\propto\mu_x^V. \end{align} $$

Here and in the following, we write $\nu \propto \nu ^{\prime }$ for two measures $\nu ,\nu ^{\prime }$ if there exists $c>0$ with $\nu =c\nu ^{\prime }$ .

One way to characterize the leafwise measures is through the notion of subordinate $\sigma $ -algebras:

Definition 3.1. (Subordinate $\sigma $ -algebras)

A $\sigma $ -algebra $\mathcal {A}$ of Borel subsets of $\widetilde X$ is subordinate to V if $\mathcal {A}$ is countably generated, for every $x \in \widetilde X$ the atom $[x]_{\mathcal {A}}$ of x with respect to $\mathcal {A}$ is contained in the leaf $x+V$ , and for a.e. x

$$ \begin{align*} x+B_\epsilon^V\subseteq [x]_{\mathcal{A}}\subseteq x+B_\rho^V\quad\mbox{for some } \epsilon>0\mbox{ and }\rho>0. \end{align*} $$

For these x, we define the shape $S_x$ of the atom $[x]_{\mathcal {A}}$ (from the point of view of x) as the subset of V satisfying $x+S_x=[x]_{\mathcal {A}}$ .

Let $\mathcal {A}$ be a countably generated $\sigma $ -algebra on $\widetilde {X}$ that is subordinate to V. Then the leafwise measures for $\mu $ with respect to V and the conditional measures of $\mu $ with respect to $\mathcal {A}$ satisfy that for a.e. $x\in \widetilde {X}$ , the conditional measure $\mu _x^{\mathcal {A}}$ at x with respect to $\mathcal {A}$ equals the normalized push forward $x+(\mu _x^V|_{S_x})$ of the restriction $\mu _x^V|_{S_x}$ under the addition map $v \mapsto x+v$ from $V \to \widetilde X$ .

A slightly subtler but important feature of the system of leafwise measures is that while they may be (and typically are) infinite measures, they have certain a priori restrictions on how fast they grow: There exists a concrete function $f_V$ on V (only depending on V) that is integrable with respect to $\mu _x^V$ for every x in a set of full measure (that we may as well assume already contains the conull set $\widetilde {X}^{\prime }$ ), see [Reference Einsiedler and Lindenstrauss10, Theorem 6.30]. In fact, $f_V$ can be chosen with very mild (polynomial-like) decay properties.

In particular, if $x \in {\widetilde {X}}^{\prime }$ and $v \in V$ satisfies

(3.2) $$ \begin{align} (\mu_x^V+v)\propto\mu_x^V, \end{align} $$

this in fact implies the formally stronger conclusion that $\mu _x^V$ is translation invariant by the same v, that is,

(3.3) $$ \begin{align} (\mu_x^V+v)=\mu_x^V, \end{align} $$

for otherwise $\mu _x^V$ would have exponential growth, which would contradict the polynomial-like growth condition. Finally we recall the following proposition.

3.2 Entropy and leafwise measures

Suppose now $T:G\to G$ is a group automorphism preserving $\Gamma $ and V. Recall that we have restricted ourselves without loss of generality to the case where X is the Pontryagin dual to an m-dimensional vector space L over $\mathbb Q $ . Fixing a choice of basis in L, we can restrict our attention to the case of $G=\mathbb A^m$ , $\Gamma =\mathbb Q^m$ , and hence the automorphism T is defined by a rational matrix $\mathbf T \in \operatorname {GL}_m(\mathbb Q)$ . We note however that some of the definitions below, e.g. the ‘sufficiently fine’ condition in (3.7), depend on the choice of basis used to give the isomorphism $L\cong \mathbb Q^m$ (which induces an isomorphism $\Gamma \cong \mathbb Q^m$ ).

We denote the resulting automorphism of $X=G/\Gamma $ also by T, and consider an extension $\widetilde {T}=T\times T_\Omega :\widetilde {X}\to \widetilde {X}$ with $T_\Omega :\Omega \to \Omega $ measurable. Furthermore, suppose the probability measure $\mu $ on $\widetilde {X}$ is invariant under $\widetilde {T}$ . Then the characterizing properties of the leafwise measures $\mu _x^V$ imply the equivariance formula

(3.4) $$ \begin{align} \mu_{\widetilde{T}x}^V\propto T_*(\mu_x^V) \end{align} $$

for a.e. $x\in {\widetilde {X}}^{\prime }$ , see e.g. [Reference Einsiedler and Lindenstrauss10, Lemma 7.16].

We say that a closed subgroup $V<\mathbb A^m$ is S-linear where S is a finite set of places of $\mathbb Q$ (that is, a set of prime numbers or infinity) if for each $\sigma \in S$ there is a subspace $V_\sigma <\mathbb Q_\sigma ^m$ so that V is the direct product of the $V_\sigma $ for $\sigma \in S$ . Below we will frequently use the stable horospherical subgroup

$$ \begin{align*} U_T^-=\{a \in \mathbb A^m: T^n a \to 0 \text{ as}\ n \to \infty\} \end{align*} $$

for T and the unstable horospherical subgroup $U_T^+=U_{T^{-1}}^-$ . If

(3.5) $$ \begin{align} S = \{ p: \mathbf T \not\in \operatorname{GL} (m, \mathbb{Z} _ p) \} \cup \{ \infty \}, \end{align} $$

then the horospherical subgroups $U_T^-$ and $U_T^+$ are S-linear for this S.

3.3 Increasing subordinate $\sigma $ -algebras and entropy

We continue with the notation of §3.2. We say that a countably generated $\sigma $ -algebra $\mathcal {A}$ on $\widetilde {X}$ is increasing with respect to $\widetilde T$ if $\mathcal {A} \subseteq \widetilde T\mathcal {A}$ (modulo $\mu $ ), that is, the atom $[\widetilde T x]_{\mathcal {A}}$ almost surely (a.s.) contains $\widetilde T ([x]_{\mathcal {A}})$ .

Let $\mathcal {P}$ be a finite partition of $\widetilde X$ , which we identify with the corresponding finite algebra of sets. For any $\epsilon> 0$ , let

$$ \begin{align*} \partial ^ V _ \epsilon \mathcal{P} = \{ \widetilde x \in \widetilde X: \widetilde x + B _ \epsilon ^ V \not \subseteq [\widetilde x] _{\mathcal{P}} \}. \end{align*} $$

The following lemma follows quickly from monotonicity of the function $r\in [0,\infty )\mapsto \mu (B_r(x))$ for $x\in \widetilde {X}$ (and its almost sure differentiability) together with compactness of $\widetilde {X}$ .

Lemma 3.3. [Reference Einsiedler and Lindenstrauss10, Lemma 7.27]

For any probability measure $\widetilde \mu $ on $\widetilde X$ , there exists a finite partition $\mathcal {P}$ of $ \widetilde X$ into arbitrarily small sets such that for some fixed C and for every $\epsilon>0$

(3.6) $$ \begin{align} \mu(\partial _ \epsilon ^ V\mathcal{P})<C \epsilon. \end{align} $$

For more details, see [Reference Einsiedler and Lindenstrauss10, §7]. A partition $\mathcal {P}$ satisfying the conclusion of the above lemma will be said to have thin boundaries. We will assume throughout that any finite partition $\mathcal {P} $ of X we will consider below is sufficiently fine in the sense that

(3.7) $$ \begin{align} P-P \subset \pi_X \bigg(\prod_ {v \in S} B _ {\mathbb Q _ v ^ m} (r _ v) \times \prod_ {v \not \in S} \mathbb{Z} _ v ^ m \bigg) \quad\text{for every}\ P \in \mathcal{P} ,\end{align} $$

with $r _ v = 0.1 \max ({ \| {\mathbf T} \|}_v,{ \| {\mathbf T^{-1}} \|}_v)^{-1}$ (with respect to the operator norm on $\operatorname {GL} (\mathbb Q _ v)$ ).

For any $\sigma $ -algebra $\mathcal {A}$ and $-\infty \leq k_0<k_1 \leq \infty $ set

$$ \begin{align*} \mathcal{A}^{k_0}=\widetilde T^{{-k_0}}\mathcal{A}\mbox{ and }\mathcal{A}^{(k_0,k_1)}=\bigvee_{k_0 \leq i \leq k_ 1} \widetilde T^{{-i}}\mathcal{A} \end{align*} $$

(for $k_0$ or $k_1 = \pm \infty $ strict inequality instead of $\leq $ should be used).

An easy Borel–Cantelli argument gives that if $\mathcal P$ is a sufficiently fine finite partition of X with small boundaries in the sense of (3.6) and (3.7), then

$$ \begin{align*} \mathcal{C} _ {\mathcal{P}} = \mathcal P ^{(0,\infty)} \end{align*} $$

is a countably generated $\sigma $ -algebra satisfying one of the two conditions required by Definition 3.1 for $V = U _ T ^ -$ , namely for a.e. x it holds that there is an $\epsilon> 0$ so that $x + B _ {U _ T ^ -} (\epsilon ) \subset [x]_{\mathcal {P}}$ . (With a bit more care, using a countable partition $\mathcal {P}$ with finite entropy, one can get that $\mathcal {P} ^ {({0,\infty })}$ is actually $U _ T ^ -$ -subordinate; we achieve a similar goal by a cruder approach below.)

A modified version of this increasing $\sigma $ -algebra $\mathcal {P} ^ {(0,{\infty })}$ can be used to construct for any S-linear T-normalized subgroup $V<U_T^-$ of $\mathbb A^m$ an increasing V subordinate $\sigma $ -algebra $\mathcal {C} _ V$ on $\widetilde {X}=X\times \Omega $ so that moreover $ \mathcal {C} _ V = \tilde T ^{-1} \mathcal {C} _ V \vee \mathcal P$ . Indeed, first we construct starting from a (sufficiently fine, with small boundaries) finite partition $\mathcal P$ on X a $\sigma $ -algebra $\mathcal {P} _V$ on $\widetilde {X}$ as follows: for each $P \in \mathcal {P}$ , lift it to a subset $\widetilde P \subset \mathbb A ^ m$ contained in a translate of $\prod _ {v \in S} B _ {\mathbb Q _ v ^ m} (r _ v) \times \prod _ {v \not \in S} \mathbb {Z} _ v ^ m$ (up to translation by an element of $\mathbb Q ^ m$ , this lift is uniquely defined). Now take the countably generated $\sigma $ -ring $\widetilde { \mathcal {Q}} _ P$ of subsets C of $\widetilde P$ with the property that if $x \in C$ then $(x +V) \cap \widetilde P \subseteq C$ . Since $\pi _ X$ is a bijection from $\widetilde P$ to P, the image $\mathcal {Q} _ P$ of $ \widetilde {\mathcal {Q}} _ P$ in X is a countably generated $\sigma $ -ring, and now we define $\mathcal {P} _ V$ to be the $\sigma $ -algebra of subsets of X generated by the $\sigma $ -rings $\mathcal {Q} _ P\times \mathcal {B}_\Omega $ for all $P \in \mathcal P$ . Then $\mathcal {C}_V=\mathcal {P} _ V ^ {({0,\infty })}$ is a T-increasing, countably generated $\sigma $ -algebra subordinate to V satisfying $\mathcal {C}_V= \widetilde T^{-1} \mathcal {C}_V \vee \mathcal {P}$ . Note also that by the way $\mathcal {C} _ V$ is defined, there is a fixed $\rho> 0$ so that $[x] _ {\mathcal C _ V} \subseteq x + B _ \rho ^ V$ for all $x \in \widetilde {X}$ . For more details, the reader is referred again to [Reference Einsiedler and Lindenstrauss10, §7].

For $V\leq U^-_T$ as above and $\mathcal {C}_V$ as above, we define the entropy contribution of V to be

(3.8) $$ \begin{align} {\sf h}_{\widetilde \mu}(\widetilde T,V)={\sf H}_{\widetilde \mu}(\mathcal{C}_V\mid \widetilde T^{-1}\mathcal{C}_V). \end{align} $$

We also need the conditional form of this definition: if $\mathcal Y$ is a $\widetilde T$ -invariant $\sigma $ -algebra, then the entropy contribution of V conditional on $\mathcal Y$ is defined to be

(3.8^′) $$\begin{align} h _ {\widetilde \mu} (\widetilde T, V \mid \mathcal Y) = H _ {\widetilde \mu} (\mathcal C _ V \mid {\widetilde T} ^{-1} \mathcal C _ V \vee \mathcal Y), \end{align}$$

where, as usual, we will identify a factor Y of $\widetilde X$ with the corresponding $\widetilde T$ -invariant $\sigma $ -algebra $\mathcal Y$ of subsets of $\widetilde X$ . Formally the conditional entropy contribution is included in the previous case, replacing $\Omega $ by $\Omega \times Y$ , but notationally it will be useful to allow additional explicit conditioning. The following propositions shows that—as implied by the notation— ${\sf h}_{\widetilde \mu }(\widetilde T,V)$ does not depend on the choice of $\mathcal {P}$ and $\mathcal {C}_V$ .

Proposition 3.4. Let $V\leq U^-_T$ be an S-linear subgroup normalized by T, and let $\mathcal {C}_V$ be as above. Then

(3.9) $$ \begin{align} \operatorname{vol}(\widetilde T,V,\widetilde x)=\lim_{|N| \rightarrow \infty}\frac{1}{N} \log{\widetilde \mu}_{\widetilde x}^V(T^{-N}(B_1^V(0))) \end{align} $$

exists almost everywhere and ${\sf h}_{\widetilde \mu }(\widetilde T,V)=\int \operatorname {vol}(\widetilde T,V,\widetilde x)\,{d}{\widetilde \mu }(\widetilde x)$ . Moreover, outside a set of $\widetilde \mu $ -measure zero, $\operatorname {vol}(\widetilde T,V,x)=0$ if and only if $\widetilde \mu _{x}^V$ is trivial. Furthermore, ${\sf h}_{\widetilde \mu }(\widetilde T,V) \leq h_{\widetilde \mu }(\widetilde T \mid \Omega )$ , with equality holding for $V=U_T^-$ . In particular, $h_\mu (\widetilde T\mid \Omega )>0$ if and only if $\widetilde \mu _x^{U_T^-}$ is not almost everywhere trivial.

By the remark above, this proposition also covers the case of entropy contributions conditional on a factor. This proposition is essentially well known, and is e.g. heavily used by Ledrappier and Young in [Reference Ledrappier and Young25, Reference Ledrappier and Young26] (though we are using a version of these results relative to the factor $\Omega $ of $\widetilde {X}$ ). For proof, we refer the reader to [Reference Einsiedler and Lindenstrauss10, §7] where an exposition in the spirit of this paper can be found. (In [Reference Einsiedler and Lindenstrauss10, §7]. It is assumed that the acting group acts in a semisimple way on the leaves, which does not necessarily hold in our case as we are explicitly allowing actions with non-trivial Jordan form. However, the arguments of [Reference Einsiedler and Lindenstrauss10, §7] can be easily modified to handle this situation; we leave the details to the readers.)

Assuming that $\mu $ is invariant and ergodic under a $\mathbb {Z}^d$ -action $\widetilde {\alpha }$ with $\widetilde T=\widetilde \alpha ^{\mathbf {n}}$ for some $\mathbf {n}\in \mathbb {Z}^d$ , the value of the limit in (3.9) defines an invariant function for $\widetilde \alpha $ , hence is almost everywhere constant, and so equals $h_\mu (\widetilde T, V\mid \Omega )$ for a.e. $x\in \widetilde {X}$ . In particular, assuming $\widetilde \alpha $ is ergodic, the following three statements are equivalent: (i) $h_\mu (\widetilde T\mid \Omega )>0$ ; (ii) $\mu _x^{U_T^-}$ is non-trivial almost everywhere; (iii) $\mu _x^{U_T^-}$ is non-trivial on a set of positive measure.

The entropy contributions for $V < U^+_T$ (also denoted $ h _ {\widetilde \mu } (\widetilde T, V )$ ) are defined similarly, and satisfy that

$$ \begin{align*} h _ {\widetilde \mu} (\widetilde T, V ) = h _ {\widetilde \mu} (\widetilde T^{-1}, V ).\end{align*} $$

Let $0 \to L _ 1 \to L \to K \to 0$ be an exact sequence of finite dimensional vector space over $\mathbb Q$ and let $0 \to Y \to X \to X _ 1 \to 0$ be the corresponding dual exact sequence of adelic solenoids. Let $T : X \to X$ be the dual map to a linear map in $\operatorname {GL} (L)$ fixing $L _ 1$ . Then T also induces a map $T _ 1: X _ 1 \to X _ 1$ . Let $V _ {Y, \mathbb A} < \mathbb A ^ m $ be the rational subspace projecting modulo $\mathbb Q ^ m$ to Y. Let $T _ \Omega : \Omega \to \Omega $ be a continuous map on the compact metric space $\Omega $ , and denote $\widetilde X = X \times \Omega $ , $\widetilde T = T \times T _ \Omega $ , $\widetilde X _ 1 = X _ 1 \times \Omega $ , etc. We let $\pi $ denote both the projection from $\mathbb A^m \to \mathbb A^m/V _ {Y, \mathbb A}$ as well as the corresponding projection $X \to X_1$ .

Proposition 3.5. (Entropy contribution of factor and fiber, cf. [Reference Einsiedler and Lindenstrauss11, Proposition 6.4] or [Reference Einsiedler and Lindenstrauss8, Proposition 3.1])

With the notation above, let $\widetilde \mu $ be a $\widetilde T$ -invariant measure on X, let V be a S-linear subgroup of $U _ T ^ - <\mathbb A^m$ , and let $V_1$ be a S-linear subgroup of $U _ {T _ 1}^{-}<\mathbb A^m/V _ {Y, \mathbb A} $ so that $V \leq \pi ^{-1}(V_1)$ . Let $\widetilde \mu _ 1 = \pi _ {*} \widetilde \mu $ . Then

(3.10) $$ \begin{align} h _ {\widetilde \mu} (\widetilde T, V) \leq h _ {\widetilde \mu _ 1} (\widetilde T _ 1, V _ 1) + h _ {\widetilde \mu} (\widetilde T, V \cap V _ {Y, \mathbb A}) ,\end{align} $$

with equality holding for $V=U_T^-$ and $V_1=U_{T _ 1} ^{-}$ .

Note that by definition, $h _ {\widetilde \mu } (\widetilde T, V) = h _ {\widetilde \mu } (\widetilde T, V\mid \Omega )$ and similarly for the other terms in (3.10).

4.1 Extension to adelic action

Suppose that $\alpha $ is a $\mathbb {Z}^d$ -action by automorphisms on a solenoid X as in Theorem 1.3. By definition, this means that the Pontryagin dual $\widehat {X}$ is isomorphic to a subgroup $V\subseteq \mathbb Q^m$ for some $m\in \mathbb {N}$ . We may assume that m is minimal, that $\widehat {X}=V$ , and by applying some linear automorphism if necessary, we may also assume that the standard basis vectors of $\mathbb Q^m$ belong to V. By Pontryagin duality, we also have that X is isomorphic to the quotient of $X_m=\widehat {\mathbb Q^m}=\mathbb A^m/\mathbb Q^m$ modulo the annihilator $K=V^\perp <X_m$ of V.

Moreover, for every $\mathbf {n}\in \mathbb {Z}^d$ the dual $\widehat {\alpha }^{\mathbf {n}}$ of the automorphism $\alpha ^{\mathbf {n}}$ is an automorphism of $V\subseteq \mathbb Q^m$ . Using the standard basis of $\mathbb Q^m$ (contained in V by the above assumption), we can represent this dual automorphism by a rational matrix and extend it to linear automorphism of $\mathbb Q^m$ (since $V\otimes _{\mathbb Q}\mathbb Q$ can be identified with $\mathbb Q^m$ ). This shows that the dual action extends to a linear representation of $\mathbb {Z}^d$ on $\mathbb Q^m$ .

We take the transpose of this representation (that is, of each of the matrices defining $\widehat {\alpha }^{\mathbf {e}_j}$ for $j=1,\ldots ,d$ ) to define an action $\widetilde {\alpha }$ of $\mathbb {Z}^d$ by automorphisms on $\mathbb Q^m$ and $\mathbb A^m$ . As discussed in §2.2, this defines an adelic action of $\mathbb {Z}^d$ by automorphisms of $X_m=\mathbb A^m/\mathbb Q^m$ . Moreover, since $V\subseteq \mathbb Q^m$ is invariant under $\widehat {\alpha }$ , the annihilator $K=V^\perp $ is a closed invariant subgroup for $\widetilde {\alpha }$ and the induced action on $X_m/K$ is isomorphic to the original action on X.

Extending the above discussion, the following lemma allows us to switch our attention to the setting of an adelic $\mathbb {Z}^d$ -action $\alpha $ on $X_m$ for some $m\in \mathbb {N}$ .

4.2 Choosing a good finite index subgroup $\Lambda <\mathbb {Z}^d$

For the proof of the lemma and some of the following arguments, we first recall the Jordan decomposition. Given a matrix $A\in \operatorname {GL}_m(\mathbb Q)$ , there exist matrices $D,U\in \operatorname {GL}_m(\mathbb Q)$ so that $A=DU=UD$ , D is semisimple (that is, is diagonalizable over $\overline {\mathbb Q}$ ), and U is unipotent (that is, has only $1$ as eigenvalue). This decomposition is unique and if A commutes with a matrix B, then D and U as above also commute with B. Moreover, a subspace $V<\mathbb Q$ is invariant under A if and only if V is invariant under both D and U.

By applying this to each of the matrices $\alpha ^{\mathbf {e}_i}$ for $i=1,\ldots ,d$ , we obtain two representations of $\mathbb {Z}^d$ on $\mathbb Q^m$ : the first representation $\alpha _{\mathrm {diag}}$ is semisimple, and the second $\alpha _{\mathrm {uni}}$ is by unipotent matrices, the two representations commute, and we have $\alpha ^{\mathbf {n}}=\alpha _{\mathrm {diag}}^{\mathbf {n}}\alpha _{\mathrm {uni}}^{\mathbf {n}}$ for all $\mathbf {n}\in \mathbb {Z}^d$ .

Proof Proof of Lemma 4.3

We first consider an $\mathbb A$ -irreducible action $\alpha $ . As the proof of Lemma 2.2 shows, $\alpha $ corresponds in this case to a global field $\mathbb {K}$ generated by d elements $\zeta _1,\ldots ,\zeta _d$ (obtained directly from the matrix representations of $\widetilde \alpha ^{\mathbf {e}_j}$ ). Restricting the action to a finite index subgroup results in replacing $\zeta _1,\ldots ,\zeta _d$ by d monomial expressions $\xi _1,\ldots ,\xi _d$ (corresponding to a basis of $\Lambda <\mathbb {Z}^d$ ) in the numbers $\zeta _1,\ldots ,\zeta _d$ . This in turn may result in $\xi _1,\ldots ,\xi _d$ generating instead of $\mathbb {K}$ a subfield $\mathbb {L}$ of $\mathbb {K}$ . In this case, the $\mathbb A$ -irreducible representation of $\mathbb {Z}^d$ on $\mathbb {K}$ obtained in the proof of Lemma 2.2 becomes, when restricted to $\Lambda $ , isomorphic to a direct sum of $[\mathbb {K}:\mathbb {L}]$ many copies of the $\mathbb A$ -irreducible representation defined by multiplication by $\xi _1,\ldots ,\xi _d$ on $\mathbb {L}$ . If this indeed happens, we may choose $\Lambda $ so that $\mathbb {L}$ is minimal in dimension. Hence for any finite index subgroup $\Lambda ^{\prime }<\Lambda $ , the monomial expressions in the variables $\xi _1,\ldots ,\xi _d$ corresponding to a basis of $\Lambda ^{\prime }$ will still generate the same field $\mathbb {L}$ and so the $[\mathbb {K}:\mathbb {L}]$ many $\mathbb A$ -irreducible representations for the restriction to $\Lambda $ will remain irreducible for the restriction to $\Lambda ^{\prime }$ . This proves the first part of the lemma in the $\mathbb A$ -irreducible case.

Let now $\alpha $ be a general adelic action. Let

$$ \begin{align*} V_0=\{0\}<V_1<V_2<\cdots<V_r=\mathbb Q^m \end{align*} $$

be as in Lemma 2.4, equation (2.5), with the action induced by $\widehat \alpha $ on $V_i/V_{i-1}$ irreducible over $\mathbb Q$ , and let $\mathbb {K}_i$ be the corresponding finite extension of $\mathbb Q$ as in Proposition 2.2 for $V_i/V_{i-1}$ (or more precisely, for the dual adelic solenoid) for $i=1,\ldots ,r$ . Applying the above discussion on each irreducible quotient by passing to a finite index subgroup $\Lambda < \mathbb {Z}^d$ , we may assume that these quotients remain irreducible even if we pass to a further finite index subgroup of $\Lambda $ , establishing the first part of the lemma.

Now let M be the least common multiple of the orders of the (finitely many) roots of unity in some finite degree Galois extension of $\mathbb Q$ containing the fields $\mathbb {K}_1,\ldots ,\mathbb {K}_r$ . We claim that if $\Lambda _1 = M \Lambda $ and if $V < \mathbb Q ^ m $ is invariant under $\widehat {\alpha }(\Lambda ^{\prime })$ for some $\Lambda ^{\prime } < \Lambda _1$ , then V is invariant under $\widehat {\alpha } (\Lambda _ 1)$ .

Indeed, as discussed above, for any $\Lambda ^{\prime } \leq \mathbb {Z} ^ d$ , the space $V=Y^\perp $ is $\widehat {\alpha } (\Lambda ^{\prime })$ -invariant if and only if it is invariant under both $\widehat {\alpha }_{\mathrm {diag}} (\Lambda ^{\prime }) $ and $\widehat {\alpha }_{\mathrm {uni}} (\Lambda ^{\prime })$ .

Since the map $\mathbf n \mapsto \widehat {\alpha }_{\mathrm {uni}} (\mathbf n)$ is polynomial, and since any finite index subgroup of $\mathbb {Z} ^ d$ is Zariski dense in affine d-dimensional space, a space V is $\widehat {\alpha }_{\mathrm {uni}} (\Lambda ^{\prime })$ -invariant if and only if it is $\widehat {\alpha }_{\mathrm {uni}} (\mathbb {Z} ^ d)$ -invariant (hence in particular $\widehat {\alpha }_{\mathrm {uni}} (\Lambda _1)$ -invariant).

Suppose V is $\widehat {\alpha }_{\mathrm {diag}} (\Lambda ^{\prime }) $ -invariant but not $\widehat {\alpha }_{\mathrm {diag}} (\Lambda _1) $ -invariant. Let $\mathbf n = M \mathbf n ^{\prime } \in \Lambda _ 1$ so that $\widehat {\alpha }_{\mathrm {diag}} (\mathbf n)$ does not fix V. Since $\widehat {\alpha }_{\mathrm {diag}} (\Lambda ^{\prime }) $ leaves V invariant, it follows that there is some $k \in \mathbb {N}$ so that $\widehat {\alpha }_{\mathrm {diag}} (k\mathbf n) $ leaves V invariant.

By Proposition 2.2, for every i, the action induced by $\alpha (\mathbf n ^{\prime })$ on each $V_i/V_{i-1}$ can be identified with multiplication by some $\xi \in \mathbb {K}_i$ on $\mathbb {K}_i$ . If $\widehat {\alpha }_{\mathrm {diag}} (\mathbf n)=\widehat {\alpha }_{\mathrm {diag}} (\mathbf n^{\prime })^M$ does not fix V but $\widehat {\alpha }_{\mathrm {diag}} (k \mathbf n)$ does, this implies that there is a j and $\xi ^{\prime }\in \mathbb {K}_j$ as well as embeddings $\sigma ,\sigma ^{\prime }$ of $\mathbb {K}_i$ and $\mathbb {K}_j$ into $\mathbb {C}$ so that $\sigma (\xi )^ M \neq \sigma ^{\prime }(\xi ^{\prime }) ^ M$ but $\sigma (\xi ) ^ {k M} = \sigma (\xi ^{\prime }) ^ {k M}$ . Then $\sigma (\xi ) \sigma ^{\prime } (\xi ^{\prime }) ^{-1}$ is an element of the compositum of $\sigma (\mathbb {K}_i)$ , $\sigma ^{\prime }(\mathbb {K}_j)$ that is a root of unity of order not dividing M—a contradiction.

In particular, Lemma 4.3 shows that it is possible to restrict any adelic action to a finite index subgroup so that each of the $\mathbb A$ -irreducible adelic actions associated to its restriction are in fact totally $\mathbb A$ -irreducible.

4.3 Reduction to Theorem 4.1

Using the above preparations, we are now ready to explain the following reduction step.

Proof Proof of Theorem 1.3 assuming Theorem 4.1

By Lemma 4.2 it suffices to consider adelic actions for the proof of Theorem 1.3. So let $d\geq 2$ , let $\alpha $ be an adelic $\mathbb {Z}^d$ -action without virtually cyclic factors, and let $\mu $ be an $\alpha $ -invariant and ergodic probability measure.

By Lemma 4.3 there exists a finite index subgroup $\Lambda <\mathbb {Z}^d$ so that the restriction of $\alpha $ to $\Lambda $ satisfies the assumptions to Theorem 4.1. Note however, that the measure $\mu $ might not be ergodic with respect to $\alpha _\Lambda $ . Hence we may have to apply the ergodic decomposition. Since $\Lambda $ has finite index in $\mathbb {Z}^d$ , this ergodic decomposition simply takes the form

$$ \begin{align*} \mu=\frac1J(\mu_1+\cdots+\mu_J), \end{align*} $$

where the probability measures $\mu _j$ are $\alpha _\Lambda $ -invariant and ergodic for $j=1,\ldots ,J$ . Since $\mu $ is invariant under the full action $\alpha $ , we also have that for every $\mathbf {n}\in \mathbb {Z}^d$ and every index $j\in \{1,\ldots ,J\}$ there exists an index k with $\alpha ^{\mathbf {n}}_*\mu _j=\mu _k$ . Since ergodic measure are either equal or singular to each other, we may also assume that the measure $\mu _1,\ldots ,\mu _J$ are mutually singular to each other. By ergodicity of $\mu $ with respect to $\alpha $ , we also have that for every pair of indices $j,k$ there exists some $\mathbf {n}\in \mathbb {Z}^d$ so that $\alpha ^{\mathbf {n}}\mu _j=\mu _k$ .

We now apply Theorem 4.1 to $\mu _1$ and the restriction $\alpha _\Lambda $ . Therefore there exists a closed subgroup $G_1<X_m$ so that $\mu _1$ is invariant under translation by elements in $G_1$ and for any $\mathbf {n}\in \Lambda $ , we have $h_{\mu _1}(\alpha ^{\mathbf {n}}_{X_m/G_1})=0$ . Applying the above transitivity claim we obtain the theorem.

4.4 Standing assumptions for the proof of Theorem 4.1

Since we have shown that Theorem 4.1 implies Theorem 1.3, our aim for the the next sections is to show the former. Hence we will assume that $\alpha $ is an adelic $\mathbb {Z}^d$ -action on $X_m$ satisfying the assumptions of Theorem 4.1. Furthermore we assume that $\mu $ is an $\alpha $ -invariant and ergodic probability measure.

Suppose that $\mu $ is translation invariant under an adelic subgroup $Y<X_m$ , then by invariance of $\mu $ under $\alpha $ , it is also translation invariant under $\alpha ^{\mathbf {n}}(Y)$ . Taking the closed subgroup generated by these, it follows that there exists a maximal adelic subgroup $Y<X_m$ so that $\mu $ is invariant under translation by elements of Y and, moreover, that Y is $\alpha $ -invariant. We may replace $X_m$ by $X_m/Y$ for this maximal $\alpha $ -invariant adelic subgroup Y and consider the push forward of $\mu $ under the canonical projection map $X_m\rightarrow X_m/Y$ . If this new measure has zero entropy, Theorem 4.1 already holds for this action. Hence we may and will assume for the proof of Theorem 4.1 that:

1. (i) $\mu $ is not invariant under any adelic subgroup; and

2. (ii) there exists some $\mathbf {n}\in \mathbb {Z}^d$ so that $h_\mu (\alpha ^{\mathbf {n}})>0$ .

Our aim is to derive a contradiction from these assumptions.

To be able to apply the method introduced in [Reference Einsiedler and Lindenstrauss7] (relying on arithmetic properties of $\mathbb A$ -irreducible actions), we start by applying Lemma 2.4 to the adelic action $\alpha $ on $X_m$ . We now think of the chain of invariant subgroups as defining a chain of factor maps,

$$ \begin{align*} X=X_{(0)}=X_m&\rightarrow X_{(1)}=X/Y_1\rightarrow\cdots\\ &\rightarrow X_{(j)}=X/Y_{j}\rightarrow X_{(j+1)}=X/Y_{j+1}\rightarrow\cdots \rightarrow X_{(r)}=\{0\}. \end{align*} $$

Applying the Rokhlin entropy addition formula inductively to these factors, we have

$$ \begin{align*} h_\mu(\alpha^{\mathbf{n}})=\sum_{j=0}^{r-1}h_\mu(\alpha^{\mathbf{n}}_{X_{(j)}}\mid X_{(j+1)}) \end{align*} $$

for all $\mathbf {n}\in \mathbb {Z}^d$ , where we write $\mu $ for the invariant measure on all of the factors, write $\alpha ^{\mathbf {n}}_{X_{(j)}}$ for the induced action on the factor $X_{(j)}$ , and write $h_\mu (\alpha ^{\mathbf {n}}_{X_{(j)}}\mid X_{(j+1)})$ for the conditional entropy of $\alpha ^{\mathbf {n}}_{X_{(j)}}$ conditioned on the next factor $X_{(j+1)}$ .

By assumption (ii) above, there exists some $\mathbf {n}\in \mathbb {Z}^d$ so that $h_\mu (\alpha ^{\mathbf {n}})>0$ . Hence, we may choose the minimal $s\in \{0,\ldots ,r-1\}$ so that $h_\mu (\alpha ^{\mathbf {n}}_{X_{(s)}}\mid X_{(s+1)})>0$ for some $\mathbf {n}\in \mathbb {Z}^d$ . We define $Y_{\mathrm {base}}=Y_{s+1}$ , $X_{\mathrm {base}}=X_{(s+1)}$ , and will always consider conditional entropy over the factor $X_{\mathrm {base}}$ . We also define $Y_{\mathrm {pos}}=Y_s$ (contained in $Y_{\mathrm {base}}$ ) and $X_{\mathrm {pos}}=X/Y_{\mathrm {pos}}$ (which factors on $X_{\mathrm {base}}$ ). In this sense, the factor $X_{{\mathrm {pos}}}$ will be important for us as it is a ‘positive entropy extension’ of $X_{\mathrm {base}}$ and at the same time, an ‘adelic extension with $\mathbb A$ -irreducible fibers’. In fact by choice of s, there exists some $\mathbf {n}\in \mathbb {Z}^d$ so that $h_\mu (\alpha ^{\mathbf {n}}_{X_{{\mathrm {pos}}}}\mid X_{\mathrm {base}})>0$ , we have $X_{{\mathrm {pos}}}=X/Y_{{\mathrm {pos}}}$ , $X_{\mathrm {base}}=X/Y_{\mathrm {base}}$ , and that the action induced on the fibers $Y_{{\mathrm {irred}}}=Y_{\mathrm {base}}/Y_{{\mathrm {pos}}}$ is (totally) $\mathbb A$ -irreducible. Finally, as we have chosen s minimally, the original system X is a zero entropy extension of $X_{{\mathrm {pos}}}$ in the sense that ${\sf h}_\mu (\alpha ^{\mathbf {n}}\mid X_{{\mathrm {pos}}})=0$ for all $\mathbf {n}\in \mathbb {Z}^d$ .

To summarize the factor maps,

(4.1) $$ \begin{align} X\rightarrow X_{{\mathrm{pos}}}\rightarrow X_{\mathrm{base}} =X_{{\mathrm{pos}}}/Y_{{\mathrm{irred}}} \end{align} $$

describe the original action as a zero-entropy extension of $X_{{\mathrm {pos}}}=X/Y_{\mathrm {pos}}$ , and $X_{\mathrm {base}}=X/Y_{\mathrm {base}}$ as the quotient of $X_{{\mathrm {pos}}}$ by an $\alpha $ -invariant $\mathbb A$ -irreducible adelic subgroup $Y_{{\mathrm {irred}}} < X_{{\mathrm {pos}}}$ so that $X_{{\mathrm {pos}}}$ is a positive entropy extension over $X_{\mathrm {base}}$ . More precisely,

$$ \begin{align*} h_\mu(\alpha^{\mathbf n} \mid X_{{\mathrm{pos}}})=0 \quad\text{for all}\ \mathbf{n}\in\mathbb{Z}^d \end{align*} $$

but

$$ \begin{align*} h_\mu(\alpha^{\mathbf n}_{X _{{\mathrm{pos}}}}\mid X_{\mathrm{base}})>0 \quad\text{for some}\ \mathbf{n} \in \mathbb{Z}^d. \end{align*} $$

To the exact sequence $0 \to Y _{{\mathrm {irred}}} \to X _{{\mathrm {pos}}} \to X _{\mathrm {base}} \to 0$ , there is attached an exact sequence of $\mathbb Q$ -vector spaces $0 \to L _{\mathrm {base}}\to L _{{\mathrm {pos}}} \to L _{{\mathrm {irred}}} \to 0$ where $L_{\mathrm {base}}=\widehat {X_{\mathrm {base}}}$ , $L_{{\mathrm {pos}}}=\widehat {X_{{\mathrm {pos}}}}$ , and $L _{{\mathrm {irred}}}=\widehat {Y_{{\mathrm {irred}}}}$ can be identified as $\mathbb Q$ -vector space with a number field $\mathbb {K}$ . Viewing $X _{{\mathrm {pos}}}$ as a quotient of $\mathbb A ^{m_1}$ amounts to choosing a $\mathbb Q$ -basis $v _ 1, \ldots , v _ {m_1}$ for the vector space $L_{\mathrm {pos}}^*$ .

Identifying $\mathbb {K}$ as a $\mathbb Q$ -vector space with its dual using the trace form, we have an embedding $\mathbb {K} \to L _{{\mathrm {pos}}} ^{*}$ , and we will always choose our $\mathbb Q$ -basis so that $v _ 1, \ldots , v _ {[\mathbb {K}: \mathbb Q]}$ will be a basis for $\mathbb {K} < L ^{*} _{{\mathrm {pos}}}$ .

6.1 Lyapunov subgroups over $\mathbb Q_\sigma $

Fix $\sigma $ a place of $\mathbb Q$ , that is, either $\sigma =\infty $ or $\sigma =p$ a prime. We consider the linear maps $\alpha ^{\mathbf {n}}$ for $\mathbf {n}\in \mathbb {Z}^d$ on $\mathbb Q_\sigma ^m$ and are mostly interested in the asymptotic behavior of its elements with respect to the norm defined by

$$ \begin{align*} \|v\|_\sigma=\max_{j=1,\ldots,m}|v_j|_\sigma \end{align*} $$

for all $v\in \mathbb Q_\sigma ^m$ . We will also consider this behavior restricted to an $\alpha $ -invariant $\mathbb Q_\sigma $ -linear subspace V (even when not explicitly stated, V will always be assumed to be $\alpha $ -invariant). For this, we will initially ignore $\alpha _{\mathrm {uni}}$ (with polynomial behavior) and focus on $\alpha _{\mathrm {diag}}$ (with exponential behavior).

Since $\alpha _{\mathrm {diag}}$ is semisimple, $\mathbb Q^m_\sigma $ (as well as any $\alpha $ -invariant subspace V) is a direct sum of $\mathbb Q_\sigma $ -irreducible linear subspaces. On each of these irreducible subspaces, the action of $\alpha _{\mathrm {diag}}$ is isomorphic to the action defined by multiplication by d elements $\zeta _1,\ldots ,\zeta _d\in \mathbb {K}$ on a local field $\mathbb {K}$ (similar to the discussion in the proof of Proposition 2.2). Recall that we extend the the norm on $\mathbb Q_\sigma $ to $\mathbb {K}$ ; this extended norm is denoted by $|\cdot |_\sigma $ . We refer to the linear functional $\chi :\mathbb {Z}^d\to \mathbb {R}$ given by

$$ \begin{align*} \chi : (n_1,\ldots,n_d)\mapsto \sum_i n_i\log|\zeta_i|_\sigma\in\mathbb{R} \end{align*} $$

as the Lyapunov weight associated with the invariant subspace $\mathbb {K}$ , and denote the pairing of a functional $\chi $ and a vector $\mathbf {n} \in \mathbb {Z}^d$ (or $\mathbb {R}^d$ ) by $\chi \cdot \mathbf {n}$ . We note that for a vector v in this subspace, we have

(6.1) $$ \begin{align} \|\alpha_{\mathrm{diag}}^{\mathbf{n}} v\|_\sigma\asymp e^{\chi\cdot \mathbf{n}}\|v\|_\sigma \end{align} $$

for all $\mathbf {n}\in \mathbb {Z}^d$ , where we write $\asymp $ to indicate that we can bound each of the two terms by a multiple of the other. Here the implicit constants only depend on the action and not on the vector v or on $\mathbf {n}$ . We will call $\mathbb {K}$ an $\mathbb Q_\sigma $ -irreducible eigenspace (for $\alpha _{\mathrm {diag}}$ ) with Lyapunov weight $\chi $ .

It follows that $\mathbb Q_\sigma ^m$ (respectively V) is isomorphic to a finite direct product of local fields $\mathbb {K}$ extending $\mathbb Q_\sigma $ so that the linear maps $\alpha _{\mathrm {diag}}^{\mathbf {e}_1},\ldots ,\alpha _{\mathrm {diag}}^{\mathbf {e}_d}$ are written in diagonal form using this isomorphism. Moreover, we obtain in this way finitely many Lyapunov weights arising from the action on $\mathbb Q_\sigma ^m$ . Note however that these Lyapunov weights are all functionals into $\mathbb {R}$ (independent of the place $\sigma $ ).

6.2 Coarse Lyapunov weights and subgroups

We now apply the above for each place $\sigma $ of $\mathbb Q$ . Let S be a finite set of places containing $\infty $ so that $\alpha _{\mathrm {diag}}^{\mathbf {e}_1},\ldots ,\alpha _{\mathrm {diag}}^{\mathbf {e}_d}$ all belong to $\operatorname {GL}_m(\mathbb {Z}_\sigma )$ for $\sigma \notin S$ . For $\sigma \notin S$ , the only Lyapunov weight for the action of $\alpha _{\mathrm {diag}}$ on $\mathbb Q_\sigma $ is the zero weight. Hence by varying $\sigma $ over all places of $\mathbb Q$ , we only obtain finitely many non-zero Lyapunov weights.

We say that two non-zero Lyapunov weights $\chi ,\chi ^{\prime }$ (possibly arising from different places of $\mathbb Q$ ) are equivalent if there exists some $t>0$ so that $\chi ^{\prime }=t\chi $ . We will denote the equivalence class of a non-zero Laypunov weight $\chi $ by $[\chi ]$ and will refer to $[\chi ]$ as a coarse Lyapunov weight.

For a coarse Lyapunov weight $[\chi ]$ , we define the coarse Lyapunov subgroup $W^{[\chi ]}$ to be the S-linear subspace defined as the subgroup generated by all irreducible eigenspaces with Lyapunov weight $\chi ^{\prime }$ equivalent to $\chi $ . Alternatively, we can use (6.1) to see that the coarse Lyapunov subgroup could also be defined as an intersection of stable horospherical subgroups for $\alpha _{\mathrm {diag}}$ , namely

$$ \begin{align*} W^{[\chi]}=\bigcap_{\mathbf{n}\in\mathbb{Z}^d: \chi\cdot\mathbf{n}<0}U^-_{\alpha_{\mathrm{diag}}^{\mathbf{n}}}; \end{align*} $$

this relation to the horospherical subgroups is the reason for the dynamical importance of the coarse Lyapunov subgroups.

Because of the polynomial nature of $\alpha _{\mathrm {uni}}$ , we have that $U^-_{\alpha _{\mathrm {diag}}^{\mathbf {n}}}=U^-_{\alpha ^{\mathbf {n}}}$ for all $\mathbf {n}\in \mathbb {Z}^d$ and hence $W^{[\chi ]}$ can be defined directly in terms of the action $\alpha $ by

$$ \begin{align*} W^{[\chi]}=\bigcap_{\mathbf{n}\in\mathbb{Z}: \chi\cdot \mathbf{n}<0}U^-_{\alpha^{\mathbf{n}}}. \end{align*} $$

In particular, $W^{[\chi ]}$ is invariant under $\alpha ^{\mathbf {n}}$ for all $\mathbf {n}\in \mathbb {Z}^d$ .

Conversely, any stable horospherical group is a product of coarse Lyapunov subgroups: indeed, for any $\mathbf {n}\in \mathbb {Z}^d$ , we have

(6.2) $$ \begin{align} U^-_{\alpha^{\mathbf{n}}}=U^-_{\alpha_{\mathrm{diag}}^{\mathbf{n}}}=\bigoplus_{[\chi]: \chi\cdot\mathbf{n}<0}W^{[\chi]}, \end{align} $$

where the direct sum runs over all the coarse Lyapunov subspaces $W^{[\chi ]}$ satisfying that $\chi \cdot \mathbf {n}<0$ .

Given an $\alpha $ -invariant S-linear subgroup $V<U^-_{\alpha ^{\mathbf {n}}}$ , we define $V^{[\chi ]}=V\cap W^{[\chi ]}$ for any non-zero coarse Lyapunov weight $[\chi ]$ . These also satisfy that V is the direct sum of $V^{[\chi ]}$ for all $[\chi ]$ with $\chi \cdot \mathbf {n}<0$ .

6.3 Product structure of leafwise measures

Recall that we assume that $\mu $ is as in Theorem 4.1. Let $\Omega $ be an arbitrary compact metric space as in §3.1 equipped with an action of $\mathbb {Z}^{d}$ , let $\widetilde {X}=X\times \Omega $ , and consider an invariant probability measure $\widetilde {\mu }$ on $\widetilde {X}$ projecting to $\mu $ .