INVARIANT SETS AND MEASURES OF NONEXPANSIVE GROUP AUTOMORPHISMS

ELON LINDENSTRAUSS AND KLAUS SCHMIDT

Abstract. We prove that the restriction of a probability measure invariant under a nonhyperbolic, ergodic and totally irreducible automorphism of a compact connected abelian group to the leaves of the central foliation is severely restricted. We also prove a topological analogue of this result: the intersection of every proper closed invariant subset with each central leaf is compact.

1. Introduction

A continuous automorphism α of an additive compact abelian group X is expansive if there exists a neighbourhood N(0) of the identity 0 ∈ X with ⋂_{n∈ℤ} α^{n}(N(0)) = {0}, irreducible if every closed α-invariant subgroup Y ⊊ X is finite, totally irreducible if every nonzero power of α is irreducible, and ergodic if it is topologically transitive (and hence ergodic with respect to the normalized Haar measure λ_{X} of X).

In this paper we study the collection of invariant measures of a nonhyperbolic, ergodic and totally irreducible automorphism of the n-torus 𝕋^{n} or, more generally, of a nonexpansive, ergodic and totally irreducible automorphism α of a compact connected abelian group X. Every nonhyperbolic, ergodic and irreducible automorphism α of 𝕋^{n} is partially hyperbolic in the usual sense^{1} with the additional property that its derivative Dα preserves the length of vectors in E_{c}. For an arbitrary nonexpansive, ergodic and totally irreducible continuous automorphisms α of a compact connected abelian group X these ‘sub-bundles’ can be more complicated objects (due to the fact that the group need not be locally connected), but an analogue of this strong form of partial hyperbolicity also holds in this more general situation.

2000 Mathematics Subject Classification. 37A05, 37A45, 37B05, 37C40.

Key words and phrases. Nonexpansive group automorphisms, nonhyperbolic toral automorphisms, invariant measures.

1A C^{2} diffeomorphism f of a Riemannian manifold M is partially hyperbolic if there is a Df-invariant splitting T M = E_{s} ⊕ E_{c} ⊕ E_{u} of the tangent manifold T M of M in which at least two of the sub-bundles are nontrivial, so that Df uniformly expands all vectors in E_{u}, uniformly contracts all vectors in E_{s}, and the vectors in E_{c} are neither expanded as strongly as any vector in E_{u} nor contracted as strongly as any vector in E_{s}.

Let α be a nonexpansive, ergodic and totally irreducible automorphism of a compact connected abelian group X. The normalized Haar measure λ_{X} of X is obviously invariant under α, and Y. Katznelson [6] proved that the measure-preserving system (X, α, ℬ_{X}, λ_{X}) (where ℬ_{X} denotes the Borel sigma-algebra of X) is measure-theoretically isomorphic to a Bernoulli shift.

There is another family of — admittedly not very interesting — α-invari-ant ergodic probability measures on X: let X^{(0)} ⊂ X be the dense central subgroup of α defined in (3.3), on which α acts isometrically. Then the closure of the α-orbit of any element x ∈ X^{(0)} is a compact α-invariant subset of X^{(0)} (and hence of X) on which α acts with a unique α-invariant measure denoted by λ̃_{x}.

It is not immediate how to construct other invariant measures; in fact, the main result in this paper shows that all α-invariant probability measures μ ≠ λ_{X} on X satisfy a somewhat surprising rigidity phenomenon related to the scarcity of invariant measures under a multidimensional abelian semigroup of toral endomorphisms. This scarcity of invariant measures was conjectured by H. Furstenberg and is still open, though there are important partial results by several authors including D. Rudolph [9] for the one-dimensional case and A. Katok and R. Spatzier [5] in the higher-dimensional case.

In order to describe this rigidity property we use a construction from [5] to define a system of ‘conditional’ measures on the leaves of the central foliation induced by an α-invariant measure μ on X. In general, if we start with an α-invariant probability measure μ on X, these leaf measures will only be sigma-finite. Indeed, for μ = λ_{X}, the induced measure on each central leaf is the (infinite) Haar measure on the leaf. Our main result is that the leaf measures are finite for any α-invariant probability measure μ on X which does not contain a copy of λ_{X} in its ergodic decomposition.

Theorem (Theorem 5.1). Let α be a nonexpansive, ergodic and totally irreducible automorphism of a compact connected abelian group X with normalized Haar measure λ_{X}, and let μ be an α-invariant probability measure on X which is singular with respect to λ_{X}. Then the conditional measure ρ_{x} on the central leaf through x (defined in (4.20)) is finite for almost every x ∈ X.

Both the statement and the proof of Theorem 5.1 are modelled on Host’s proof of Rudolph’s Theorem in [3] and its generalization in [4].

The following two definitions can easily be adapted to the general setting of partially hyperbolic maps.

Definition 1.1. Two α-invariant probability measures μ_{1}, μ_{2} on X are centrally equivalent if they have an invariant joining ν (i.e. an (α × α)-invariant measure ν on X × X which projects to μ_{1} and μ_{2}, respectively) so that, for ν-a.e. (x, y) ∈ X × X, x and y lie on the same central leaf; in other words,

x − y ∈ X^{(0)} for ν-a.e. (x, y) ∈ X × X,

where X^{(0)} ⊂ X is the central subgroup of α defined in (3.3).

Definition 1.2. An α invariant probability measure μ on X is virtually hyperbolic if there exists an α-invariant Borel set Z ⊂ X with μ(Z) = 1 which intersects every central leaf in at most one point, i.e. with Z∩(x+Z) = ∅ for every x ∈ X^{(0)}.

In Section 6 we prove that Theorem 5.1 implies the following result.

Theorem 1.3. Let α be a nonexpansive, ergodic and totally irreducible automorphism of a compact connected abelian group X with normalized Haar measure λ_{X}, and let μ be an α-invariant probability measure on X which is singular with respect to λ_{X}. Then the following conditions are satisfied.

(1) There is a virtually hyperbolic α-invariant probability measure μ′ on X which is centrally equivalent to μ;

(2) If μ is weakly mixing (or, more generally, if the point spectrum of the action of α on L^{2}(X, 𝒮, μ) contains no eigenvalue of α of absolute value 1), then μ is virtually hyperbolic;

(3) If μ is ergodic, but not necessarily weakly mixing, we write, for every x ∈ X^{(0)}, λ̃_{x} for the unique α-invariant probability measure on X^{(0)} — and hence on X — concentrated on the compact orbit closure \overline{\{α^{n}x : n ∈ ℤ\}} of x under α. Then μ is an ergodic component of μ′ ∗ λ̃_{x_{0}} for some x_{0} ∈ X^{(0)}.^{2}

Finally, in Section 7 we prove the following topological analogue of the Theorems 1.3 and 5.1.

Theorem (Theorem 7.1). Let α be a nonexpansive, ergodic and totally irreducible automorphism of a compact connected abelian group X. Then any closed α-invariant subset Y ⊊ X intersects every central leaf in a compact subset of the leaf.

Acknowledgements. This research has been supported in part by NSF grant DMS 0140497 (E.L.) and FWF Project P16004-N05 (K.S.). During part of this work, both authors received support from the American Institute of Mathematics and NSF grant DMS 0222452. We would furthermore like to express our gratitude to the Mathematics Department, University of Washington, Seattle, the Newton Institute, Cambridge and ETH Zürich for hospitality during parts of this work, and Doug Lind for some interesting and useful discussions.

2More precisely: for every x_{0}, the ergodic decomposition of μ′ ∗ λ̃_{x_{0}} gives a probability measure ν on the space M_{1}(X) of probability measures on X, where we consider M_{1}(X)

as a compact metric space using the usual weak^{∗} topology. Then for an appropriate choice of x_{0}, the measure μ is in the support of ν.

2. Irreducible group automorphisms

Let α and β be continuous automorphisms of compact abelian groups X and Y, respectively. Then α and β are conjugate if there exists a continuous group isomorphism ϕ: X ⟶ Y with

β ∘ ϕ = ϕ ∘ α,  (2.1)

and β is a factor of α if there exists a continuous surjective group homomorphism ϕ: X ⟶ Y satisfying (2.1). The map ϕ in (2.1) is called an (algebraic) conjugacy or an (algebraic) factor map. The automorphisms α and β are weakly conjugate if each of them is a factor of the other, and finitely equivalent if each of them is a factor of the other with a finite-to-one factor map.

We recall a few basic facts about irreducible ergodic automorphisms of compact abelian groups. Let R_{1} = ℤ[u^{±1}] be the ring of Laurent polynomials with integral coefficients. We write every h ∈ R_{1} as

h = ∑_{m∈ℤ}h_{m}u^{m}  (2.2)

with h_{m} ∈ ℤ for every m ∈ ℤ and h_{m} = 0 for all but finitely many m.

Let α be an automorphism (always assumed to be continuous) of a compact abelian group X with (additive) dual group X̂, and let α̂ be the dual automorphism of X̂ defined by

⟨α̂a, x⟩ = ⟨a, αx⟩

for every x ∈ X and a ∈ X̂, where ⟨a, x⟩ denotes the value of a ∈ X̂ at x ∈ X. For every h = ∑_{n∈ℤ} h_{n}u^{n} ∈ R_{1}, x ∈ X and a ∈ X̂ we set

h(α)(x) = ∑_{n∈ℤ}h_{n}α^{n}x,  h(α̂)(a) = ∑_{n∈ℤ}h_{n}α̂^{n}a,  (2.3)

and note that

⟨h(α̂)(a), x⟩ = ⟨ĥ(α)(a), x⟩ = ⟨a, h(α)(x)⟩.  (2.4)

The dual group X̂ is a module over the ring R_{1} with operation

h · a = h(α̂)(a)  (2.5)

for h ∈ R_{1} and a ∈ X̂. In particular,

u^{m} · a = α̂^{m}a  (2.6)

for m ∈ ℤ and a ∈ X̂. This module is called the dual module M = X̂ of α. Conversely, if M is an R_{1}-module, we obtain an automorphism α_{M} on the compact abelian group

X_{M} = M̂  (2.7)

whose dual automorphism is defined by

α̂_{M}a = u · a  (2.8)

for every a ∈ M.

Examples 2.1 ([11]). (1) Let M = R_{1}. Since R_{1} is isomorphic to the direct sum ∑_{ℤ} ℤ of copies of ℤ, indexed by ℤ, the dual group X = R̂_{1} is isomorphic to the cartesian product 𝕋^{ℤ} of copies of 𝕋 = ℝ/ℤ. We write a typical element x ∈ 𝕋^{ℤ} as x = (x_{n}) with x_{n} ∈ 𝕋 for every n ∈ ℤ and choose the following identification of X_{R_{1}} = R̂_{1} and 𝕋^{ℤ}: for every x = (x_{n}) in 𝕋^{ℤ} and h = ∑_{n∈ℤ} h_{n}u^{n} ∈ R_{1},

⟨x, h⟩ = e^{2πi ∑_{n∈ℤ} h_{n}x_{n}}.  (2.9)

Under this identification the automorphism α_{R_{1}} on X_{R_{1}} = 𝕋^{ℤ} becomes the shift

(τx)_{m} = x_{m+1}  (2.10)

with m ∈ ℤ and x = (x_{m}) ∈ X_{R_{1}} = 𝕋^{ℤ}.

(2) Let I ⊂ R_{1} be an ideal, and let M = R_{1}/I. Since M is a quotient of the additive group R_{1} by an α̂_{R_{1}}-invariant subgroup, the dual group X_{M} is the α_{R_{1}}-invariant subgroup

X_{R_{1}/I} = I^{⊥} = {x ∈ X_{R_{1}} = 𝕋^{ℤ} : ⟨x, h⟩ = 1 for every h ∈ I}

= {x ∈ 𝕋^{ℤ} : ∑_{n∈ℤ}h_{n}x_{m+n} = 0 (mod 1)

for every h ∈ I and m ∈ ℤ}

= {x ∈ 𝕋^{ℤ} : h(τ)(x) = 0 for every h ∈ I},

(2.11)

and α_{R_{1}/I} is the restriction of τ = α_{R_{1}} to X_{R_{1}/I} ⊂ 𝕋^{ℤ} = X_{R_{1}}.

We can express (2.11) as

X_{R_{1}/I} = X̂/I = I^{⊥} = ⋂_{h∈I} ker(h(τ)).  (2.12)

If I = (f) = fR_{1} is the principal ideal generated by some f ∈ R_{1}, then (2.12) becomes

X_{R_{1}/(f)} = { \atop X/(f)} = (f)^{⊥} = ker(f(τ)).  (2.13)

(3) Let α be the automorphism of the m-torus X = 𝕋^{m} = ℝ^{m}/ℤ^{m} defined by a matrix A ∈ GL(m, ℤ). Then the dual module M = X̂ is equal to ℤ^{m} with operation f · m = f(A^{⊤})(m) for every f ∈ R_{1} and m ∈ ℤ^{m} (cf. (2.5)), where A^{⊤} ∈ GL(m, ℤ) is the transpose matrix of A.

The automorphism α is irreducible if and only if the characteristic polynomial f = f_{0} + ⋯ + f_{m−1}u^{m−1} + u^{m} of A is irreducible, and α is conjugate to α_{R_{1}/(f)} if and only if A is conjugate in GL(m, ℤ) to the companion matrix

C_{f} = ( \begin{matrix}0 & 1 & ··· & 0 & 0 \\ 0 & 0 & ··· & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ··· & 0 & 1 \\ −f_{0} & −f_{1} & ... & −f_{m−2} & −f_{m−1} \end{matrix} ) ∈ GL(m, ℤ).  (2.14)

Theorem 2.2. Let α be an irreducible automorphism of an infinite compact connected abelian group X. Then there exists a unique irreducible polynomial f = f_{0} + ⋯ + f_{n}u^{n} ∈ R_{1} with the following properties.

(1) n ≥ 1, f_{n} > 0 and f_{0} ≠ 0;

(2) α is finitely equivalent to α_{R_{1}/(f)}, where (f) = fR_{1} ⊂ R_{1} is the ideal generated by f (cf. Example 2.1 (2));

(3) α is ergodic if and only if f is not cyclotomic (i.e. f does not divide u^{m} − 1 for any m ≥ 1);

(4) α is expansive if and only if f has no roots of absolute value 1.

(5) α is totally irreducible if and only if f has no two distinct roots whose

ratio is a root of unity. Conversely, if f = f_{0}+⋯+f_{n}u^{n} ∈ R_{1} is an irreducible polynomial satisfying condition (1) above, then the group X_{R_{1}/(f)} in (2.11) is connected and the automorphism α_{R_{1}/(f)} of X_{R_{1}/(f)} is irreducible.

Proof. The statements (1)–(4) and the converse follow from [11, Proposition 2.7 and Theorem 29.2].

For the proof of (5) we note that α is totally irreducible if and only if α_{R_{1}/(f)} is totally irreducible. Since α_{R_{1}/(f)}^{m} is dual to multiplication by u^{m} on X̂ = R_{1}/(f) = M, α_{R_{1}/(f)}^{m} is irreducible if and only if the subgroup

N = {h(u^{m}) : h ∈ R_{1}}/(f) ⊂ R_{1}/(f) = M

has finite index in R_{1}/(f). As the group M is torsion-free, the latter condition is equivalent to the statement that N ⊗_{ℤ}ℚ = M ⊗_{ℤ}ℚ ≅ ℚ^{n}, and hence to the condition that the elements {(1+(f)), (u^{m}+(f)), …, (u^{m(n−1)}+(f))} in M are rationally independent. In other words, α_{R_{1}/(f)}^{m} is reducible if and only if one can find a nonzero element (k_{0}, …, k_{n−1}) ∈ ℤ^{n} with

g(u^{m}) = k_{0} + k_{1}u^{m} + ⋯ + k_{n−1}u^{m(n−1)} ∈ (f),  (2.15)

where we may assume without loss of generality that the resulting polynomial g ∈ R_{1} is irreducible. By evaluating (2.15) on any root θ of f we obtain that g(θ^{m}) = 0 for every root θ of f, and Galois theory shows that the degree of g is equal to the number of distinct elements in the set Ω_{f}^{(m)} = {θ^{m} : θ is a root of f}. This proves that α_{R_{1}/(f)}^{m} is irreducible if

and only if the cardinality of Ω_{f}^{(m)} is equal to n, which implies (5). □

Example 2.1 (2) gives an explicit representation — up to finite equivalence — of every irreducible automorphism of a compact connected abelian group X. For an alternative description we follow [2] (for background see [10], [11, Section 7] and [12]).

Let α be an irreducible automorphism of an infinite compact connected abelian group X, and let f ∈ R_{1} be the irreducible polynomial appearing in Theorem 2.2. We fix a root θ ∈ ℚ̄ of f, denote by K = ℚ(θ) the algebraic number field generated by θ, and write P^{(K)}, P_{f}^{(K)}, and P_{∞}^{(K)}, for the sets of places (= equivalence classes of valuations), finite places and infinite places of K. For every place v of K and every valuation ϕ ∈ v, the v-adic completion K_{v} of K (i.e. the completion of K with respect to metric δ(a, b) = ϕ(a − b)^{γ} for some suitable γ > 0) is a locally compact, metrizable field and hence a locally compact additive group. We fix a Haar measure λ_{v} on the additive group K_{v} and denote by mod_{K_{v}}: K_{v} ⟶ ℝ the map satisfying

λ_{v}(aB) = mod_{K_{v}}(a)λ_{v}(B)  (2.16)

for every a ∈ K_{v} and every Borel set B ⊂ K_{v}. The restriction of mod_{K_{v}} to K is a valuation in v, denoted by | · |_{v}.

Let

P = {v ∈ P_{f}^{(K)} : |θ|_{v} ≠ 1}, S = P_{∞}^{(K)} ∪ P.  (2.17)

For every infinite place v ∈ P_{∞}^{(K)}, the v-adic completion K_{v} is either equal to ℝ or to ℂ (in particular, K_{v} = ℂ for any v ∈ S^{(0)}). We write

ι_{v}: K ⟶ K_{v}(= ℝ or ℂ)  (2.18)

for the embedding of K in its completion K_{v} and use the same symbol ι_{v} to denote the corresponding identification of K_{v} with ℝ or ℂ.

The set

W = ∏_{v∈S} K_{v}  (2.19)

is a locally compact algebra over K with respect to coordinate-wise addition, multiplication and scalar multiplication (with scalars in K). We write every w ∈ W as w = (w_{v}) = (w_{v}, v ∈ S) with w_{v} ∈ K_{v} for every v ∈ S and define

∥w∥ = max_{v∈S} |w_{v}|_{v}.  (2.20)

Let β̄ be the automorphism of W given by

β̄w = (θw_{v})  (2.21)

for every w = (w_{v}) ∈ W.

We put

ℛ = {a ∈ K : |a|_{v} ≤ 1 for every v ∈ P^{(K)} ∖ S} ⊃ 𝔬_{K},  (2.22)

where 𝔬_{K} is the ring of integers in K, and denote by

ι: K ⟶ W  (2.23)

the diagonal embedding a ↦ ι(a) = (a, …, a), a ∈ K. By abuse of notation we identify each K_{v}, v ∈ S, with the subgroup

{w ∈ W : w_{v′} = 0 for every v′ ≠ v} ⊂ W.

Theorem 2.3. Suppose that α is an automorphism of an infinite compact connected abelian group X. Then α is irreducible if and only if there exist an element θ ∈ ℚ̄^{×} = ℚ̄ ∖ {0} and a finitely generated ℤ[θ^{±1}]-submodule L ⊂ K = ℚ(θ) such that α is algebraically conjugate to the automorphism β_{(θ,L)} on the quotient group

Y_{L} = W/ι(L)  (2.24)

induced by β̄ (cf. (2.17)–(2.21)).

(1) The following conditions are equivalent. (a) α is ergodic,

(b) θ is not a root of unity.

(2) The following conditions are equivalent. (a) α is expansive,

(b) The orbit of θ under the action of the Galois group Gal[ℚ̄ : ℚ]

does not intersect 𝕊 = {z ∈ ℂ : |z| = 1}.

(3) The following conditions are equivalent. (a) X ≅ 𝕋^{n} for some n ≥ 1,

(b) S = P_{∞}^{(K)},

(c) θ is an algebraic unit.

(4) The following conditions are equivalent. (a) α is totally irreducible,

(b) The orbit of θ under the action of the Galois group Gal[ℚ̄ : ℚ] does not contain two distinct elements whose ratio is a root of unity.

Proof. [2, Corollary 3.5], [11, Theorem 7.1 and Propositions 7.2–7.3] and Theorem 2.2 in this paper. □

Corollary 2.4. In the notations of Theorem 2.3, for any θ ∈ ℚ̄^{×} and finitely generated ℤ[θ^{±1}]-submodule L ⊂ K = ℚ(θ), and any S′ ⊊ S there is an automorphism ψ of Y_{L} commuting with β_{(θ,L)} so that ψ expands the subgroup

W_{S′} = {w ∈ W : w_{v} = 0 for every v ∈ S \ S′} .

Proof. Let ℛ ⊃ 𝔬_{K} be the ring of S-integers as in (2.22). We claim that there is a τ ∈ ℛ so that

|τ|_{v} > 1  for all v ∈ S′.  (2.25)

Indeed, if θ a unit in 𝔬_{K}, or more generally if S′ ⊊ P_{∞}^{(K)}, (2.25) is a direct consequence of the Dirichlet unit theorem, and the general case follows from the extension of this theorem to the S-arithmetic context that can be proved in the same way [12, Chap. 4, Th. 9].

Since L is a finitely generated ℤ[θ^{±1}]-submodule, and θ generates K it follows that L is commensurable to ℛ, i.e. [L : L ∩ ℛ], [ℛ : L ∩ ℛ] < ∞. Define τ̄ : W → W using τ as in (2.21); since τ ∈ ℛ the map τ̄ preserves ℛ, and so a suitable power of τ̄, which without loss of generality we can assume to be τ̄ itself, preserves L.

It follows that ψ = β_{(τ,L)} is an automorphism of Y_{L} commuting with β_{(θ,L)} with the required properties. □

3. Structure and examples of nonexpansive automorphisms

Let α be a nonexpansive irreducible ergodic automorphism of a compact connected abelian group X. We apply the Theorem 2.3 and assume that

α = β_{(θ,L)},  X = W/ι(L),  (3.1)

for some θ ∈ ℚ̄^{×} and some finitely generated ℤ[θ^{±1}]-submodule L ⊂ K = ℚ(θ). Denote by λ_{X} the normalized Haar measure of X and write

π: W ⟶ X = W/ι(L)  (3.2)

for the quotient map (cf. (2.17)–(2.24)). In the notation of (2.17) and (2.19) we set

S^{(0)} = {v ∈ S : |θ|_{v} = 1} ⊂ P_{∞}^{(K)},
W^{(0)} = {w = (w_{v}) ∈ W : w_{v} = 0 for every v ∈ S ∖ S^{(0)}}

≅ ∏_{v∈S^{(0)}} K_{v} ≅ ℂ^{|S^{(0)}|},

X^{(0)} = π(W^{(0)}).

(3.3)

The central subgroup group X^{(0)} ⊂ X is α-invariant and dense by irreducibility. Furthermore, since |L/ι(ℛ)| < ∞ (cf. (2.22) and [2]) and ι(ℛ) ∩ W^{(0)} = {0} by the product formula ([1, Theorem 10.2.1]), L ∩ W^{(0)} = {0}.

Examples 3.1. (1) Let α be a nonexpansive irreducible ergodic automorphism of X = 𝕋^{m} defined by a matrix A ∈ GL(m, ℤ) with real eigenvalues θ_{1}, …, θ_{m_{1}} and complex eigenvalues θ_{m_{1}+1}, θ̄_{m_{1}+1}, …, θ_{m_{1}+m_{2}}, θ̄_{m_{1}+m_{2}}, where m = m_{1} + 2m_{2}, and where θ̄_{i} is the complex conjugate of θ_{i} for i = m_{1} + 1, …, m_{1} + m_{2}. We fix an eigenvalue θ of A, set K = ℚ(θ), and obtain that S = P_{∞}^{(K)}, W ≅ ℝ^{m_{1}} × ℂ^{m_{2}}, and that

W^{(0)} = ⨁_{\begin{matrix}j=m_{1}+1,...,m_{1}+m_{2} \\ |θ_{j}|=1 \end{matrix}} ℂ

is the subspace of W ≅ ℝ^{m} on which A acts isometrically. Since α is ergodic, dim_{ℝ}(W^{(0)}) ≤ dim_{ℝ}(W) − 2 = m − 2.

Take, for example, the irreducible ergodic and nonexpansive automorphism α of X = 𝕋^{4} determined by the matrix

A = ( \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ −1 & 1 & 1 & 1 \end{matrix} ) ∈ GL(4, ℤ).

If θ > 1 is the dominant eigenvalue of A, then the algebraic number field K = ℚ[θ] has two real places v_{1}, v_{2} (corresponding to the real roots θ_{1} = θ and θ_{2} = θ^{−1} of the characteristic polynomial f = u^{4} − u^{3} − u^{2} − u + 1 of A) and one complex place v_{3} (corresponding to the two complex roots θ_{3} and θ̄_{3} of f of absolute value 1). Then S^{(0)} = {v_{3}}, W^{(0)} ≅ K_{v_{3}} = ℂ, and the central subgroup X^{(0)} ⊂ X of α is a densely embedded copy of ℂ.

For another example of this form we take the automorphism α of X = 𝕋^{6} defined by the matrix

B = ( \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ −1 & 1 & 1 & 1 & 1 & 1 \end{matrix} ) ∈ GL(6, ℤ)

with dominant eigenvalue θ > 1. The algebraic number field K = ℚ[θ] has two real places v_{1}, v_{2} (corresponding to the real roots θ_{1} = θ and θ_{2} = θ^{−1} of the characteristic polynomial f = u^{6} − u^{5} − u^{4} − u^{3} − u^{2} − u + 1 of B) and two complex places v_{3}, v_{4} (corresponding to the four complex roots θ_{3}, θ_{4} and θ̄_{3}, θ̄_{4} of f of absolute value 1). Then S^{(0)} = {v_{3}, v_{4}}, W^{(0)} ≅ K_{v_{3}}⊕K_{v_{4}} = ℂ^{2}, and the central subgroup X^{(0)} ⊂ X of α is a densely embedded copy of ℂ^{2}.

(2) Let f = 5u^{2}−6u+5 ∈ R_{1}, and let α = α_{R_{1}/(f)} be the automorphism of the compact connected abelian group X = X_{R_{1}/(f)} defined in (2.11). Since f is irreducible and all roots of f have absolute value 1 (they are of the form θ = \frac{3}{5} ±i· \frac{4}{5}), α is ergodic and nonexpansive by Theorem 2.2. If θ is a root of f and K = ℚ(θ), then P ⊂ P_{f}^{(K)}, S^{(0)} = P_{∞}^{(K)}, W = W^{(0)} × ∏_{v∈P} K_{v}, where W^{(0)} ≅ ℂ and ∏_{v∈P} K_{v} is zero-dimensional, and X ≅ W/L for some discrete co-compact β̄-invariant subgroup L ⊂ W. In this example the central subgroup X^{(0)} ⊂ X is a densely embedded copy of ℂ.

(3) Let f = 6u^{4} + 3u^{3} + 10u^{2} + 6u + 6 ∈ R_{1}, and let α = α_{R_{1}/(f)} be the automorphism of the compact connected abelian group X = X_{R_{1}/(f)} defined in (2.11). Again f is irreducible, all roots of f have absolute value 1, and α is ergodic and nonexpansive by Theorem 2.2. If θ is a root of f and K = ℚ(θ), then P ⊂ P_{f}^{(K)}, S^{(0)} = P_{∞}^{(K)}, W = W^{(0)} × ∏_{v∈P} K_{v}, where W^{(0)} ≅ ℂ^{2} and ∏_{v∈P} K_{v} is zero-dimensional, and X ≅ W/L for some discrete co-compact β̄-invariant subgroup L ⊂ W. Here the central subgroup X^{(0)} ⊂ X is a densely embedded copy of ℂ^{2}.

The group W^{(0)} ≅ ℂ^{|S^{(0)}|} in (3.3) is an algebra with respect to coordinatewise addition and multiplication. We define a map ι_{0} : K ⟶ W^{(0)} by setting

ι(a)_{v} = \begin{cases}ι_{v}(a) & if v ∈ S^{(0)}, \\ 0 & if v ∈ S ∖ S^{(0)} \end{cases}

for every a ∈ K (cf. (2.18)), set

ξ = ι_{0}(θ),  (3.4)

where θ is the algebraic number appearing in Theorem 2.3 and (3.1), and denote by

Γ = \overline{\{ξ^{m} : m ∈ ℤ\}}  (3.5)

the closure of the multiplicative subgroup {ξ^{m} : m ∈ ℤ} ⊂ W^{(0)}. Then Γ is a compact abelian multiplicative subgroup of W^{(0)}. For every γ = (γ_{v}) ∈ Γ we denote by M_{γ}: W^{(0)} ⟶ W^{(0)} multiplication by γ, i.e.

M_{γ}w = (γ_{v}w_{v})  (3.6)

for every w = (w_{v}) ∈ W^{(0)}.

Proposition 3.2. If α is totally irreducible, then for any two distinct elements v, v′ ∈ S^{(0)}, the natural projection of Γ to K_{v} ⊕ K_{v′} is surjective.

Proof. Let ξ_{vv′} be defined by

(ξ_{vv′})_{ν} = \begin{cases}θ & for ν = v, v′, \\ 0 & otherwise. \end{cases}

Clearly the projection of Γ to K_{v} ⊕ K_{v′} is equal to \overline{\{ξ_{vv′}^{m} : m ∈ ℤ\}}. Let ξ_{v} = ι_{v}(θ) ∈ ℂ and ξ_{v′} = ι_{v′}(θ) ∈ ℂ. Since v, v′ ∈ S^{(0)} we know that |ξ_{v}| = |ξ_{v′}| = 1 (cf. (2.18)).

In order to prove our claim it suffices to show that, for any nonzero element (m, m′) ∈ ℤ^{2},

ξ_{v}^{m}ξ_{v′}^{m′} ≠ 1.  (3.7)

That ξ_{v}^{m} ≠ 1 for m ≠ 0 follows from ergodicity (ξ_{v} is a root of an irreducible polynomial with integer coefficients, which is noncylotomic if α is to be ergodic). To prove (3.7) for the case where both m, m′ ≠ 0, we note that since ξ_{v} and ξ_{v′} are conjugate under the Galois group of the splitting field of the polynomial f, we also have that ξ_{v′}^{m} = ξ_{3}^{m′} for some ξ_{3} ∈ ℂ with f(ξ_{3}) = 0 (it could be that ξ_{3} = ξ_{v}). We can now apply the same argument for ξ_{3} and obtain that ξ_{3}^{m} = ξ_{4}^{m′} for some root ξ_{4} ∈ ℂ of f, etc. Since f has finitely many roots, we will eventually get an equation of the form ξ_{j}^{m^{k}} = ξ_{j}^{(−m′)^{k}} for some positive integer k and some root ξ_{j} of f. As all roots of f are conjugate under the Galois group, this shows that

θ^{m^{k}} = θ^{(−m′)^{k}}.

If m^{k} ≠ (−m′)^{k} then θ is a root of unity, which is a contradiction to ergodicity. Otherwise m = ±m′, and either ξ_{v}^{m} = ξ_{v′}^{m} or ξ_{v}^{m} = ξ_{v′}^{−m}.

First suppose that ξ_{v}^{m} = ξ_{v′}^{m}. Since v and v′ are inequivalent valuations, ξ_{v} ≠ ξ_{v′}, and hence ξ_{v}ξ_{v′}^{−1} is a nontrivial root of unity, contrary to the hypothesis that α is totally irreducible (cf. Theorem 2.2).

If ξ_{v}^{m} = ξ_{v′}^{−m}, then the complex conjugate ξ′ = \overline{ξ_{v′}} of ξ_{v′} is again a root of f satisfying that ξ_{v}^{m} = ξ′^{m}, and the same argument as above shows that ξ_{v}ξ′^{−1} is a nontrivial root of unity. Again this violates the total irreducibility of α. □

4. Conditional measures on the leaves of the central foliation

We assume that α and X are of the form (3.1) and use the notation of (2.17)–(2.24). Write F for the foliation of X by the cosets of the central subgroup X^{(0)} = π(W^{(0)}) ⊂ X (cf. (3.3)), and fix a nonatomic α-invariant probability measure μ on the Borel field 𝒮 = ℬ_{X} of X. Note that we do not make any assumptions regarding ergodicity of μ.

Since the central subgroup X^{(0)} is dense by irreducibility, the foliation of X into cosets of X^{(0)} has no Borel cross-section, and one cannot generally decompose μ directly into a family of measures supported on the individual leaves of F. In order to overcome this difficulty we break up each of these leaves into countably many atoms of an appropriate sub-sigma-algebra 𝒜 ⊂ 𝒮, decompose the measure μ with respect to this sigma-algebra, and recombine the conditional measures supported by the individual atoms on each leaf into a leaf-measure.

It will be necessary to work not just with one such sigma-algebra 𝒜 but with a sequence (𝒜^{(k)}, k ≥ 1) of sigma-algebras whose atoms consist of larger and larger pieces of leaves of F. In order to describe these sigma-algebras we fix an integer q > 1 with |q|_{v} = 1 for every v ∈ P and set Λ = \frac{1}{q}L ⊂ K. Then ι(Λ) is a discrete co-compact subgroup of W (cf. (2.23)), and we choose a Borel set Δ ⊂ W with compact closure such that

Δ ∩ (Δ + ι(a)) = ∅ for every nonzero a ∈ Λ,

⋃_{a∈Λ} Δ + ι(a) = W. (4.1)

The first equation in (4.1) implies that the restriction to Δ of the map π: W ⟶ X in (3.2) is injective, and that π(Δ) is therefore a Borel subset of X. After replacing Δ by Δ + w_{0} for some w_{0} ∈ W, if necessary, we may take it that the sets

Q = {π(Δ + ι(a)) : a ∈ Λ}  (4.2)

form a Borel partition of X into N = |L/qL| sets with the following properties.

(i) μ(∂Q) = 0 for every Q ∈ Q; (ii) For every a ∈ Λ, the restriction of the map π in (3.2) to Δ + ι(a) is

injective, and π(Δ + ι(a)) = π(Δ + ι(a′)) if and only if a − a′ ∈ L; (iii) For every Q ∈ Q and w ∈ W, the set W^{(0)} ∩ (π^{−1}(Q) − w) is a

countable union of sets with disjoint and compact closures. Let T_{y} denote the map

T_{y}x = x + y  (4.3)

for every x, y ∈ X. We denote by B_{W^{(0)}}(w, r) the ball of radius r > 0 around w in W^{(0)}; while it will not be important for us which norm we use in W^{(0)}, the natural norm to take is

∥w∥ = max_{v∈S^{(0)}} |w_{v}|  (4.4)

(cf. (2.20)). Finally we write B_{F}(x, r) for the ball of radius r around x in the leaf x + π(W^{(0)}) of F, i.e.

B_{F}(x, r) = x + π(B_{W^{(0)}}(0, r)) = T_{x} ∘ π(B_{W^{(0)}}(0, r)).  (4.5)

Proposition 4.1. There exist a sequence of fundamental domains (Δ^{(n)}, n ≥ 1) for ι(Λ) and a corresponding sequence of partitions (Q^{(n)}, n ≥ 1) of X in (4.2) with the properties (i)–(iii) above, such that

∑_{Q∈Q^{(n)}} μ((Q + π(B_{W^{(0)}}(0, n))) △ Q) ≤ 2^{−n}  (4.6)

for every n ≥ 1.

Proof. Let Δ ⊂ W be a fundamental domain for Λ such that the corresponding partition Q satisfies conditions (ii)–(iii) above and such that

λ_{X}(∂Q) = 0  for every Q ∈ Q.  (4.7)

Let β_{(τ,L)} be an automorphism of X commuting with α expanding W^{(0)} as in Corollary 2.4. Set Δ_{n} = β_{(τ,L)}^{k_{n}}(Δ) with corresponding partitions Q_{n}. Clearly these also satisfy (ii)–(iii); furthermore, since Q satisfied (4.7), if k_{n} increases fast enough we can guarantee that

∑_{Q∈Q_{n}} λ_{X}((Q + π(B_{W^{(0)}}(0, n))) △ Q) ≤ 2^{−n}

for every n ≥ 1. Since for any two Borel sets Q, B ⊂ X,

∫ μ((Q + s + B) △ (Q + s)) dλ_{X}(s)
= ∫∫ |1_{Q+s+B}(x) − 1_{Q+s}(x)| dμ(x) dλ_{X}(s)
= λ_{X}((Q + B) △ Q),

where 1_{Q+s+B} and 1_{Q+s} are the indicator function of the sets Q + s + B and Q + s, there is a sequence (x_{n}, n ≥ 1) so that the translated partitions Q^{(n)} = Q_{n} + x_{n} satisfy (4.6) and the conditions (i)–(iii) for every n ≥ 1. For later use we choose a bounded sequence (w_{n}, n ≥ 1) in W with π(w_{n}) = x_{n} for every n ≥ 1 and set Δ^{(n)} = Δ_{n} + w_{n}, n ≥ 1. □

Definition 4.2. Let 𝒜 ⊂ 𝒮 be a countably generated sigma-algebra, and let C ⊂ 𝒜 be a countable algebra which generates 𝒜. The atom [x]_{𝒜} of a point x ∈ X in 𝒜 is defined as

[x]_{𝒜} = ⋂_{C∈C:x∈C} C = ⋂_{A∈𝒜:x∈A} A.

Lemma 4.3. For every n ≥ 1, let Δ^{(n)} be the fundamental domain for ι(Λ) ⊂ W and Q^{(n)} the partition of X described in Proposition 4.1. Then there exist a countably generated sigma-algebra 𝒜^{(n)} ⊂ 𝒮 with

[x]_{𝒜^{(n)}} = π((Δ^{(n)} + ι(a)) ∩ (W^{(0)} + w))  (4.8)

for every x ∈ X, where [x]_{𝒜_{Q^{(n)}}} is the atom of 𝒜^{(n)} containing x and a ∈ Λ and w ∈ W satisfy that π(w) = x and w ∈ Δ^{(n)} + ι(a).

Proof. Fix a ∈ Λ for the moment. We set W′ = (∏_{v∈S∖S^{(0)}} K_{v}), denote by κ: W = W^{(0)} × W′ ⟶ W′ the second coordinate projection (cf. (3.3)), and write ℬ_{W′} for the (countably generated) Borel field of W′. The sigma-algebra 𝒜 = {κ^{−1}(B)∩(Δ+ι(a)) : B ∈ ℬ_{W′}} of subsets of Δ+ι(a) is again countably generated, and its atoms are of the form (Δ+ι(a))∩(W^{(0)}+w), w ∈ Δ+ι(a). Since the restriction of π to Δ + ι(a) is injective, π maps 𝒜 to a countably generated sigma-algebra 𝒜_{Q} of subsets of Q = π(Δ + ι(a)) ∈ Q^{(n)} whose atoms are of the required form. The sigma-algebra 𝒜^{(n)} is defined as the unique sub-sigma-algebra of 𝒮 = ℬ_{X} which contains the partition Q^{(n)} and induces 𝒜_{Q} on each Q ∈ Q^{(n)}. □

For any countably generated sigma-algebra 𝒜 ⊂ 𝒮 we consider the decomposition of μ with respect to the sigma-algebra 𝒜, i.e. a set of probability measures {μ_{x}^{𝒜} : x ∈ X} on X with the following properties.

(1) For all x, x′ ∈ X with [x]_{𝒜} = [x′]_{𝒜} ,

μ_{x}^{𝒜} = μ_{x′}^{𝒜} and μ_{x}^{𝒜} ([x]_{𝒜} ) = 1,

(2) For every B ∈ 𝒮, the map x ↦ μ_{x}^{𝒜} (B) is Borel (and hence 𝒜measurable),

(3) For every bounded Borel map f: X ⟶ ℝ,

∫ f dμ_{x}^{A} = E_{μ}(f|A)(x)

for μ-a.e. x ∈ X, where E_{μ}(·|·) denotes conditional expectation. In order to make notation less cumbersome we set, for every n ∈ ℤ and

k ≥ 1,

𝒜_{n}^{(k)} = α^{−n}(𝒜_{Q^{(k)}}),  𝒜^{(k)} = 𝒜_{0}^{(k)} = 𝒜_{Q^{(k)}},  (4.9)

and denote by {μ_{x}^{𝒜_{n}^{(k)}} : x ∈ X} and {μ_{x}^{𝒜^{(k)}∨𝒜_{n}^{(k)}} : x ∈ X} the decompositions of μ with respect to the sigma-algebras 𝒜_{n}^{(k)} and 𝒜^{(k)} ∨ 𝒜_{n}^{(k)}, respectively.

Definition 4.4. A Borel measure ρ on W^{(0)} is locally finite if ρ(C) < ∞ for every compact set C ⊂ W^{(0)}. Let M_{∞}(W^{(0)}) be the set of all locally finite (and hence sigma-finite) Borel measures on W^{(0)}, furnished with the smallest topology in which the map ρ ↦ ∫ f dρ from M_{∞}(W^{(0)}) to ℝ is continuous for every continuous map f: W^{(0)} ⟶ ℝ with compact support. In this topology M_{∞}(W^{(0)}) is a separable metrizable space.

For every w ∈ W^{(0)} we denote by

T̄_{w}v = v + w,  (4.10)

the translation by w on W^{(0)}. The maps ρ ↦ ρT̄_{w} and ρ ↦ ρβ̄ are homeomorphisms of M_{∞}(W^{(0)}) for every w ∈ W^{(0)}, where β̄ is defined in (2.21).

For the next theorem, we take r_{0} to be large enough so that

[x]_{𝒜^{(1)}} ⊂ B_{F}(x, r_{0})  (4.11)

for all x ∈ X (cf. (4.5)).

Proposition 4.5. There is a Borel map x ↦ ρ_{x} from X to M_{∞}(W^{(0)}) and an α-invariant Borel set X′ of full μ-measure with the following properties (for notation we refer to (2.21), (4.9) and (4.10)).

(1) For every x ∈ X′, every bounded Borel set B ⊂ W^{(0)} and every sufficiently large k,

ρ_{x}(B) = \frac{1}{μ_{x}^{𝒜^{(k)}} (B_{F}(x, r_{0}))}μ_{x}^{𝒜^{(k)}} (T_{x} ∘ π(B));  (4.12)

(2) For every x ∈ X,

ρ_{x} = ρ_{αx}β̄;  (4.13)

(3) There exists a Borel map K_{μ}: X × W^{(0)} ⟶ ℝ so that, for every x ∈ X′ and every w ∈ W^{(0)} with x + π(w) ∈ X′,

e^{K_{μ}(x,w)}ρ_{x−π(w)} = ρ_{x}T̄_{w}.  (4.14)

We begin the proof of Proposition 4.5 with a lemma.

Lemma 4.6. There exists a Borel set X′ ⊂ X with μ(X′) = 1, which is invariant under T̄_{w} for every w ∈ W^{(0)}, so that for all x ∈ X′ and r > 0,

B_{F}(x, r) ⊂ [x]_{𝒜^{(k)}}  (4.15)

for every sufficiently large k.

Proof. The set

N_{r,k} = {x ∈ X : B_{F}(x, r) ⊄ [x]_{𝒜^{(k)}} }.

is equal to

⋃_{\begin{matrix}Q,Q′∈Q^{(k)} \\ Q≠Q′ \end{matrix}}(Q ∩ (Q′ + π(B_{W^{(0)}}(0, r)))) = ⋃_{Q′∈Q^{(k)}}((Q′ + π(B_{W^{(0)}}(0, r))) ∖ Q′)

⊂ ⋃_{Q′∈Q^{(k)}}((Q′ + π(B_{W^{(0)}}(0, r))) △ Q′).

and hence Borel. Since ∑_{k≥1} μ(N_{r,k}) < ∞ for every r > 0 by (4.6), it follows that

X′ = X ∖ ⋃_{r>0} ⋂_{n≥1} ⋃_{k≥n} N_{r,k}

is a Borel set of full measure. From the definition of X′ it is also clear that any x ∈ X′ satisfies (4.15), and that X′ consists of a union of full F leaves, i.e. that it is invariant under T̄_{w} for any w ∈ W^{(0)}. □

Proof of Proposition 4.5. We take X′ to be the set of all x ∈ X with the following properties.

(1) For every r > 0 and n ∈ ℤ, and for every sufficiently large k ≥ 1 (depending on r and n),

B_{F}(x, r) ⊂ [x]_{𝒜_{n}^{(k)}} ;  (4.16)

(2) For every k, l ≥ 1 and n ∈ ℤ,

μ_{x}^{𝒜^{(k)}} ([x]_{𝒜^{(k)}∨𝒜_{n}^{(l)}} ) > 0,  μ_{x}^{𝒜_{n}^{(l)}} ([x]_{𝒜^{(k)}∨𝒜_{n}^{(l)}} ) > 0,  (4.17)

\begin{aligned}μ_{x}^{𝒜^{(k)}∨𝒜_{n}^{(l)}} & = \frac{1}{μ_{x}^{𝒜^{(k)}} ([x]_{𝒜^{(k)}∨𝒜_{n}^{(l)}} )} · μ_{x}^{𝒜^{(k)}} |_{[x]_{𝒜^{(k)}∨𝒜{(l) \atop n}}} \\ & = \frac{1}{μ_{x}^{𝒜_{n}^{(l)}} ([x]_{𝒜^{(k)}∨𝒜_{n}^{(l)}} )} · μ_{x}^{𝒜_{n}^{(l)}} |_{[x]_{𝒜^{(k)}∨𝒜{(l) \atop n}}} , \end{aligned}  (4.18)

where μ_{x}^{𝒜^{(k)}} |_{[x]_{𝒜^{(k)}∨𝒜{(l) \atop n}}} and μ_{x}^{𝒜_{n}^{(l)}} |_{[x]_{𝒜^{(k)}∨𝒜{(l) \atop n}}} are the restrictions of

μ_{x}^{𝒜^{(k)}} and μ_{x}^{𝒜_{n}^{(l)}} to the atom [x]_{𝒜^{(k)}∨𝒜_{n}^{(l)}} of x in 𝒜^{(k)} ∨ 𝒜_{n}^{(l)};

(3) For every k ≥ 1 and n ∈ ℤ,

μ_{x}^{𝒜_{n}^{(k)}} = μ_{α^{n}x}^{𝒜^{(k)}} α^{n}.  (4.19)

Note that by (4.11) and (4.17) (with l = 1 and n = 0),

μ_{x}^{𝒜^{(k)}} (B_{F}(x, r_{0})) > 0

for every k ≥ 1 and x ∈ X′. Furthermore, by (4.16) and (4.18) (again with n = 0),

\frac{1}{μ_{x}^{𝒜^{(k)}} (B_{F}(x, r_{0}))}μ_{x}^{𝒜^{(k)}} |_{[x]_{𝒜^{(k)}∨𝒜^{(l)}}} = \frac{1}{μ_{x}^{𝒜^{(l)}} (B_{F}(x, r_{0}))}μ_{x}^{𝒜^{(l)}} |_{[x]_{𝒜^{(k)}∨𝒜^{(l)}}}

for all x ∈ X′ and all sufficiently large k, l, so that

ρ_{x}(B) = lim_{k→∞} \frac{1}{μ_{x}^{𝒜^{(k)}} (B_{F}(x, r_{0}))}μ_{x}^{𝒜^{(k)}} (T_{x} ∘ π(B))  (4.20)

exists for every x ∈ X′ and every Borel set B ⊂ W^{(0)}.

Equation (4.14) easily follows from the fact that, for every x ∈ X′, every w ∈ W^{(0)} with x − π(w) ∈ X′, and every sufficiently large k,

y = x − π(w) ∈ [x]_{𝒜^{(k)}} ,

and hence

μ_{y}^{𝒜^{(k)}} = μ_{x}^{𝒜^{(k)}} .

The sequence

log \frac{μ_{x−π(w)}^{𝒜^{(k)}}(B_{F}(x − π(w), r_{0}))}{μ_{x}^{𝒜^{(k)}} (B_{F}(x, r_{0}))} = log \frac{μ_{x}^{𝒜^{(k)}} (B_{F}(x − π(w), r_{0}))}{μ_{x}^{𝒜^{(k)}} (B_{F}(x, r_{0}))}

is eventually constant, and we set

K_{μ}(x, w) = \begin{cases}lim_{k→∞} log \frac{μ_{x}^{𝒜^{(k)}} (B_{F} (x−π(w),r_{0}))}{μ_{x}^{𝒜^{(k)}} (B_{F} (x,r_{0}))} & for x ∈ X′ and w ∈ W^{(0)} \\ & with x − π(w) ∈ X′, \\ 0 & otherwise. \end{cases}

Equation (4.13) is immediate from (4.16), (4.18) and (4.19) (with k = l and n = 1).

Finally we extend the map x ↦ ρ_{x} to X by setting ρ_{x} = 0 for every x ∈ X ∖ X′ and note that the resulting map from X to M_{∞}(W^{(0)}) is Borel. □

5. Finiteness of the central leaf measures

Theorem 5.1. Let α be a nonexpansive, ergodic and totally irreducible automorphism of a compact connected abelian group X with normalized Haar measure λ_{X}, and let μ be an α-invariant probability measure on X which is singular with respect to λ_{X}. Then there exists a Borel set X′ ⊂ X with μ(X′) = 1 such that ρ_{x}(W^{(0)}) < ∞ for every x ∈ X′ (cf. (4.12)).

We begin the proof of Theorem 5.1 with a series of lemmas in which we denote the l-th derivative of a map f by f^{(l)}.

Lemma 5.2. For every s ≥ 1 we can find a constant A_{s} > 0 such that, for every polynomial p(x) = ∑_{l=0}^{2s−1} a_{l}x^{l} of degree ≤ 2s − 1 and every ε > 0,

sup_{t∈(−ε,ε)} |p(t)| ≥ A_{s} · max_{0≤l≤2s−1}(ε^{l}|a_{l}|).

Proof. The statement of the lemma is clearly unchanged by rescaling p and ε, so that we may assume that ε = 1 and max_{l} |a_{l}| = 1. We can now set

A_{s} = inf{ sup_{t∈(−1,1)} |p(t)| : p(x) = ∑_{l=0}^{2s−1} a_{l}x^{l} with max_{l} |a_{l}| = 1} > 0.  □

Lemma 5.3. Let ε > 0, s ≥ 1, and let A_{s} > 0 be the constant appearing in Lemma 5.2. Then

sup_{t∈(−ε,ε)} |f(t)| ≥ \frac{A_{s}B}{2(2s − 1)!}.  (5.1)

for every B > 0 and every map f: (−ε, ε) ⟶ ℝ with 2s derivatives at every point such that

max_{0≤l≤2s−1} |f^{(l)}(t)| ≥ \frac{B}{ε^{l}} for every t ∈ (−ε, ε)  (5.2)

and

sup_{t∈(−ε,ε)} |f^{(2s)}(t)| < \frac{A_{s}B}{ε^{2s}} .  (5.3)

Proof. Consider the Taylor expansion

p(x) = ∑_{l=0}^{2s−1} \frac{f^{(l)}(0)}{l!}x^{l}

of f of degree 2s−1. From Lemma 5.2 we know that there is some t ∈ (−ε, ε) with |p(t)| ≥ \frac{A_{s}B}{(2s−1)!}, and Taylor’s Theorem allows us to find a ξ ∈ [0, 1] with

f(t) = p(t) + \frac{f^{(2s)}(ξt)}{(2s)!}t^{2s}.

Thus

|f(t)| ≥ |p(t)| − ε^{2s} · (sup_{t∈(−ε,ε)} |f^{(2s)}(t)|) / (2s)!

≥ \frac{A_{s}B}{(2s − 1)!} − \frac{A_{s}B}{(2s)!} ≥ \frac{A_{s}B}{2(2s − 1)!}.  □

Lemma 5.4. Let p(t) = ∑_{k=1}^{s}(a_{k} cos(2πm_{k}t)+b_{k} sin(2πm_{k}t)) be a trigonometric polynomial, where the m_{k}, k = 1, …, s, are distinct positive integers. Let ∥p∥ = max_{k=1,...,s}(|a_{k} + ib_{k}|). Then there exists a constant c_{2} > 0, which depends on s and M = max_{k=1,...,s} |m_{k}|, but not on the coefficients a_{k}, b_{k}, k = 1, …, s, such that

|∫_{0}^{1} e^{ip(t)} dt| ≤ c_{2} · ∥p∥^{−1/2s}.  (5.4)

Proof. We first claim that, unless all coefficients a_{k} and b_{k} are 0, the derivative p′ of p does not have zeros of order > 2s − 1. Indeed,

p^{(2l)}(t) = ∑_{k=1}^{s}(−1)^{l}(2πm_{k})^{2l}(a_{k} cos(2πm_{k}t) + b_{k} sin(2πm_{k}t))

for every l ≥ 0. If p^{(2l)}(t_{0}) = 0 for l = 1, …, s, the nonsingularity of the Vandermonde matrix (due to our hypothesis that the m_{k} are all distinct) implies that

a_{k} cos(2πm_{k}t_{0}) + b_{k} sin(2πm_{k}t_{0}) = 0

for k = 1, …, s. Similarly, if p^{(2l−1)}(t_{0}) = 0 for l = 1, …, s, then

−a_{k} sin(2πm_{k}t_{0}) + b_{k} cos(2πm_{k}t_{0}) = 0

for k = 1, …, s, and by combining these statements we get that a_{k} = b_{k} = 0 for k = 1, …, s. In fact, this argument gives more: since one can bound the norm of the inverse of the Vandermonde matrix for all choices 0 < m_{1} < ⋯ < m_{s} ≤ M by some function of M, there exists a constant c′_{M} > 0 depending only on M, such that

max_{1≤l≤2s} |p^{(l)}(t)| ≥ c′_{M}∥p∥

for every t ∈ ℝ and every choice of the coefficients a_{k}, b_{k} in p.

Trivially there exists, for every l ≥ 0 and M ≥ 1, a constant c′_{l,M} > 0 such that

|p^{(l)}(t)| ≤ c′_{l,M}∥p∥

for every l ≥ 0 and t ∈ ℝ.

In order to complete the proof of Lemma 5.4 we recall the van der Corput Lemma in [7, p. 220]: if ϕ is a real-valued function on an interval [a, b] ⊂ ℝ with a monotonic derivative satisfying that ϕ′(t) > A > 0 for every t ∈ [a, b], then

|∫_{a}^{b} e^{iϕ(t)} dt| ≤ \frac{4}{A}.

Since a trigonometric polynomial of degree M such as p′′(t) can have at most 2M roots in the interval [0, 1), the interval [0, 1] can be divided into at most 2M + 1 subintervals I_{1}, I_{2}, …, on each of which p′ is monotonic. By applying the van der Corput Lemma on each of these subintervals separately we have that, for any A > 0,

|∫_{0}^{1} e^{ip(t)} dt| ≤ \frac{8M + 4}{A} + λ(\{0 ≤ t ≤ 1 : |p′(t)| < A\}),  (5.5)

where λ is the Lebesgue measure on ℝ.

It remains to estimate λ({0 ≤ t ≤ 1 : |p′(t)| < A}). In fact, we claim that there exists a constant c′′ > 0 with

λ(\{0 ≤ t ≤ 1 : |p′(t)| < A\}) ≤ c′′ (\frac{A}{∥p∥})^{\frac{1}{2s−1}}  (5.6)

for every A > 0. Estimates of this kind can be found e.g. in [8]; for completeness we provide a proof below.

As p′ is monotonic on every subinterval I_{k},

I′_{k} = I_{k} ∩ {0 ≤ t ≤ 1 : |p′(t)| < A}

is connected and hence an interval, and we apply Lemma 5.3 with f = p′ on some sufficiently small subinterval I′′_{k} ⊂ I′_{k}. The conditions (5.2) and (5.3) are clearly satisfied for some B = ∥p∥·λ(I′′_{k})^{2s−1}, and Lemma 5.3 guarantees the existence of a constant c_{1} > 0 with

A ≥ sup_{t∈I′_{k}} p′(t) ≥ c_{1}∥p∥λ(I′_{k})^{2s−1}

or

λ(I′_{k}) ≤ c_{1}^{− \frac{1}{2s−1}} · (\frac{A}{∥p∥})^{\frac{1}{2s−1}} .

By summing over k we obtain (5.6).

According to (5.5) and (5.6),

|∫_{0}^{2π} e^{2πip(t)} dt| ≤ \frac{2}{A} + c′′ · (\frac{A}{∥p∥})^{\frac{1}{2s−1}} ,

and by taking A = ∥p∥^{1/2s} we get (5.4). □

We derive from this the following estimate.

Lemma 5.5. For every nontrivial character a ∈ X̂ there exists a constant c_{a} > 0 with

∫_{Γ}⟨a, π(M_{γ}w)⟩ dγ ≤ c_{a} · min(1, ∥w∥^{−1/2s})

for every w ∈ W^{(0)}, where s = |S^{(0)}|, and where Γ and M_{γ} are defined in (3.5) and (3.6).

Proof. We recall that W = ∏_{v∈S} K_{v}, consider each K_{v} (by abuse of notation) as an additive subgroup of W, and identify X with W/ι(L) as in (2.24). Let a be a nontrivial character of X = W/ι(L), and let f: w ↦ f(w) = ⟨a, π(w)⟩ be the corresponding character of W. We write f_{0} = f|_{W^{(0)}} for the restriction of f to W^{(0)}. The isomorphisms ι_{v}: K_{v} ⟶ ℂ, v ∈ S^{(0)}, in (2.18) allow us to write f_{0} : W^{(0)} ⟶ ℂ as the map

w ↦ f_{0}(w) = e^{2πi ∑_{v∈S^{(0)}} ℜ(a_{v}ι_{v}(w_{v}))},

where a_{v} ∈ ℂ for every v ∈ S^{(0)}, and where ℜ denotes the real part. Since the image π(K_{v}) of K_{v} is dense in X for every v by irreducibility, f_{0}|_{K_{v}} is a nontrivial character, hence a_{v} ≠ 0 for every v ∈ S^{(0)}.

Let v, v′ be distinct elements of S^{(0)}. By Proposition 3.2, the projection of Γ to K_{v} ⊕ K_{v′} ⊂ W^{(0)} maps Γ onto the set

{w ∈ W : |w_{v}| = |w_{v′}| = 1 and all other coordinates are 0}.

Hence there exists a closed one-dimensional subgroup

Γ_{0} = {(ι_{v}^{−1}(z^{m_{v}}))_{v∈S^{(0)}} : |z| = 1} ⊂ Γ ⊂ W^{(0)}

such that the integers m_{v}, v ∈ S^{(0)}, are all distinct.

Now we use Lemma 5.4 to check that there exists a constant c with

|∫_{Γ_{0}} f(M_{γ}w) dγ| = |∫_{0}^{1} e^{2πiℜ(∑_{v∈S^{(0)}} a_{v}ι_{v}(w_{v})e^{2πimvt})}dt| ≤ c · ∥w∥^{−1/2s}

for every w ∈ W^{(0)}, and by integrating over Γ we see that

|∫_{Γ} f(M_{γ}w) dγ| = |∫_{Γ} ∫_{Γ_{0}} f(M_{γ_{0}}M_{γ}w) dγ_{0} dγ|
≤ c · ∫_{Γ} ∥M_{γ}w∥^{−1/2s}dγ = c · ∥w∥^{−1/2s}.  □

Lemma 5.6. For every nontrivial character a ∈ X̂ there exists a constant c_{a} > 0 such that

\begin{aligned}∫_{Γ}|∫_{W^{(0)}} & ⟨a, π(M_{γ}w)⟩ dτ̄(w)|^{2}dγ \\ & ≤ c_{a} · ∫ min(1, ∥w − w′∥^{−1/2s}) dτ̄(w) dτ̄(w′) \end{aligned}  (5.7)

for every probability measure τ̄ on W^{(0)}, where s = |S^{(0)}|.

Proof. By Fubini’s theorem,

∫Γ|∫_{W^{(0)}}⟨a, π(M_{γ}w)⟩ dτ̄(w)|^{2}dγ
= ∫_{Γ} ∫_{W^{(0)}} ∫_{W^{(0)}}⟨a, π(M_{γ}w)⟩\overline{⟨a, π(M_{γ}w′)⟩} dτ̄(w) dτ̄(w′) dγ

= ∫_{W^{(0)}} ∫_{W^{(0)}} ∫_{Γ}⟨a, π(M_{γ}(w − w′))⟩ dγ dτ̄(w) dτ̄(w′).

From Lemma 5.5 we know that there exists a constant c_{a} > 0 with

∫_{Γ}⟨a, π(M_{γ}(w − w′))⟩ dγ ≤ c_{a} min(1, ∥w − w′∥^{−1/2s})

for every w ≠ w′ in W^{(0)}, and by integrating we obtain (5.7). □

Corollary 5.7. Let τ̄ be a probability measure on W^{(0)}, x_{0} ∈ X, and let ρ = (τ̄π^{−1})T_{−x_{0}} (so ρ is supported on the central leaf through x_{0}). For every N ∈ ℕ we set

ρ_{N} = \frac{1}{N} ∑_{i=0}^{N−1} ρα^{i}

Then for every nontrivial character a ∈ X̂

lim sup_{N→∞} |∫ ⟨a, x⟩ dρ_{N}(x)|^{2} ≤ c_{a} · ∫ min(1, ∥w − w′∥^{−1/2s}) dτ̄(w) dτ̄(w′),

where c_{a} is as in Lemma 5.6.

Proof. By Cauchy-Schwarz,

|∫ ⟨a, x⟩ dρ_{N}(x)|^{2} = |\frac{1}{N} ∑_{i=0}^{N−1} ∫ ⟨a, x⟩ dρα^{i}|^{2} ≤ \frac{1}{N} ∑_{i=0}^{N−1} |∫ ⟨a, x⟩ dρa^{i}(x)|^{2}

= \frac{1}{N} ∑_{i=0}^{N−1} |∫_{W^{(0)}}⟨a, π(w)⟩ dτ̄β̄^{i}(w)|^{2}

The map

γ ↦ |∫_{W^{(0)}}⟨a, π(w)⟩ d(τ̄M_{γ})(w)|

from Γ to ℝ^{+} is continuous and bounded, so by the unique ergodicity of the action of β̄ on Γ

\frac{1}{N} ∑_{i=0}^{N−1} |∫_{W^{(0)}}⟨a, π(w)⟩ dτ̄β̄^{i}(w)|^{2} → ∫_{Γ} |∫_{W^{(0)}}⟨a, π(w)⟩ dτ̄M_{γ}|^{2} dγ.

We can now apply Lemma 5.6 to conclude the proof of this corollary. □

Proof of Theorem 5.1. Consider the α-invariant Borel set

B = {x : ρ_{x}(W^{(0)}) = ∞}.

We will show that

μ′ = \frac{1}{μ(B)}μ|_{B}= λ_{X}  (5.8)

whenever μ(B) > 0.

Assume therefore that μ(B) > 0, and let X′ ⊂ X be the α-invariant Borel set of full measure described in Proposition 4.5. From (4.14) it follows that, if x ∈ B ∩ X′, then any other point in X′ ∩ (x − π(W^{(0)})) also lies in B ∩ X′. Hence

μ_{x}^{𝒜^{(k)}} (B) ∈ {0, 1}

for μ-a.e. x ∈ X and every k ≥ 1. We can thus choose, for every k ≥ 1, a set B^{(k)} ∈ 𝒜^{(k)} with

μ(B^{(k)} △ B) = 0.

We fix temporarily a large number r > 0 and a small ε > 0. According to (4.15) there exist an increasing sequence (n_{k}, k ≥ 1) of natural numbers and a Borel set D ⊂ B with μ′(D) > 1 − ε so that, for any x ∈ D and k ≥ 1,

[x]_{𝒜^{(n_{k})}} + π(B_{W^{(0)}}(0, r)) ⊂ [x]_{𝒜^{(n_{k+1})}} ,
0 < μ_{x}^{𝒜^{(n_{k+1})}} ([x]_{𝒜^{(n_{k})}} + π(B_{W^{(0)}}(0, r))) < ε

(in the second of these conditions we use the fact that ρ_{x}(W^{(0)}) = ∞ for every x ∈ B). For every K ≥ 1 and x ∈ X we set

τ_{x}^{K} = \frac{1}{K} ∑_{k=1}^{K} μ_{x}^{𝒜^{(n_{k})}} .

Since B^{(n_{k})} ∈ 𝒜^{(n_{k})} is equal to B (mod μ) and hence also (mod μ′), and since μ′ is α-invariant, we have that

μ′ = ∫ (μ_{x}^{𝒜^{(n_{k})}} α^{n}) dμ′(x)

for every k ≥ 1 and n ∈ ℤ, and hence that

μ′ = ∫ (τ_{x}^{K}α^{n})dμ′(x)  (5.9)

for every K ≥ 1 and n ∈ ℤ. We define τ̄_{x}^{K} ∈ M_{∞}(W^{(0)}) by

τ̄_{x}^{K}(C) = τ_{x}^{K}(T_{x} ∘ π(C))

for every Borel set C ⊂ W^{(0)}, where π and T_{x} are taken from (3.2) and (4.3). For any x ∈ D,

M(x, r) = (τ̄_{x}^{K} × τ̄_{x}^{K})({(w_{1}, w_{2}) ∈ (W^{(0)})^{2} : ∥w_{1} − w_{2}∥ < r})

= \frac{1}{K^{2}} · ∑_{k=1}^{K} ∑_{k′=1}^{K}(μ_{x}^{𝒜^{(n_{k})}} × μ_{x}^{𝒜^{(n_{k′})}} )(\{(x + π(w_{1}), x + π(w_{2})) :

(w_{1}, w_{1}) ∈ (W^{(0)})^{2} and ∥w_{1} − w_{2}∥ < r})

≤ \frac{1}{K^{2}} · ∑_{k=1}^{K} (μ_{x}^{𝒜^{(n_{k})}} × μ_{x}^{𝒜^{(n_{k})}} )(X × X)

+ \frac{2}{K^{2}} · ∑_{1≤k<k′≤K}(μ_{x}^{𝒜^{(n_{k})}} × μ_{x}^{𝒜^{(n_{k′})}} )(\{(y, y + π(w)) :

y ∈ [x]_{𝒜^{(n_{k})}} , ∥w∥ < r})

≤ \frac{1}{K} + \frac{2}{K^{2}} · ∑_{k<k′} μ_{x}^{𝒜^{(n_{k′})}} ([x]_{𝒜^{(n_{k})}} + π(B_{W^{(0)}}(0, r))) < \frac{1}{K} + ε.

Using this, we see that for any x ∈ D,

\begin{matrix}∫_{W^{(0)}} ∫_{W^{(0)}} min(1 & , ∥w − w′∥^{−1/2s}) dτ̄_{x}^{K}(w) dτ̄_{x}^{K}(w′) \\ ≤ M(x, r) + r^{−1/2s}(1 − M(x, r)) \\ ≤ K^{−1} + ε + r^{−1/2s}. \end{matrix}  (5.10)

Equation (5.9) shows that, for an arbitrary positive integer K,

|∫_{X}⟨a, x⟩ dμ′|^{2} = lim_{N→∞} |∫_{X} ∫_{X}⟨a, y⟩ d [\frac{1}{N} ∑_{n=0}^{N−1} τ̄_{x}^{K}α^{n}](y) dμ′(x)|^{2}

≤ ∫_{X} lim sup_{N→∞} |∫_{X}⟨a, y⟩ d [\frac{1}{N} ∑_{n=0}^{N−1} τ̄_{x}^{K}α^{n}](y)|^{2} dμ′(x),

(5.11)

where the first limit is actually over a constant sequence. By Corollary 5.7, (5.11) and (5.10),

|∫_{X}⟨a, x⟩ dμ′(x)| ≤ ∫_{X∖D} lim sup_{N→∞} |∫_{X}⟨a, y⟩ d [\frac{1}{N} ∑_{n=0}^{N−1} τ̄_{x}^{K}α^{n}](y)|^{2} dμ′(x)

+ ∫_{D} lim sup_{N→∞} |∫_{X}⟨a, y⟩ d [\frac{1}{N} ∑_{n=0}^{N−1} τ̄_{x}^{K}α^{n}](y)|^{2} dμ′(x)

≤ μ′(X ∖ D) + c_{a} · (K^{−1} + ε + r^{−1/2s})

≤ ε + c_{a} · (K^{−1} + ε + r^{−1/2s}).

Since ε, K, r were arbitrary we see that ∫_{X}⟨a, x⟩ dμ′(x) = 0 for every a ∈ X̂, and that μ′ is therefore equal to λ_{X}. □

6. Virtually hyperbolic measures and central equivalence

In this section, we deduce Theorem 1.3 from Theorem 5.1. For any locally compact metric space Y, we let M_{f}(Y) ⊂ M_{∞}(Y) denote the finite Borel measures on Y.

Lemma 6.1. There is a Borel map c_{m} : M_{f}(ℝ^{d}) ⟶ ℝ^{d} which commutes with the action of the isometry group of ℝ^{d} and is invariant under scalar multiplication: that is, if F : ℝ^{d} ⟶ ℝ^{d} is an isometry of ℝ^{d} and t > 0, then

c_{m}(ρ) = F ∘ c_{m}(tρF).  (6.1)

Remark 6.2. For measures ρ ∈ M_{f}(ℝ^{d}) which have finite first moments, the vector of moments

c_{m}(ρ) = (∫ x_{1} dρ(x_{1}, …, x_{d}), …, ∫ x_{d} dρ(x_{1}, …, x_{d})) ∈ ℝ^{d}

would satisfy all these requirements. Unfortunately, there are measures for which this naive definition of center of mass does not make sense; the lemma should be interpreted as an alternative, generalized notion of a center of mass which works for any measure in M_{f}(ℝ^{d}).

Proof. For a given r, ε > 0, and for every ρ ∈ M_{f}(ℝ^{d}), let

S_{r,ε}(ρ) = {x ∈ ℝ^{d} : ρ(B(x, r)) ≥ ε},
a_{r,ε}(ρ) = ∫_{S_{r,ε}(ρ)} x dρ(x), m_{r,ε}(ρ) = ρ(S_{r,ε}(ρ)).

Note that the maps ρ ↦ m_{r,ε}(ρ) and ρ ↦ \frac{a_{r,ε}(ρ)}{m_{r,ε}(ρ)}, the latter when defined, are invariant under the action of isometry group of ℝ^{d} on M_{f}(ℝ^{d}). In order to get a map c_{m} which is defined everywhere we arbitrarily fix r > 0, set

n_{r}(ρ) = min {n : m_{r,1/n}(ρ) > 0}, a_{r}(ρ) = a_{r,1/n_{r}}(ρ), m_{r}(ρ) = m_{r,1/n_{r}}(ρ),

and put c_{m}(ρ) = \frac{a_{r}(ρ)}{m_{r}(ρ)}. □

Proof of Theorem 1.3. We first show that for every α-invariant measure μ which is singular with respect to the Haar measure λ_{X} there is a virtually hyperbolic measure μ′ which is centrally equivalent to it. Indeed, consider the map τ : X ⟶ X defined by

τ(x) = π ∘ c_{m}(ρ_{x}) + x.

Let X′ be the subset of full measure of X in Proposition 4.5. Then for any x ∈ X′ we have that

τ ∘ α(x) = π ∘ c_{m}(μ_{αx}) + αx = α(π ∘ c_{m}(ρ_{x})) + αx = α ∘ τ(x)

where the second equality follows from (4.13) and (6.1). Similarly, by (4.14),

\begin{aligned}τ(x) & = π ∘ c_{m}(ρ_{x}) + x = π ∘ c_{m}(ρ_{y−π(w)}) + x \\ & = π ∘ c_{m}(e^{K_{μ}(y,w)} · ρ_{y−π(w)}) + x \\ & = π ∘ c_{m}(ρ_{y}T̄_{w}) + x = π ∘ c_{m}(ρ_{y}) + x + π(w) = τ(y) \end{aligned}  (6.2)

for any x, y ∈ X′ with y − x = π(w) ∈ X^{(0)}. By setting μ′ = μτ^{−1} we get a new α-invariant probability measure on X which is clearly centrally equivalent to μ. The α-invariant set Y = τ(X′) ⊂ X is analytic and has full μ′-measure, and (6.2) implies that Y intersects each central leaf in at most one point. By choosing an α-invariant Borel subset Z ⊂ Y with μ′(Z) = 1 we see that μ′ is virtually hyperbolic.

We now specialize to the case where μ is ergodic. The map τ^{∗}: X′ ⟶ W^{(0)} defined by τ^{∗}(x) = −c_{m}(ρ_{x}) (so x = τ(x) + π ∘ τ^{∗}(x)) satisfies that τ^{∗}(αx) = β̄τ^{∗}(x) ⊂ Γτ^{∗}(x) for every x ∈ X′ (where Γ is defined in (3.5)), and the ergodicity of μ and the compactness of Γ together imply that there exist an element w^{∗} ∈ W^{(0)} so that for μ-almost every x ∈ X it holds that τ^{∗}(x) ∈ Γw^{∗} = {M_{γ}w^{∗} : γ ∈ Γ}.

For every γ ∈ Γ we define a map τ_{γ} : X′ → X by

τ_{γ}(x) = τ(x) + M_{γ}τ^{∗}(x),

and let μ_{γ} = μτ_{γ}^{−1}. Then μ_{e} = μ with e ∈ Γ the identity, and

μ′ ∗ λ̃_{π(w^{∗})} = ∫_{Γ} μ_{γ}dγ,

with λ̃_{π(w^{∗})} as in Theorem 1.3. Since the identity is in the support of Haar measure on Γ, and since the map γ ↦ μ_{γ} is continuous with respect to the weak^{∗} topology, μ an ergodic component of μ′ ∗ λ̃_{π(w^{∗})} in the sense of Theorem 1.3.

For every γ ∈ Γ we write M_{γ}w^{∗} as M_{γ}w^{∗} = (γ_{v}w_{v}^{∗}, v ∈ S) with w_{v}^{∗} = 0 for every v ∈ S ∖ S^{(0)} (cf. (3.3)). similarly we set τ^{∗}(x) = (τ^{∗}(x)_{v}) for every x ∈ X′′. Then there exists, for every v ∈ S^{(0)} with w_{v}^{∗} ≠ 0, a well-defined map f_{v}: X′ ⟶ ℂ with

τ^{∗}(x)_{v} = f_{v}(x)w_{v}^{∗}

for every x ∈ X′′, where we are identifying K_{v} with ℂ (cf. (2.18)). Since f_{v} is obviously an eigenfunction of α for every v ∈ S^{(0)} with w_{v}^{∗} ≠ 0, we have arrived at the following alternative: either the map x ↦ π ∘ c_{m}(x) = x−τ(x) is zero almost everywhere, which implies that μ = μ′, hence virtually hyperbolic, or μ is not weakly mixing; indeed, this argument shows that the point spectrum of μ (more precisely: the point spectrum of the action of α on L^{2}(X, 𝒮, μ)) contains some eigenvalue of α of absolute value 1. □

7. Central leaves and closed invariant subsets

This section is devoted to proving the following topological analogue to Theorem 5.1.

Theorem 7.1. Let α be a nonexpansive, ergodic and totally irreducible automorphism of a compact connected abelian group X. Then any closed αinvariant subset Y ⊊ X intersects every central leaf in a compact subset of the leaf.

The key to this theorem is the following lemma in which we call a subset A ⊂ W^{(0)} R-separated if ∥x−y∥ ≥ R for any two distinct elements x, y ∈ A.

Lemma 7.2. Let α and X be as in Theorem 7.1. Then for any ε > 0 there exist positive integers R, K so that for any R-separated subset A ⊂ W^{(0)} with at least K elements, the set

Ã = ⋃_{n=1}^{∞} α^{−n}(π(A) + x_{0})

is ε-dense in X.

Proof. Let {f_{1}, …, f_{k}} be a partition of unity of X (i.e. a set of nonnegative continuous functions so that ∑_{i=1}^{k} f_{i} ≡ 1) so that the support of each f_{i} has diameter at most ε. Clearly, to show that Ã is ε-dense it is sufficient to find some probability measure ρ supported on Ã so that

∫_{X} f_{i}dρ > 0

for every i = 1, …, k. Since the linear span of X̂ is dense in C(X), there exists a finite subset Ξ ⊂ X̂ containing the identity element 0 ∈ X̂ so that for each i we can find an approximation

f̃_{i}(x) = ∑_{a∈Ξ} u_{i,a}⟨a, x⟩

to f_{i} in the linear span of Ξ so that

∥f_{i} − f̃_{i}∥_{∞} < ∥f_{i}∥_{1}/100.

Let Ξ′ = Ξ \ {0}.

We denote by c_{a} the constant in Lemma 5.6 and Corollary 5.7 and define R, K by

R^{2s} = K = 100 max_{i} (\begin{matrix}∑ & a∈Ξ & |u_{i,a}c_{a}| \\ ∥f_{i}∥ & 1 \end{matrix}) .

Now suppose that A ⊂ W^{(0)} is an R-separated set of cardinality ≥ K and x_{0} ∈ X is arbitrary. We define

τ̄ = \frac{1}{|A|} ∑_{w∈A} δ_{w}, ρ = (τ̄π^{−1})T_{−x_{0}}, ρ_{N} = \frac{1}{N} ∑_{i=0}^{N−1} ρα^{i},

where δ_{w} is the point-mass at w. For every N, ρ_{N} is supported on Ã, and if N is large enough, then

|∫ ⟨a, x⟩ dρ_{N}(x)|^{2} ≤ 2c_{a} ∫∫ min(1, ∥w − w′∥^{−1/2s}) dτ̄(w) dτ̄(w′)
≤ 2c_{a}(K^{−1} + R^{−1/2s}) = 4c_{a}K^{−1}

for every a ∈ Ξ.

For N sufficiently large we obtain that

|∫ f̃_{i}dρ_{N} − ∫ f̃_{i}dx| ≤ ∑_{a∈Ξ′} |u_{i,a} ∫ ⟨a, x⟩dρ_{N} |

≤ 4K^{−1} ∑_{a∈Ξ′} |u_{i,a}c_{a}| ≤ ∥f_{i}∥_{1} /25.

But then

|∫ f_{i}dρ_{N} | ≥ |∫ f̃_{i}dρ_{N} | − ∥f_{i}∥_{1} /100 ≥ |∫ f̃_{i}dx| − ∥f_{i}∥_{1} /20 ≥ ∥f_{i}∥_{1} /2 > 0

for i = 1, …, k, and we are done. □

Proof of Theorem 7.1. Suppose that Y ⊊ X is α-invariant and closed, and that the intersection of Y with some central leaf X^{(0)} + x_{0} is not compact. Fix a w_{0} ∈ π^{−1}(x_{0}) and take C = [π^{−1}(Y) − w_{0}] ∩ W^{(0)}.

By our assumptions, C is a closed unbounded subset of W^{(0)}. Let ε > 0 be arbitrary, and let K, R be as in Lemma 7.2. Take A to be a finite Rseparated subset of C of cardinality ≥ K. Then the set Ã ⊂ X defined in that lemma is a subset of Y and is ε-dense. So Y is ε-dense, and since ε was arbitrary, Y = X. □

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[10] K. Schmidt, Automorphisms of compact abelian groups and affine varieties, Proc. London Math. Soc. 61 (1990), 480–496.

[11] K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser Verlag, Basel-Berlin-Boston, 1995.

[12] A. Weil, Basic Number Theory, Springer Verlag, Berlin-Heidelberg-New York, 1974.

Elon Lindenstrauss: Department of Mathematics, Stanford University, Stanford, CA 94305, USA

E-mail address: elonl@cims.nyu.edu

Current address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012

Klaus Schmidt: Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria

and Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9,

A-1090 Vienna, Austria

E-mail address: klaus.schmidt@univie.ac.at