SYMBOLIC REPRESENTATIONS OF NONEXPANSIVE GROUP AUTOMORPHISMS

ELON LINDENSTRAUSS AND KLAUS SCHMIDT

Abstract. If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then α has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in X. In spite of this we are able to construct a symbolic space V and a class of shift-invariant probability measures on V each of which corresponds to an α-invariant probability measure on X. Moreover, every α-invariant probability measure on X arises essentially in this way.

The last part of the paper deals with the connection between the twosided beta-shift V_{β} arising from a Salem number β and the nonhyperbolic ergodic toral automorphism α arising from the companion matrix of the minimal polynomial of β, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures on V_{β} and certain α-invariant probability measures on X. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.

1. Introduction

For expansive automorphisms α of compact connected abelian groups X, the attempt to find symbolic representations of the dynamical system (X, α) has a long and extensive history. In addition to the classical ‘geometric’ constructions of Markov partitions (e.g. in [1], [4] and [25]), there are explicit algebraic constructions of continuous equivariant finite-to-one maps from a sofic shift onto X. The first such construction for arbitrary irreducible hyperbolic toral automorphisms was given by R. Kenyon and A. Vershik in [10] (irreducibility is explained in Definition 2.1); a different, but related, general construction for irreducible expansive automorphisms of tori and solenoids was given by the second named author in [21]. In certain cases, this map can be chosen to be one-to-one almost everywhere (cf. [21] and [24]; of course, since X is connected and a sofic shift completely disconnected one cannot hope to find a map which is one-to-one everywhere). The existence of such a map gives an explicit essentially one-to-one map between shiftinvariant measures on a sofic shift and α-invariant measures on X.

2000 Mathematics Subject Classification. 37A05, 37A45, 37C15, 37C29, 37H05. Key words and phrases. Partially hyperbolic group automorphisms, invariant measures,

Markov partitions, Beta-shifts.

The key idea in the construction of these maps is to find a surjective equivariant map from some symbolic system V onto X (it turns out to be natural to set V either equal to the space ℓ^{∞}(ℤ, ℤ) of bounded integer sequences or to some sufficiently large compact shift-invariant subset of ℓ^{∞}(ℤ, ℤ)). Following an idea originally introduced by A. Vershik in [27]– [29] one may, for example, take a point x ∈ X which is homoclinic to 0 (i.e. which satisfies that lim_{|n|→∞} α^{n}x = 0) and send any integer sequence v = (…, v_{−1}, v_{0}, v_{1}, …) ∈ ℓ^{∞}(ℤ, ℤ) to the point

ξ(v) = ∑_{n∈ℤ}v_{n}α^{−n}x ∈ X.

The resulting map ξ: ℓ^{∞}(ℤ, ℤ) ⟶ X is equivariant (i.e. ξ ∘ σ̄ = α ∘ ξ), and it is not hard to see that it is surjective. From this map ξ one obtains a surjective map from the collection of shift-invariant probability measures on ℓ^{∞}(ℤ, ℤ) onto the α-invariant measures on X. The more refined construction of [21] alluded to earlier is obtained by restricting this map to a carefully chosen sofic subshift V ⊂ ℓ^{∞}(ℤ, ℤ) on which ξ is surjective and almost oneto-one. Other interesting and, indeed, more canonical examples arise when α is the automorphism of X = 𝕋^{m} defined by the companion matrix of the minimal polynomial of a Pisot unit β (i.e of an algebraic integer β > 1 whose conjugates all have absolute values < 1). In this case the corresponding twosided beta-shift V_{β} ⊂ ℓ^{∞}(ℤ, ℤ) is sofic, and the map ξ: V_{β} ⟶ 𝕋^{m} defined above is surjective, finite-to-one and conjectured to be almost one-to-one (cf. [21]–[24]).

In this paper we investigate to what extent one can find a suitable substitute for this construction in the nonexpansive case. This question is motivated by the somewhat exotic behaviour of invariant probability measures of irreducible nonhyperbolic ergodic toral automorphisms described in [13]: if μ is a probability measure on X = 𝕋^{n} which is invariant under an irreducible nonhyperbolic ergodic toral automorphism α, but which is completely singular with respect to Lebesgue measure, then there exists an α-invariant Borel set B ⊂ X which intersects μ-a.e. coset of the dense central subgroup X^{(0)} ⊂ X, on which α acts isometrically, in at most one point. If μ is weakly mixing then one may assume in addition that μ(B) = 1. Any natural ‘symbolic model’ of such an automorphism would enable one to construct such measures quite explicitly.

The first difficulty one encounters in the search for symbolic models of an irreducible ergodic nonexpansive automorphism α of a compact connected abelian group X is that every continuous equivariant map ϕ: Y ⟶ X from a mixing shift of finite type Y with finite or countably infinite alphabet (or from a two-sided beta-shift) to X maps the shift space to a single point (cf. Corollaries 4.7–4.11 and Remark 4.12); in particular, (X, α) cannot have finite or countably infinite Markov partitions in any reasonable sense. The reason for this is that these automorphisms have no nonzero homoclinic points (Theorem 4.1).

It is, however, possible to define a continuous map ξ̃ from the noncompact space Ỹ = ℓ^{∞}(ℤ, ℤ) × X^{(0)} to X which is equivariant with respect to an isometric cocycle extension σ̃: Ỹ ⟶ Ỹ of the shift on ℓ^{∞}(ℤ, ℤ). This map is surjective (though far from injective), and allows us in particular to map shift-invariant probability measures on Ỹ to α-invariant measures on X. Indeed, we show the following (cf. Proposition 4.18; central equivalence is explained in Definition 4.15).

Theorem 1.1. Any α-invariant probability measure on X is centrally equivalent to the push-forward under ξ̃ of a σ̃-invariant probability measure on Ỹ, which may further be taken to be compactly supported.

We emphasize that this is true even for Lebesgue (or Haar) measure (since central equivalence preserves entropy, any α-invariant probability measure on X which is centrally equivalent to Lebesgue measure must be equal to Lebesgue measure). Since Ỹ is a noncompact extension of ℓ^{∞}(ℤ, ℤ), not every shift invariant measure on ℓ^{∞}(ℤ, ℤ) can be lifted to a σ̃-invariant probability measure on Ỹ. The measures which can be lifted are precisely those for which the cocycle appearing in the definition of σ̃ is a coboundary (cf. Theorem 4.13 and Proposition 4.18). The fact that it is natural to consider only those measures on a symbolic model for which this cocycle is trivial can be viewed as a manifestation of some weak form of measure rigidity for nonexpansive group automorphisms.

The main drawback of Theorem 1.1 is that the same measure on X can be obtained as the push-forward of many measures on Ỹ; furthermore, it is quite hard to understand properties such as the entropy of the resulting measures in terms of the properties of the original measure. In order to resolve such difficulties one would like to replace ℓ^{∞}(ℤ, ℤ) by a smaller closed subshift, just like in the hyperbolic case.

In the case of toral automorphisms corresponding to Pisot numbers (i.e. of irreducible hyperbolic toral automorphisms with one-dimensional unstable manifolds) there is a natural candidate: the beta-shift V_{β} corresponding to the unique ‘large’ eigenvalue β of the automorphism. Motivated by this question we devote Section 6 to a problem which has also provided much of the original motivation for this research: the connection between the twosided beta-shift V_{β} arising from a Salem number β and the nonhyperbolic ergodic toral automorphism α defined by the companion matrix of the minimal polynomial of β (a Salem number is an algebraic unit β > 1 whose conjugates all have absolute values ≤ 1, with at least one conjugate of absolute value = 1). In contrast to the Pisot case, which is reasonably well understood (though some important questions in this construction are still unresolved, as described on page 24), the beta-shifts associated with Salem numbers still hold many mysteries. For example, it is not known whether they are always sofic (cf. [5]–[7] and [17]). Not surprisingly, the dynamical interpretation of two-sided beta-shifts arising from Salem numbers is much more complicated than in the Pisot case.

By restricting the map ξ̃: Ỹ ⟶ X described above to the space Ỹ_{β} = V_{β} × X^{(0)} we obtain a map from σ̃-invariant probability measures on Ỹ_{β} (or, equivalently, from shift invariant probability measures on V_{β} satisfying the cocycle condition mentioned above) to α-invariant measures on X. In particular, the following theorem follows from the main result of Section 6 (cf. Theorem 6.4).

Theorem 1.2. For any σ̃-invariant probability measure μ̃ on Ỹ_{β}, the entropy of the push-forward ξ̃_{∗}(μ̃) is equal to that of μ̃.

By constructing in Section 7 shift-invariant probability measures on V_{β} satisfying a strong form of the cocycle condition with entropies arbitrarily close to log β we obtain from Theorem 1.2 α-invariant probability measures on X which are singular with respect to Lebesgue measure and whose entropies are arbitrarily close to log β, the topological entropy of (X, α).

The β-shift V_{β} has a unique measure of maximal entropy log β. In Proposition 6.2, we prove that this measure does not satisfy our coboundary condition, and so it follows from Theorem 1.2 that Lebesgue measure on X cannot be presented as ξ̃_{∗}(μ̃) with μ̃ a measure on V_{β} as above. The main question highlighted by our work is the following:

Question 1. Can every α-invariant probability measure on X which is completely singular with respect to Lebesgue measure be presented as ξ̃_{∗}(μ̃) for an invariant probability measure μ̃ on Ỹ_{β}?

At present, we have no evidence in either direction. Even if the answer turns out to be negative, it would be interesting to understand the relation between the space of measures obtained by the construction of Theorem 1.2 and the space of all invariant measures.

It follows from Theorem 1.2 and the results of Section 7 that ξ̃_{∗}(Ỹ_{β}) ⊂ X is fairly large; for example, it can be shown that its Hausdorff dimension is the same as that of X. However, we do not even know the answer to the following natural question:

Question 2. Is ξ̃_{∗}(Ỹ_{β}) = X?

In the notation of Section 4, Question 2 can be rephrased as follows: is the β-shift V_{β} a pseudo-cover of X?

We end this introduction with a comment on a technical simplification we adopt throughout this paper: every irreducible automorphism α of a compact connected abelian group X is finitely equivalent to a group automorphism of the special form α_{R_{1}/(f)} described in (2.6)–(2.7). By restricting ourselves to automorphisms of this special form we avoid some minor notational and technical complications in the statements of our results due to the presence of finite-to-one factor maps, but our discussion here can be translated to the general case without any difficulty.

2. Homoclinic points of irreducible group automorphisms

Definition 2.1. Let α be a continuous automorphism of a compact abelian group X with identity element 0 = 0_{X}. The automorphism α is irreducible if every closed α-invariant subgroup Y ⊊ X is finite. A point x ∈ X is αhomoclinic (or simply homoclinic) if lim_{|n|→∞} α^{n}x = 0. The set Δ_{α}(X) of homoclinic points in X is an α-invariant subgroup.

Recall that two continuous automorphisms α and β of compact abelian groups X and Y are finitely equivalent if there exist continuous, surjective, equivariant and finite-to-one group homomorphisms ϕ: X ⟶ Y and ψ: Y ⟶ X. In order to describe all irreducible automorphisms of compact abelian groups up to finite equivalence we use notation from [13]. Let R_{1} = ℤ[u^{±1}] be the ring of Laurent polynomials with integral coefficients. Every h ∈ R_{1} is of the form

h = ∑_{k∈ℤ}h_{k}u^{k}  (2.1)

with h_{k} ∈ ℤ for every k ∈ ℤ and h_{k} = 0 for all but finitely many k. Fix an irreducible polynomial

f = f_{0} + ⋯ + f_{m}u^{m} ∈ R_{1}  (2.2)

with m > 0, f_{m} > 0 and f_{0} ≠ 0, denote by Ω_{f} the set of roots of f, and set

\begin{matrix}Ω_{f}^{−} = \{ω ∈ Ω_{f} : |ω| < 1\}, Ω_{f}^{(0)} = \{ω ∈ Ω_{f} : |ω| = 1\}, \\ Ω_{f}^{+} = \{ω ∈ Ω_{f} : |ω| > 1\}. \end{matrix}  (2.3)

We write 𝕋 = ℝ/ℤ for the circle group, define the shift τ: 𝕋^{ℤ} ⟶ 𝕋^{ℤ} by

τ(x)_{n} = x_{n+1}  (2.4)

for every x = (x_{n}) ∈ 𝕋^{ℤ}, and set

h(τ) = ∑_{k∈ℤ}h_{k}τ^{k}: 𝕋^{ℤ} ⟶ 𝕋^{ℤ}  (2.5)

for every h ∈ R_{1} of the form (2.1). Consider the closed, shift-invariant subgroup

X = X_{R_{1}/(f)} = {x ∈ 𝕋^{ℤ} : ∑_{n∈ℤ} f_{n}x_{k+n} = 0 (mod 1) for every k ∈ ℤ}
= {x ∈ 𝕋^{ℤ} : f(τ)(x) = 0} = ker f(τ),  (2.6)

and write

α = α_{R_{1}/(f)}  (2.7)

for the restriction of τ to X ⊂ 𝕋^{ℤ} (cf. [13, (2.3) and (2.11)]). By [20, Theorem 7.1 and Propositions 7.2–7.3], α is nonexpansive if and only if Ω_{f}^{(0)} ≠ ∅, and ergodic if and only if f is not cyclotomic (i.e. if and only if f does not divide u^{m} − 1 for any m ≥ 1). In view of this we adopt the following terminology.

Definition 2.2. The polynomial f in (2.2) is hyperbolic if Ω_{f}^{(0)} = ∅, nonhyperbolic if Ω_{f}^{(0)} ≠ ∅, and cyclotomic if Ω_{f}^{(0)} contains a root of unity.

According to [19], every irreducible automorphism α of a compact abelian group X is finitely equivalent to an automorphism of the form α_{R_{1}/(f)} for some irreducible polynomial f ∈ R_{1}. Note that the automorphisms α and α_{R_{1}/(f)} are expansive if and only if f is hyperbolic, and ergodic if and only if f is not cyclotomic.

For the remainder of this article we assume that the irreducible polynomial f in (2.2) is noncyclotomic.

We denote by ∥ · ∥_{1} and ∥ · ∥_{∞} the norms on the Banach spaces ℓ^{1}(ℤ, ℝ) and ℓ^{∞}(ℤ, ℝ) and write ℓ^{1}(ℤ, ℤ) ⊂ ℓ^{1}(ℤ, ℝ) and ℓ^{∞}(ℤ, ℤ) ⊂ ℓ^{∞}(ℤ, ℝ) for the subgroups of integer-valued functions. By viewing every h = ∑_{n∈ℤ} h_{n}u^{n} ∈ R_{1} as the element (h_{n}) ∈ ℓ^{1}(ℤ, ℤ) we can identify R_{1} with ℓ^{1}(ℤ, ℤ).

We furnish the space ℓ^{∞}(ℤ, ℝ) with the topology of coordinate-wise convergence. In this topology ℓ^{∞}(ℤ, ℝ) is a metrizable topological group, ℓ^{∞}(ℤ, ℤ) ⊂ ℓ^{∞}(ℤ, ℝ) is a closed subgroup, and the shift-invariant sets

\begin{matrix}B_{r}(ℓ^{∞}(ℤ, ℝ)) = \{w ∈ ℓ^{∞}(ℤ, ℝ) : ∥w∥_{∞} ≤ r\}, \\ B_{r}(ℓ^{∞}(ℤ, ℤ)) = B_{r}(ℓ^{∞}(ℤ, ℝ)) ∩ ℓ^{∞}(ℤ, ℤ) \end{matrix}  (2.8)

are compact for every r ≥ 0: on these sets our topology coincides with the weak^{∗}-topology.

As in [8] we denote by σ̄ the shift

(σ̄w)_{n} = w_{n+1}  (2.9)

on ℓ^{∞}(ℤ, ℝ), observe that σ̄: ℓ^{∞}(ℤ, ℝ) ⟶ ℓ^{∞}(ℤ, ℝ) is a continuous group automorphism, and define, for every h = ∑_{k∈ℤ} h_{k}u^{k} ∈ R_{1}, a continuous group homomorphism

h(σ̄) = ∑_{k∈ℤ}h_{k}σ̄^{k}: ℓ^{∞}(ℤ, ℝ) ⟶ ℓ^{∞}(ℤ, ℝ)  (2.10)

(cf. (2.5)). The map ρ: ℓ^{∞}(ℤ, ℝ) ⟶ 𝕋^{ℤ}, given by

ρ(w)_{n} = w_{n} (mod 1)  (2.11)

for every w = (w_{n}) ∈ ℓ^{∞}(ℤ, ℝ) and n ∈ ℤ, is a continuous surjective group homomorphism with

ρ ∘ σ̄ = τ ∘ ρ,  (2.12)

and the set

W_{f} = ρ^{−1}(X) = f(σ̄)^{−1}(ℓ^{∞}(ℤ, ℤ)) ⊂ ℓ^{∞}(ℤ, ℝ),  (2.13)

is a closed and shift-invariant subgroup with ker ρ = ℓ^{∞}(ℤ, ℤ) ⊂ W_{f}.

The kernel

W_{f}^{(0)} = ker f(σ̄) ⊂ W_{f}  (2.14)

is obviously finite-dimensional and the restriction of σ̄ to the complexification ℂ ⊗_{ℝ} W_{f}^{(0)} of W_{f}^{(0)} is linear. Hence σ̄ has a nonzero eigenvector v ∈ ℂ ⊗_{ℝ} W_{f}^{(0)} with eigenvalue ω ∈ ℂ, say, and f(ω) = 0. As σ̄ is an isometry on W_{f}^{(0)} we conclude that ω ∈ Ω_{f}^{(0)}. Conversely, if ω ∈ Ω_{f}^{(0)}, then we set v_{n} = ω^{n} for every n ∈ ℤ and obtain that v = (v_{n}) ∈ ℂ ⊗_{ℝ} W_{f}^{(0)}.

This shows that W_{f}^{(0)} = ker f(σ̄) ⊂ W_{f} ⊂ ℓ^{∞}(ℤ, ℝ) is the linear span of the vectors {ℜ(w(ω)), ℑ(w(ω)) : ω ∈ Ω_{f}^{(0)}} with

w(ω)_{n} = ω^{n},  ℜ(w(ω))_{n} = ℜ(ω^{n}),  ℑ(w(ω))_{n} = ℑ(ω^{n}) (2.15)

for every n ∈ ℤ and ω ∈ Ω_{f}^{(0)}, where ℜ and ℑ denote the real and imaginary parts. By (2.12),

X^{(0)} = ρ(ker f(σ̄)) = ρ(W_{f}^{(0)})  (2.16)

is an α-invariant subgroup of X, and the irreducibility of α implies that the closure of X^{(0)} is either equal to {0} (if α is expansive), or to X (if α is nonexpansive). The group X^{(0)} ⊂ X in (2.16) is isomorphic to W_{f}^{(0)}, since ρ is injective on W_{f}^{(0)}, and coincides with the central subgroup of X defined in [13, (3.3)] on which α acts isometrically.

We write

\frac{1}{f(u)} = \frac{1}{f_{m}} ∑_{ω∈Ω_{f}} \frac{b_{ω}}{u − ω}

for the partial fraction decomposition of 1/f with b_{ω} ∈ ℂ for every ω ∈ Ω_{f} and define elements w^{Δ_{±}} and w^{Δ_{0}} in ℓ^{∞}(ℤ, ℝ) by

w_{n}^{Δ_{+}} = \begin{cases}\frac{1}{f_{m}} · ∑_{ω∈Ω_{f}^{−}}  b_{ω}ω^{n−1} & if n ≥ 1, \\ \frac{1}{f_{m}} · ∑_{ω∈Ω_{f}^{(0)}∪Ω_{f}^{+}} −b_{ω}ω^{n−1} & if n ≤ 0, \end{cases}

w_{n}^{Δ_{−}} = \begin{cases}\frac{1}{f_{m}} · ∑_{ω∈Ω_{f}^{−}∪Ω_{f}^{(0)}} b_{ω}ω_{i}^{n−1} & if n ≥ 1, \\ \frac{1}{f_{m}} · ∑_{ω∈Ω_{f}^{+}}  −b_{ω}ω^{n−1} & if n ≤ 0, \end{cases}
w_{n}^{Δ_{0}} = \frac{1}{f_{m}} · ∑_{ω∈Ω_{f}^{(0)}}  b_{ω}ω^{n−1}  for every n ∈ ℤ.

(2.17)

Then

w^{Δ_{0}} ∈ W_{f}^{(0)}, w^{Δ_{+}} + w^{Δ_{0}} = w^{Δ_{−}},
f(σ̄)(w^{Δ_{+}})_{n} = f(σ̄)(w^{Δ_{−}})_{n} = v_{n}^{Δ} = \begin{cases}1 & if n = 0, \\ 0 & otherwise, \end{cases}

(2.18)

where we are using the formal power series identities

∑_{n∈ℤ}w_{n}^{Δ_{+}}u^{n} = \frac{1}{f_{m}} · ( ∑_{ω∈Ω_{f}^{−}} \frac{b_{ω}u}{1 − ωu} + ∑_{ω∈Ω_{f}^{(0)}∪Ω_{f}^{+}} \frac{−b_{ω}ω^{−1}}{1 − ω^{−1}u^{−1}} )

= \frac{1}{f_{m}} · ∑_{ω∈Ω_{f}} \frac{b_{ω}}{u^{−1} − ω} = \frac{1}{f(u^{−1})}

= \frac{1}{f_{m}} · ( ∑_{ω∈Ω_{f}^{−}∪Ω_{f}^{(0)}} \frac{b_{ω}u}{1 − ωu} + ∑_{ω∈Ω_{f}^{+}} \frac{−b_{ω}ω^{−1}}{1 − ω^{−1}u^{−1}} ) = ∑_{n∈ℤ}w_{n}^{Δ_{−}}u^{n}

and

∑_{n∈ℤ}f(σ̄)(w)_{n}u^{n} = f(u^{−1}) · ∑_{n∈ℤ}w_{n}u^{n}

for every w = (w_{n}) ∈ ℓ^{∞}(ℤ, ℝ). The points w^{Δ_{±}} ∈ ℓ^{∞}(ℤ, ℝ) have the following properties.

x^{Δ_{±}} = ρ(w^{Δ_{±}}) ∈ X  by (2.18),
lim_{n→∞} w_{n}^{Δ_{+}} = lim_{n→∞} w_{−n}^{Δ_{−}} = 0  exponentially fast,  (2.19)
x^{Δ_{+}} = x^{Δ_{−}}  if and only if α is expansive.

3. A review of the expansive case

One of the key tools in attempting to find symbolic covers or representations of the automorphism α = α_{R_{1}/(f)} lies in identifying the subgroup

V_{f} = f(σ̄)(W_{f}) ⊂ ℓ^{∞}(ℤ, ℤ).  (3.1)

We first discuss the space V_{f} in the expansive setting, before moving on to the nonexpansive case.

Suppose that the polynomial f in (2.2) is hyperbolic (i.e. that Ω_{f}^{(0)} = ∅ in (2.3)). In this case

w^{Δ_{+}} = w^{Δ_{−}} = w^{Δ},  x^{Δ_{+}} = x^{Δ_{−}} = x^{Δ},  w^{Δ_{0}} = 0.  (3.2)

The point x^{Δ} is a fundamental homoclinic point of α in the sense of [12]:

Δ_{α}(X) = {h(α)(x^{Δ}) : h ∈ R_{1}}.  (3.3)

In the case where f_{m} = |f_{0}| = 1 in (2.2) and X is therefore isomorphic to 𝕋^{m} = ℝ^{m}/ℤ^{m}, the fundamental homoclinic point x^{Δ} has a convenient geometric description. The automorphism α = α_{R_{1}/(f)} of X = X_{R_{1}/(f)} ⊂ 𝕋^{ℤ} in (2.6) is algebraically conjugate to the companion matrix

M_{f} = [ \begin{matrix}0 & 1 & 0 & ... & 0 & 0 \\ 0 & 0 & 1 & ... & 0 & 0 \\ ⋮ & & ⋮ & ... & ⋮ & 0 \\ 0 & 0 & 0 & ... & 0 & 1 \\ −f_{0} & −f_{1} & −f_{2} & ... & −f_{m−2} & −f_{m−1} \end{matrix} ],  (3.4)

of f, acting on 𝕋^{m} from the left, where the isomorphism between X and 𝕋^{m} is the coordinate projection

x ↦ [ x_{0} ; x_{1} ; ⋮ ; x_{m−1} ].

We write W^{(s)} ⊂ ℝ^{m} and W^{(u)} ⊂ ℝ^{m} for the contracting and expanding subspaces of the matrix M_{f}. The quotient map π: ℝ^{m} ⟶ 𝕋^{m} is injective on W^{(s)} and W^{(u)}, and the dense subgroups X^{(s)} = π(W^{(s)}) and X^{(u)} = π(W^{(u)}) satisfy that

Δ_{α}(X) = X^{(s)} ∩ X^{(u)} = π((W^{(s)} + ℤ^{m}) ∩ W^{(u)}).

There exists a unique point y^{Δ} ∈ (W^{(s)}+e^{(1)})∩W^{(u)}, where e^{(1)} = (1, 0, …) is the first unit vector in ℝ^{m}. Since M_{f} is of the form (3.4), the orbit {M_{f}^{n}e^{(1)} : n ∈ ℤ} ⊂ ℤ^{m} generates ℤ^{m} as a group, which is easily seen to imply that the homoclinic point x^{Δ} = π(y^{Δ}) is indeed fundamental.

We return to our more general setting. From (2.19)–(3.2) it follows that

∥w^{Δ}∥_{1} = ∑_{n∈ℤ}|w_{n}^{Δ}| < ∞,

and that

ξ̄(v) = ∑_{n∈ℤ}v_{n}σ̄^{−n}w^{Δ}

is a well-defined element of ℓ^{∞}(ℤ, ℝ) for every v ∈ ℓ^{∞}(ℤ, ℤ). As in [8] we denote by

ξ̄: ℓ^{∞}(ℤ, ℤ) ⟶ ℓ^{∞}(ℤ, ℝ),  ξ = ρ ∘ ξ̄: ℓ^{∞}(ℤ, ℤ) ⟶ X  (3.5)

the resulting continuous group homomorphisms and observe that

ξ(v) = ∑_{n∈ℤ}v_{n}α^{−n}x^{Δ}  (3.6)

for every v ∈ ℓ^{∞}(ℤ, ℤ). Hence

ξ ∘ σ̄ = α ∘ ξ,

i.e. ξ is equivariant. We summarize this discussion in a theorem; the relevant proofs can be found in [8].

Theorem 3.1. Let f ∈ R_{1} be an irreducible hyperbolic polynomial, and let α = α_{R_{1}/(f)} be the expansive automorphism of the compact connected abelian group X = X_{R_{1}/(f)} defined in (2.6)–(2.7). Then

V_{f} = f(σ̄)(W_{f}) = ℓ^{∞}(ℤ, ℤ),

and the homomorphisms ξ̄: ℓ^{∞}(ℤ, ℤ) ⟶ ℓ^{∞}(ℤ, ℝ) and ξ = ρ ∘ ξ̄: ℓ^{∞}(ℤ, ℤ) ⟶ X in (3.5)–(3.6) satisfy that

ξ(ℓ^{∞}(ℤ, ℤ)) = X,
ker ξ = f(σ̄)(ℓ^{∞}(ℤ, ℤ)) ⊂ ℓ^{∞}(ℤ, ℤ),
ξ ∘ σ̄ = α ∘ ξ.

In [21] it was proved that there always exists a compact shift-invariant subset (in fact, a sofic subshift) Ṽ ⊂ ℓ^{∞}(ℤ, ℤ) such that the restriction of ξ to Ṽ is surjective and almost one-to-one (for the definition of a sofic shift we refer to [11] and [31]). In general, however, there is at present no distinguished candidate for such a set Ṽ.

In Section 5 we present an interesting special case in which there is a natural candidate for Ṽ (cf. [21]–[29]): the beta-shift.

4. Homoclinic points and coding in the nonexpansive case

Now suppose that the irreducible polynomial f in (2.2) is nonhyperbolic and not cyclotomic, and that the ergodic automorphism α = α_{R_{1}/(f)} of the compact connected abelian group X = X_{R_{1}/(f)} is therefore ergodic and nonexpansive. Since f is irreducible and has a root of absolute value 1, m is even and f_{i} = f_{m−i} for i = 0, …, m, and we assume that f_{0} = f_{m} > 0. In contrast to the expansive situation, W_{f}^{(0)} = ker f(σ̄) ≠ {0}, the central subgroup X^{(0)} = ρ(W_{f}^{(0)}) in (2.16) is dense in X, and the following theorem shows that there are no nonzero α-homoclinic points.

Theorem 4.1. Let f ∈ R_{1} be an irreducible nonhyperbolic polynomial which is not cyclotomic, and let α = α_{R_{1}/(f)} be the ergodic and nonexpansive automorphism of the compact connected abelian group X = X_{R_{1}/(f)} defined in (2.6)–(2.7). Then Δ_{α}(X) = {0}.

Corollary 4.2. Let α be an irreducible, ergodic and nonexpansive automorphism of a compact connected abelian group X. Then Δ_{α}(X) = {0}.

Proof of Theorem 4.1. The triviality of the homoclinic group Δ_{α}(X) for irreducible nonhyperbolic ergodic toral automorphisms was shown in [12]. Here we give another (and slightly more general) proof using the methods described in the previous section.

Suppose that x is a nonzero α-homoclinic point. Since the restriction of α to X^{(0)} is an isometry it is clear that X^{(0)} ∩ Δ_{α}(X) = {0} and hence that x ∉ X^{(0)}.

We choose w ∈ W_{f} ⊂ ℓ^{∞}(ℤ, ℝ) such that ρ(w) = x and lim_{|n|→∞} w_{n} = 0 (such a choice is obviously possible). Then v = f(σ̄)(w) ∈ ℓ^{∞}(ℤ, ℤ) has only finitely many nonzero coordinates and is therefore of the form h(σ̄)(v^{Δ}) for some h ∈ R_{1}, where the point v^{Δ} ∈ ℓ^{∞}(ℤ, ℤ) is defined in (2.18). We put

w^{∗} = h(σ̄)(w^{Δ_{−}}) ∈ ℓ^{∞}(ℤ, ℝ)

and observe that

w − w^{∗} ∈ ker(f(σ̄)) = W_{f}^{(0)}

by (2.18). From (2.17) we know that lim_{n→−∞} w_{n}^{∗} = lim_{n→−∞} w_{n} = 0, and hence that w = w^{∗} and lim_{n→∞} w_{n}^{∗} = 0, since every element in W_{f}^{(0)} is almost periodic. However,

w_{n}^{∗} = \frac{1}{f_{m}} ∑_{ω∈Ω_{f}^{(0)}∪Ω_{f}^{+}} b_{ω}ω^{n−1}h(ω)

for all sufficiently large positive n, which shows that

∑_{ω∈Ω_{f}^{(0)}} b_{ω}ω^{n}h(ω) = 0 for every n ≥ 0.  (4.1)

From (4.1) we see that h(ω) = 0 for every ω ∈ Ω_{f}^{(0)} or, equivalently, that h is divisible by f. We set h = fh′ with h′ ∈ R_{1}, v′ = h′(σ̄)(v^{Δ}) ∈ ℓ^{∞}(ℤ, ℤ) and w′ = h′(σ̄)(w^{Δ_{−}}) ∈ W_{f} as above, and conclude that w = w^{∗} = f(σ̄)(w′) ∈ ℓ^{∞}(ℤ, ℤ) and x = ρ(w) = 0, contrary to our choice of x. □

Proof of Corollary 4.2. There exists an irreducible, nonhyperbolic and noncyclotomic polynomial f ∈ R_{1} such that α is finitely equivalent to α_{R_{1}/(f)} (cf. (2.6)–(2.7)). If ϕ: X ⟶ X_{R_{1}/(f)} is a continuous, finite-to-one and equivariant group homomorphism, then the restriction of ϕ to Δ_{α}(X) is injective and ϕ(Δ_{α}(X)) ⊂ Δ_{α_{R_{1}/(f)}}(X_{R_{1}/(f)}) = {0} by Theorem 4.1. This proves that Δ_{α}(X) = {0}. □

Although α = α_{R_{1}/(f)} has no nonzero homoclinic points, we have at our disposal the ‘one-sided homoclinic’ points x^{Δ_{±}} in (2.19). Again it may be helpful to identify these points in the special case where f_{0} = f_{m} = 1 in (2.2) and X is therefore isomorphic to 𝕋^{m} = ℝ^{m}/ℤ^{m} (remember that f(u) = u^{m}f(u^{−1})!). As on page 8 we write W^{(s)} ⊂ ℝ^{m}, W^{(u)} ⊂ ℝ^{m} and W^{(0)} ⊂ ℝ^{m} for the contracting, expanding and isometric subspaces of the matrix M_{f} in (3.4). Then there exist unique points y^{Δ_{+}} ∈ (W^{(s)}+e^{(1)})∩(W^{(u)}+W^{(0)}) and y^{Δ_{−}} ∈ (W^{(s)} + W^{(0)} + e^{(1)}) ∩ W^{(u)}, and x^{Δ_{±}} = π(y^{Δ_{±}}) (cf. (2.17)–(2.19)).

We return to the general nonhyperbolic setting and put

ℓ^{∗}(ℤ, ℝ) = \{w = (w_{n}) ∈ ℝ^{ℤ} : sup_{n∈ℤ} \frac{|w_{n}|}{|n| + 1} < ∞\} ⊃ ℓ^{∞}(ℤ, ℝ),  (4.2)

denote by ℓ^{∗}(ℤ, ℤ) the group of integer sequences in ℓ^{∗}(ℤ, ℝ), and furnish these spaces with the topology of coordinate-wise convergence. We extend the maps σ̄, f(σ̄) and ρ in (2.9)–(2.11) to group homomorphisms

σ̄^{∗}: ℓ^{∗}(ℤ, ℝ) ⟶ ℓ^{∗}(ℤ, ℝ), f(σ̄^{∗}): ℓ^{∗}(ℤ, ℝ) ⟶ ℓ^{∗}(ℤ, ℝ),
ρ^{∗}: ℓ^{∗}(ℤ, ℝ) ⟶ 𝕋^{ℤ},

and set

\begin{aligned}W_{f}^{∗} & = \{w ∈ ℓ^{∗}(ℤ, ℝ) : f(σ̄^{∗})(w) ∈ ℓ^{∗}(ℤ, ℤ)\} \\ & = \{w ∈ ℓ^{∗}(ℤ, ℝ) : ρ^{∗}(w) ∈ X\}. \end{aligned}  (4.3)

Then W_{f}^{∗} ⊂ ℓ^{∗}(ℤ, ℝ) is a closed, σ̄^{∗}-invariant subgroup. Since

ker f(σ̄^{∗}) = {w ∈ ℓ^{∗}(ℤ, ℝ) : f(σ̄^{∗})(w) = 0} ⊂ ℓ^{∞}(ℤ, ℝ)

(cf. (2.15)), we obtain that

ker f(σ̄^{∗}) = ker f(σ̄) = W_{f}^{(0)} ⊂ W_{f}.

We define continuous group homomorphisms ξ̄^{∗}: ℓ^{∞}(ℤ, ℤ) ⟶ W_{f}^{∗} and ξ^{∗}: ℓ^{∞}(ℤ, ℤ) ⟶ X by setting

\begin{matrix}ξ̄^{∗}(v) = ∑_{n≥0} v_{n}σ̄^{−n}(w^{Δ_{−}}) + ∑_{n<0} v_{n}σ̄^{−n}(w^{Δ_{+}}), \\ ξ^{∗}(v) = ρ^{∗} ∘ ξ̄^{∗}(v) \end{matrix}  (4.4)

for every v = (v_{n}) ∈ ℓ^{∞}(ℤ, ℤ). Since the coordinates w_{n}^{Δ_{+}} and w_{−n}^{Δ_{−}} decay exponentially as n → ∞ by (2.19), each coordinate of ξ̄^{∗}(v) in (4.4) converges and

ξ̄^{∗}(ℓ^{∞}(ℤ, ℤ)) ⊂ W_{f}^{∗}.  (4.5)

According to (2.18),

\begin{matrix}f(σ̄^{∗}) ∘ ξ̄^{∗}(v) = v, \\ ξ̄^{∗} ∘ f(σ̄^{∗})(w) − w ∈ W_{f}^{(0)} \end{matrix}  (4.6)

for every v ∈ ℓ^{∞}(ℤ, ℤ) and w ∈ W_{f}^{∗} (for the second equation in (4.6) we note that the maps σ̄^{∗}, f(σ̄^{∗}) and ξ̄^{∗} can be extended to the set of sequences with polynomial growth in ℝ^{ℤ} and ℤ^{ℤ}, respectively, where they still satisfy the first equation in (4.6)). From (4.3) and (4.6) we see that

\begin{matrix}ξ̄^{∗} ∘ f(σ̄)(W_{f}) ⊂ W_{f}, \\ V_{f} = f(σ̄)(W_{f}) = \{v ∈ ℓ^{∞}(ℤ, ℤ) : ξ̄^{∗}(v) ∈ ℓ^{∞}(ℤ, ℝ)\}. \end{matrix}  (4.7)

The map ξ̄^{∗}: ℓ^{∞}(ℤ, ℤ) ⟶ ℓ^{∗}(ℤ, ℝ) can obviously not be expected to be shift-equivariant. Indeed,

d(n, v) = (σ̄^{∗})^{n} ∘ ξ̄^{∗}(v) − ξ̄^{∗} ∘ (σ̄^{∗})^{n}(v)

= \begin{cases}∑_{j=0}^{n−1} v_{j}σ̄^{n−j}w^{Δ_{0}} & if n > 0, \\ 0 & if n = 0, \\ − ∑_{j=1}^{n} v_{−j}σ̄^{j−n}w^{Δ_{0}} & if n < 0. \end{cases} (4.8)

for every n ∈ ℤ and v ∈ ℓ^{∞}(ℤ, ℤ), and the resulting map

d: ℤ × ℓ^{∞}(ℤ, ℤ) ⟶ W_{f}^{(0)}  (4.9)

satisfies the cocycle equation

d(m, σ̄^{n}v) + σ̄^{m}d(n, v) = d(m + n, v)  (4.10)

for every m, n ∈ ℤ and v ∈ ℓ^{∞}(ℤ, ℤ).

From the first formula in (4.4) it is clear there exists a constant c′ > 0 with |ξ̄^{∗}(v)|_{n} ≤ c′ · ∥v∥_{∞} for every v ∈ ℓ^{∞}(ℤ, ℤ) and n = 0, …, m − 1, where m is the degree of f. Hence

|ξ̄^{∗} ∘ f(σ̄)(w)|_{n} ≤ c′∥f(σ̄)(w)∥_{∞} ≤ c′∥f∥_{1} · ∥w∥_{∞}

for every w ∈ ℓ^{∞}(ℤ, ℝ) and n = 0, …, m − 1. Since there exists a constant c′′ > 0 with

∥w∥_{∞} ≤ c′′ · max {|w_{0}|, …, |w_{m−1}|}  (4.11)

for every w ∈ W_{f}^{(0)} by (2.15), we can find a constant c > 0 with

∥ξ̄^{∗} ∘ f(σ̄)(w)∥_{∞} ≤ c · ∥f(σ̄)(w)∥_{∞} ≤ c · ∥f∥_{1} · ∥w∥_{∞}  (4.12)

for every w ∈ ℓ^{∞}(ℤ, ℝ).

Equation (4.8) shows that the map ξ^{∗}: ℓ^{∞}(ℤ, ℤ) ⟶ X is equivariant modulo X^{(0)}, and our next result implies that ξ^{∗} is also surjective modulo X^{(0)}.

Proposition 4.3. There exists a closed, bounded, shift-invariant subset Y ⊂ V_{f} with ξ^{∗}(Y) + X^{(0)} = X.

Proof. We recall the notation B_{r}(ℓ^{∞}(ℤ, ℝ)) = {w ∈ ℓ^{∞}(ℤ, ℝ) : ∥w∥_{∞} ≤ r} (cf. (2.8)) and set

B_{r}(W_{f}) = W_{f} ∩ B_{r}(ℓ^{∞}(ℤ, ℝ)).  (4.13)

Then B_{r}(W_{f}) is a closed and bounded — and hence compact — shiftinvariant subset of ℓ^{∞}(ℤ, ℝ), and

Y_{r} = f(σ̄)(B_{r}(W_{f}))  (4.14)

is a compact shift-invariant subset of V_{f}.

For r ≥ 1/2, ρ(B_{r}(W_{f})) = X, and (4.6) guarantees that ξ̄^{∗}∘f(σ̄)(w)−w ∈ W_{f}^{(0)} for every w ∈ B_{r}(W_{f}). Hence there exists, for every x ∈ X, an element w ∈ B_{r}(W_{f}) with ρ(w) = x, and the element v = f(σ̄)(w) ∈ Y_{r} satisfies that ξ̄^{∗}(v) − w ∈ W_{f}^{(0)} and ξ^{∗}(v) − x ∈ X^{(0)}. This proves (2). □

Proposition 4.3 suggests the following definition.

Definition 4.4. A closed, bounded, shift-invariant subset V ⊂ ℓ^{∞}(ℤ, ℤ) is a pseudo-cover of X if ξ^{∗}(V) + X^{(0)} = X.

The last part of this section is devoted to the question whether — and to what extent — the non-equivariance of ξ^{∗} can be ‘corrected’. We start by showing that there is no continuous, equivariant and surjective map ϕ from ℓ^{∞}(ℤ, ℤ) (or from any shift of finite type Y ⊂ ℓ^{∞}(ℤ, ℤ)) to X.

Definition 4.5. Let T be a homeomorphism of a compact metrizable space Y, and let δ be a metric on Y. Two points x, y ∈ Y are homoclinic if lim_{|n|→∞} δ(T^{n}x, T^{n}y) = 0. The homoclinic equivalence relation Δ_{T} (Y) is defined as

Δ_{T} (Y) = {(x, y) ∈ Y^{2} : x and y are homoclinic}.

For every x ∈ Y we denote by

Δ_{T} (x) = {y ∈ Y : (x, y) ∈ Δ_{T} (Y)}

the homoclinic equivalence class of x. The homoclinic relation Δ_{T} (Y) is topologically transitive if Δ_{T} (y) is dense in Y for some y ∈ Y, and minimal if Δ_{T} (y) is dense in Y for every y ∈ Y.

Note that all these definitions are independent of the specific choice of the metric δ.

Proposition 4.6. Let α be an irreducible, ergodic and nonexpansive automorphism of a compact connected abelian group X, and let T be a homeomorphism of a compact metrizable space Y whose homoclinic relation Δ_{Y} (T) is topologically transitive. If ϕ: Y ⟶ X a continuous equivariant map then ϕ(Y) consists of a single fixed point x̄ of α in X.

Proof. For any pair y, y′ of homoclinic points in Y, the points ϕ(y) and ϕ(y′) are homoclinic in X, and hence ϕ(y) − ϕ(y′) = 0 by Corollary 4.2. If Δ_{T} (y) is dense in Y for some y ∈ Y then the continuity of ϕ implies that ϕ(Y) is a single point which must be fixed under α. □

Corollary 4.7. Let f ∈ R_{1} be an irreducible nonhyperbolic polynomial which is not cyclotomic, α = α_{R_{1}/(f)} the ergodic and nonexpansive automorphism of the compact connected abelian group X = X_{R_{1}/(f)} defined in (2.6)–(2.7), and let σ̄ be the shift (2.9) on ℓ^{∞}(ℤ, ℤ). If ϕ: ℓ^{∞}(ℤ, ℤ) ⟶ X is a continuous equivariant map, then ϕ(ℓ^{∞}(ℤ, ℤ)) consist of a single fixed point of α.

Proof. For every positive integer r we set B_{r} = {v ∈ ℓ^{∞}(ℤ, ℤ) : ∥v∥_{∞} ≤ r} = {−r, …, r}^{ℤ}. Then the restriction T = σ̄|_{B_{r}} of σ̄ to B_{r} has a topologically transitive homoclinic equivalence relation, and Proposition 4.6 implies that ϕ(B_{r}) consists of a single point. Since this is true for every r ≥ 1, ϕ(ℓ^{∞}(ℤ, ℤ)) consist of a single fixed point of α. □

Definition 4.8. Let A be a countably infinite set, P = (P (a, a′), a, a′ ∈ A)) a transition matrix with entries in {0, 1}, and X_{P} = {x = (x_{n}) ∈ A^{ℤ} : P (x_{n}, x_{n+1}) = 1 for every n ∈ ℤ} the associated shift of finite type.

The shift X_{P} and the transition matrix P are irreducible if there exist, for every a, a′ ∈ A, an n ≥ 1 and elements a_{1} = a, a_{2}, …, a_{n} = a′ in A with P (a_{i}, a_{i+1}) = 1 for i = 1, …, n − 1. If X_{P} is irreducible then the period p(X_{P}) is the highest common factor of the set of integers n ≥ 1 for which there exist elements elements a_{1}, a_{2}, …, a_{n} = a_{1} in A with P (a_{i}, a_{i+1}) = 1 for i = 1, …, n − 1.

The shift X_{P} and the matrix P are mixing if they are irreducible with period 1.

For the following corollaries we assume that α is an irreducible, ergodic and nonexpansive automorphism of a compact connected abelian group X.

Corollary 4.9. Let Y be a mixing shift of finite type with finite or countably infinite alphabet. Then every continuous equivariant map ϕ: Y ⟶ X sends Y to a single point.

Proof. If T is the shift on Y, then Δ_{Y} (T) is minimal, and our claim follows from Proposition 4.6. □

Corollary 4.10. Let Y be an irreducible shift of finite type with finite or countably infinite alphabet. Then every continuous equivariant map ϕ: Y ⟶ X sends Y to a finite set.

Proof. If p(Y) is the period of Y then the shift T on Y has the property that there exists a closed subset Y_{0} ⊂ Y such that the sets T^{k}Y_{0}, k = 0, …, n − 1 are disjoint, T^{n}Y_{0} = Y_{0}, ⋃_{k=0}^{n−1} T^{k}Y_{0} = Y, and the restriction of T^{n} to Y_{0} is mixing. Hence ϕ(Y) consists of a single periodic orbit for any continuous equivariant map ϕ: Y ⟶ X. □

Corollary 4.11. Let Y be a topologically transitive sofic shift with finite alphabet. Then every continuous equivariant map ϕ: Y ⟶ X sends Y to a finite set.

Proof. Since Y is a continuous equivariant image of a topologically transitive shift of finite type with finite alphabet, the result follows from Corollary 4.10. □

Remark 4.12. Some non-sofic shift-spaces have a topologically transitive homoclinic equivalence relation. For example, in Proposition 5.1 (3) we show that if β > 1 is a real number, and if V_{β} ⊂ {0, …, ⌈β − 1⌉}^{ℤ} is the two-sided beta-shift space defined in (5.4), then the homoclinic equivalence relation Δσ̄(V_{β}) of the beta-shift σ_{β} is topologically transitive, although V_{β} is in general not sofic.

The Corollaries 4.9–4.11 and Remark 4.12 imply that α cannot have Markov (or sofic) partitions or covers in any reasonable sense, and that there are no nontrivial continuous equivariant maps from beta-shifts to X.

In order to understand to what extent the non-equivariance of ξ^{∗} can be ‘corrected’ if we are allowed to drop continuity we set

Ỹ = ℓ^{∞}(ℤ, ℤ) × W_{f}^{(0)} ≅ ℓ^{∞}(ℤ, ℤ) × X^{(0)}  (4.15)

(where we are using the fact that the restriction ρ|_{W_{f}^{(0)}} : W_{f}^{(0)} ⟶ X^{(0)} is

a bijection) and consider the continuous surjective maps σ̃: Ỹ ⟶ Ỹ and ξ̃^{∗}: Ỹ ⟶ W_{f}^{∗}, defined by

\begin{matrix}σ̃(v, w) = (σ̄v, σ̄w + d(1, v)), \\ ξ̃^{∗}(v, w) = ξ̄^{∗}(v) + w \end{matrix}  (4.16)

for every (v, w) ∈ Ỹ = ℓ^{∞}(ℤ, ℤ) × W_{f}^{(0)}. The map σ̃ is obviously a homeomorphism, and

ξ̃^{∗} ∘ σ̃ = σ̄^{∗} ∘ ξ̃^{∗}.  (4.17)

Finally we write π̃: Ỹ ⟶ ℓ^{∞}(ℤ, ℤ) for the first coordinate projection.

Theorem 4.13. Let f ∈ R_{1} be an irreducible nonhyperbolic polynomial which is not cyclotomic, α = α_{R_{1}/(f)} the ergodic and nonexpansive automorphism of the compact connected abelian group X = X_{R_{1}/(f)} defined in (2.6)–(2.7), and let σ̃: Ỹ ⟶ Ỹ be defined by (4.15)–(4.16). For every σ̄- invariant probability measure ν on ℓ^{∞}(ℤ, ℤ) the following conditions are equivalent.

(1) There exists a σ̃-invariant probability measure ν̃ on Ỹ with π̃_{∗}ν̃ = ν;

(2) For every ε > 0 there exists a compact subset C_{ε} ⊂ W_{f}^{(0)} with

ν({v ∈ ℓ^{∞}(ℤ, ℤ)} : d(k, v) ∈ C_{ε}}) > 1 − ε for every k ∈ ℤ;  (4.18)

(3) There exists a Borel map b: ℓ^{∞}(ℤ, ℤ) ⟶ W_{f}^{(0)} with

d(1, v) = b(σ̄v) − σ̄b(v) for ν-a.e. v ∈ ℓ^{∞}(ℤ, ℤ).  (4.19)

If ν satisfies these equivalent conditions, then the Borel map ξ_{b}^{∗}: ℓ^{∞}(ℤ, ℤ) ⟶ X, defined by

ξ_{b}^{∗}(v) = ξ(v) + ρ^{∗} ∘ b(v)  (4.20)

for every v ∈ ℓ^{∞}(ℤ, ℤ), has the property that

ξ_{b}^{∗}(v) − ξ(v) ∈ X^{(0)} for every v ∈ ℓ^{∞}(ℤ, ℤ),

ξ_{b}^{∗} ∘ σ̄ = α ∘ ξ_{b}^{∗} ν-a.e., (4.21)

and the probability measure

μ = (ξ_{b}^{∗})_{∗}ν  (4.22)

on X is α-invariant.

Motivated by Theorem 4.13 we adopt the following terminology.

Definition 4.14. A shift-invariant probability measure ν on ℓ^{∞}(ℤ, ℤ) is weakly d-bounded if it satisfies the three equivalent conditions of Theorem 4.13. The probability measure ν is d-bounded if there exists a compact subset C ⊂ W_{f}^{(0)} such that

ν({v ∈ ℓ^{∞}(ℤ, ℤ) : d(k, v) ∈ C for every k ∈ ℤ}) = 1.  (4.23)

Proof of Theorem 4.13. Suppose that ν̃ is a σ̃-invariant probability measure on Ỹ. We set ν = π̃_{∗}ν̃, fix ε > 0 and choose K_{ε} > 0 with ν̃(ℓ^{∞}(ℤ, ℤ) × B(K_{ε})) > 1 − ε/2, where B(K_{ε}) = {w ∈ W_{f}^{(0)} : ∥w∥_{∞} < K_{ε}}. Since ν̃ is σ̃-invariant, ν̃(σ̃^{k}(ℓ^{∞}(ℤ, ℤ)×B(K_{ε}))∩(ℓ^{∞}(ℤ, ℤ)×B(K_{ε}))) > 1−ε for every k ∈ ℤ, which implies that

ν({v ∈ ℓ^{∞}(ℤ, ℤ) : ∥d(k, v)∥_{∞} < 2K_{ε}}) > 1 − ε for every k ∈ ℤ.

Since ε was arbitrary this shows that (1) ⇒ (2).

In order to check that (2) ⇒ (3) we choose an enumeration Ω_{f}^{(0)} = {ω_{1}, …, ω_{m_{0}}} of Ω_{f}^{(0)}, write W_{f}^{(0)} = ℂ ⊗_{ℝ} W_{f}^{(0)} for the complexification of W_{f}^{(0)} and use the basis {w(ω_{i}) : i = 1, …, m_{0}} in (2.15) to identify W_{f}^{(0)} with ℂ^{m_{0}}. Let

𝕊^{m_{0}} = {(γ_{1}, …, γ_{m_{0}}) ∈ ℂ^{m_{0}} : |γ_{i}| = 1 for i = 1, …, m_{0}}  (4.24)

and define, for every γ = (γ_{1}, …, γ_{m_{0}}) ∈ 𝕊^{m_{0}}, a linear map M_{γ}: ℂ^{m_{0}} ⟶ ℂ^{m_{0}} by setting

M_{γ}z = (γ_{1}z_{1}, …, γ_{m_{0}}z_{m_{0}})  (4.25)

for every z = (z_{1}, …, z_{m_{0}}) ∈ ℂ^{m_{0}}. We form the locally compact semi-direct product

G = ℂ^{m_{0}} ⋊ 𝕊^{m_{0}}

with group operation

(z, γ) · (z′, γ′) = (z + M_{γ}z′, γγ′)

for every z, z′ ∈ ℂ^{m_{0}} and γ, γ′ ∈ 𝕊^{m_{0}} and set

d^{∗}(k, v) = (d(k, v), ω^{k})

for every k ∈ ℤ and v ∈ ℓ^{∞}(ℤ, ℤ), where

ω^{k} = (ω_{1}^{k}, …, ω_{m_{0}}^{k}).  (4.26)

By (4.10), the resulting map d^{∗}: ℤ × ℓ^{∞}(ℤ, ℤ) ⟶ G satisfies the cocycle equation

d^{∗}(k, σ̄^{l}v) · d^{∗}(l, v) = d^{∗}(k + l, v)

for every k, l ∈ ℤ and v ∈ ℓ^{∞}(ℤ, ℤ), where we are using the fact that the shift σ̄ on W_{f}^{(0)} corresponds to M_{ω} under our identification of W_{f}^{(0)} with ℂ^{m_{0}}. Furthermore, the map d^{∗}(k, ·): ℓ^{∞}(ℤ, ℤ) ⟶ G is continuous for every k ∈ ℤ.

If the probability measure ν satisfies (2), then the cocycle d^{∗}: ℤ×ℓ^{∞}(ℤ, ℤ) ⟶ G is bounded in the sense that there exists, for every ε > 0, a compact subset C_{ε} ⊂ G with

ν({v ∈ ℓ^{∞}(ℤ, ℤ) : d^{∗}(k, v) ∉ C_{ε}}) < ε

for every k ∈ ℤ, and [18, Theorem 4.7] implies the existence of a Borel map b′: ℓ^{∞}(ℤ, ℤ) ⟶ G and of a compact subgroup K ⊂ G such that

b′(σ̄v)^{−1} · d^{∗}(1, v) · b′(v) ∈ K  (4.27)

for ν-a.e. v ∈ ℓ^{∞}(ℤ, ℤ).

Every compact subgroup K ⊂ G is of the form

K = {(w(γ), γ) : γ ∈ Γ_{0}}

for some compact subgroup Γ_{0} ⊂ 𝕊^{m_{0}} and some Borel map w: Γ_{0} ⟶ ℂ^{m_{0}} satisfying the cocycle equation

w(γγ′) = w(γ) + M_{γ}w(γ′)

As Γ_{0} is compact, this cocycle is a coboundary, i.e. there exists a t ∈ ℂ^{m_{0}} with

w(γ) = M_{γ}t − t

for every γ ∈ Γ_{0}, and

K = {(M_{γ}t − t, γ) : γ ∈ Γ_{0}}.  (4.28)

We write the map b′ in (4.27) as b′ = (b_{1}, b_{2}) with b_{1} : ℓ^{∞}(ℤ, ℤ) ⟶ ℂ^{m_{0}} and b_{2} : ℓ^{∞}(ℤ, ℤ) ⟶ 𝕊^{m_{0}}. According to (4.27)–(4.28),

(b_{1}(σ̄v), b_{2}(σ̄v))^{−1} · (d(1, v), ω) · (b_{1}(v), b_{2}(v))

= (−M_{b_{2}(σ̄v)}^{−1}b_{1}(σ̄v) + M_{b_{2}(σ̄v)}^{−1}d(1, v) + M_{b_{2}(σ̄v)^{−1}ω}b_{1}(v), b_{2}(σ̄v)^{−1}ωb_{2}(v))

= (M_{b_{2}(σ̄v)^{−1}ωb_{2}(v))}t − t, b_{2}(σ̄v)^{−1}ωb_{2}(v)))

with b_{2}(σ̄v)^{−1}ωb_{2}(v)) ∈ Γ_{0}, and hence

d(1, v) = M_{ω}(M_{b_{2}(v)}t − b_{1}(v)) − (M_{b_{2}(σ̄v)}t − b_{1}(σ̄v))
= σ̄(M_{b_{2}(v)}t − b_{1}(v)) − (M_{b_{2}(σ̄v)}t − b_{1}(σ̄v))

for ν-a.e. v ∈ ℓ^{∞}(ℤ, ℤ). We set b(v) = b_{1}(v)−M_{b_{2}(v)}t for every v ∈ ℓ^{∞}(ℤ, ℤ), view b as a map from ℓ^{∞}(ℤ, ℤ) to W_{f}^{(0)} ⊃ W_{f}^{(0)}, and obtain that

d(1, v) = b(σ̄v) − σ̄b(v)

for ν-a.e. v ∈ ℓ^{∞}(ℤ, ℤ). If

b(v) = (b(v) + \overline{b(v)})/2,

where the bar denotes complex conjugation in W_{f}^{(0)} = ℂ ⊗_{ℝ} W_{f}^{(0)}, then the resulting map b: ℓ^{∞}(ℤ, ℤ) ⟶ W_{f}^{(0)} satisfies (4.19), and (4.21) follows from (4.19) and (4.20). This completes the proof that (2) ⇒ (3).

Finally, if (3) is satisfied, then there exists a unique, and obviously σ̃- invariant, probability measure ν̃ on Ỹ with π̃_{∗}ν̃ = ν and ν̃({(v, b(v)) : v ∈ ℓ^{∞}(ℤ, ℤ)}) = 1, which proves that (3) ⇒ (1).

The final assertions (4.21) and the α-invariance of the probability measure μ in (4.22) are immediate consequences of (4.19). □

We recall the following definition from [13].

Definition 4.15. Let f ∈ R_{1} be an irreducible nonhyperbolic polynomial which is not cyclotomic, and let α = α_{R_{1}/(f)} be the ergodic and nonexpansive automorphism of the compact connected abelian group X = X_{R_{1}/(f)} in (2.6)-(2.7). Two α-invariant probability measures μ_{1}, μ_{2} on X are centrally equivalent if they have an invariant joining ϱ (i.e. an (α × α)-invariant measure ϱ on X × X which projects to μ_{1} and μ_{2}, respectively) so that, for ϱ-a.e. (x, y) ∈ X × X, x and y lie on the same central leaf. In other words,

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

where X^{(0)} ⊂ X is the central subgroup of α defined in (2.16).

It is not hard to show that any two centrally equivalent probability measures have the same entropy under α. Since Lebesgue measure is the unique measure of maximum entropy for α, it follows that the only measure centrally equivalent to Lebesgue measure is Lebesgue measure itself.

Example 4.16. If ν is a weakly d-bounded σ̄-invariant probability measure on ℓ^{∞}(ℤ, ℤ), and if b′, b′′: ℓ^{∞}(ℤ, ℤ) ⟶ W_{f}^{(0)} are two maps satisfying (4.19), then the α-invariant probability measures μ = (ξ_{b′}^{∗})_{∗}ν and μ′ = (ξ_{b′′}^{∗})_{∗}ν are centrally equivalent, since ξ_{b′}^{∗}(x) − ξ_{b′′}^{∗}(x) ∈ X^{(0)} for every x ∈ X.

If the equation (4.19) has a measurable solution b, then this solution is generally not unique. Given a weakly d-bounded σ̄-invariant probability measure we may thus try to choose b so that (ξ_{b}^{∗})_{∗}ν is as simple as possible.

Proposition 4.17. Let ν be a weakly d-bounded σ̄-invariant probability measure on ℓ^{∞}(ℤ, ℤ), and let b be a solution of (4.19). Assume that (ξ_{b}^{∗})_{∗}ν is singular with respect to Lebesgue measure. Then there is a solution b′ of (4.19) and an α-invariant Borel set Z ⊂ X so that

(1) Z intersect each coset of X^{(0)} in at most one point,

(2) (ϕ_{b′}^{∗})_{∗}ν(Z) = 1.

Proof. By [13, Theorem 1.3.(1)], there exist a probability measure μ′ on X which is centrally equivalent to μ = (ξ_{b}^{∗})_{∗}ν and a Borel set Z ⊂ X (which we may as well assume to be α-invariant) of full μ′-measure which intersects each coset of X^{(0)} in at most one point.

Since μ′ and μ are centrally equivalent,

|(ξ_{b}^{∗}(v) + X^{(0)}) ∩ Z| = 1  (4.29)

for ν-a.e. v ∈ ℓ^{∞}(ℤ, ℤ). Define b′(v) ∈ W_{f}^{(0)} by the requirement that

ξ^{∗}(v) + ρ ∘ b′(v)

is the single point in the set (4.29). This function is certainly measurable, as can be verified easily by using the joining establishing the central equivalence of μ′ and μ. It also satisfies (4.19): since Z is α-invariant,

{α ∘ ξ^{∗}(v) + ρ ∘ σ̄(b′(v))} = α(ξ_{b}^{∗}(v) + X^{(0)}) ∩ Z

= (ξ_{b}^{∗}(σ̄v) + X^{(0)}) ∩ Z = {ξ^{∗}(σ̄x) + ρ ∘ b(σ̄x)} ,

and since ρ is injective on W_{f}^{(0)},

d(1, v) = σ̄^{∗} ∘ ξ̄^{∗}(v) − ξ̄^{∗} ∘ σ̄(v) = b′(σ̄v) − σ̄b′(v).

By construction, ξ_{b′}(v) ∈ Z for every v for which b′ is well-defined (i.e. on a set of full ν-measure). □

Proposition 4.18. Let f ∈ R_{1} be an irreducible nonhyperbolic polynomial which is not cyclotomic, and let α = α_{R_{1}/(f)} be the ergodic and nonexpansive automorphism of the compact connected abelian group X = X_{R_{1}/(f)} in (2.6)-(2.7). For every α-invariant probability measure μ on X there exists a d-bounded σ̄-invariant probability measure ν on ℓ^{∞}(ℤ, ℤ) such that μ is centrally equivalent to the probability measure (ξ_{b}^{∗})_{∗}ν in Theorem 4.13.

Proof. Let μ be an α-invariant probability measure on X. We set W = {w = (w_{n}) ∈ W_{f} : 0 ≤ w_{n} < 1 for every n ∈ ℤ} (cf. (2.13)), note that the restriction ρ|_{W} of the equivariant map ρ: W_{f} ⟶ X to W is bijective, and conclude that there exists a unique σ̄-invariant probability measure μ′ on W with ρ_{∗}μ′ = μ.

The σ̄-invariant probability measure ν = f(σ̄)_{∗}μ′ is supported on Y_{r} ⊂ ℓ^{∞}(ℤ, ℤ) for some r > 0, where Y_{r} = f(σ̄)(B_{r}(W_{f})) as in the proof of Proposition 4.3. By (4.12), ξ̄^{∗}(Y_{r}) is a bounded subset of ℓ^{∞}(ℤ, ℝ). This shows that the cocycle d in (4.9) is uniformly bounded on Y_{r} and so ν is d-bounded (cf. (4.14)).

Let b: ℓ^{∞}(ℤ, ℤ) ⟶ W_{f}^{(0)} be a Borel map satisfying (4.19), and let ξ_{b}^{∗}: ℓ^{∞}(ℤ, ℤ) ⟶ X be given by (4.20). Since x−ξ_{b}^{∗}∘f(σ̄)∘(ρ|_{W})^{−1}(x) ∈ X^{(0)} for every x ∈ X, the α-invariant probability measure (ξ_{b}^{∗})_{∗}ν is centrally equivalent to μ. □

The discussion in this section shows that in the nonexpansive case we have to make a choice between continuity and equivariance: the map ξ^{∗}: ℓ^{∞}(ℤ, ℤ) ⟶ X in (4.4) is continuous, but not equivariant, and the maps ξ_{b} : ℓ^{∞}(ℤ, ℤ) ⟶ X in (4.20), which are equivariant at least on some reasonably large sets, are generally not continuous. In neither case can we expect these maps to be surjective.

If ν is a weakly d-bounded σ̄-invariant probability measure on ℓ^{∞}(ℤ, ℤ), then the Borel map ξ_{b} : ℓ^{∞}(ℤ, ℤ) ⟶ X in (4.20) is equivariant ν-a.e. and the μ = (ξ_{b}^{∗})_{∗}ν in (4.22) is therefore α-invariant, but the entropy of μ will generally be lower than that of ν.

In Proposition 4.18 we saw that we can obtain every α-invariant probability measure on X — up to central equivalence — from a d-bounded shift-invariant probability measure on ℓ^{∞}(ℤ, ℤ). However, all such measures are concentrated on the somewhat elusive set V_{f}, so that this result is of limited interest.

For this reason one would ideally like to find ‘nice’ and ‘large’ compact subshifts V ⊂ ℓ^{∞}(ℤ, ℤ) (where nice means something like a shift of finite type or a sofic shift, and large means that the subshift should be a pseudocover of X in the sense of Definition 4.4), such that the following conditions are satisfied:

(1) for every v ∈ ℓ^{∞}(ℤ, ℤ), the intersection of V with v+(f(σ̄)(ℓ^{∗}(ℤ, ℤ)) ∩ ℓ^{∞}(ℤ, ℤ)) is as small as possible,

(2) for every weakly d-bounded shift-invariant probability measure ν on V, the probability measure μ = (ξ_{b}^{∗})_{∗}ν in (4.22) has the same entropy as ν,

(3) every α-invariant probability measure is centrally equivalent to a probability measure obtained in this manner.

At this stage we have made only limited progress in this direction (cf. Section 6, where we investigate the connection between two-sided beta-shift arising from a Salem number β and the ergodic nonhyperbolic toral automorphism defined by the companion matrix if the minimal polynomial of β). One of the key difficulties one encounters in pursuing this program in any generality is the following: although the restriction to ℓ^{∞}(ℤ, ℤ) of the map f(σ̄): W_{f} ⟶ ℓ^{∞}(ℤ, ℤ) is injective, the set

{w ∈ ℓ^{∗}(ℤ, ℤ) : f(σ̄)(w) ∈ V }

need not be contained in ℓ^{∞}(ℤ, ℤ) and the set

{w ∈ ℓ^{∞}(ℤ, ℤ) : f(σ̄)(w) ∈ V }

may be unbounded, even if V ⊂ ℓ^{∞}(ℤ, ℤ) is a bounded, shift-invariant set.

Example 4.19. For every w ∈ W_{f}^{(0)} and n ∈ ℤ we set ζ(w)_{n} = ⌈w_{n}⌉, where ⌈t⌉ is again the smallest integer ≥ t for any t ∈ ℝ. The resulting map ζ: W_{f}^{(0)} ⟶ ℓ^{∞}(ℤ, ℤ) has the property that the set ζ(W_{f}^{(0)}) is unbounded, but ∥f(σ̄)(v)∥_{∞} < ∥f∥_{1} for every v ∈ ζ(W_{f}^{(0)}).

In order to verify that f(σ̄) maps some unbounded sequences in ℓ^{∗}(ℤ, ℤ) into ℓ^{∞}(ℤ, ℤ) we choose θ ∈ Ω_{h}^{(0)} (cf. (2.3)) and define, for every integer j ≥ 0, a point ω^{(j)} = (ω_{n}^{(j)}) ∈ ℓ^{∞}(ℤ, ℝ) by setting

ω_{n}^{(j)} = \begin{cases}θ^{n} + θ^{−n} & if n ≥ j, \\ 0 & otherwise. \end{cases}

Then

(f(σ̄)ω^{(j)})_{n} = 0

for n < j − d and n ≥ j, and ∥h(σ̄)ω^{(j)}∥_{∞} ≤ 2 · ∥f∥_{1}. For every n ∈ ℤ we put

ω̃ = ∑_{j=0}^{∞} ω^{(3jd)}, w_{n} = ⌈ω̃_{n}⌉.

The resulting point w in ℓ^{∗}(ℤ, ℤ) is unbounded and satisfies that ∥f(σ̄)w∥_{∞} ≤ 3 · ∥f∥_{1}.

5. Beta-shifts and their properties

We fix a real number β > 1 and consider the beta-transformation

x ↦ T_{β}x = βx (mod 1)

from the closed unit interval [0, 1] to the half-open interval I = [0, 1) (cf.

[14] and [16]).

For every x ∈ I, the beta-expansion e_{β}(x) = (e_{β}(x)_{n}, n ≥ 1) of x is defined by

e_{β}(x)_{n} = βT_{β}^{n−1}x − T_{β}^{n}x

for every n ≥ 1. Note that e_{β}(x)_{n} ∈ {0, …, ⌈β − 1⌉} for every n ≥ 1, where ⌈t⌉ is the smallest integer ≥ t for any t ∈ ℝ, and that

x = ∑_{n≥1} e_{β}(x)_{n}β^{−n}  (5.1)

for every x ∈ I.

We denote by ≺ the lexicographic order on the space ℓ_{+}^{∞} of all bounded one-sided sequences v = (v_{n}, n ≥ 1) of nonnegative integers and write σ̄_{+} for the one-sided shift (σ̄_{+}v)_{n} = v_{n+1} on ℓ_{+}^{∞}. The closed, σ̄_{+}-invariant set

V_{β}^{+} = \overline{\{e_{β}(x) : x ∈ I\}}  (5.2)

is called the beta-shift space (where the bar denotes closure); it contains a unique lexicographically maximal element e_{β}^{∗} with the following properties (cf. [14]):

σ̄_{+}^{k}e_{β}^{∗} ⪯ e_{β}^{∗} for every k ≥ 0,

∑_{n≥1} e_{β}^{∗}β^{−n} = 1, σ̄_{+}^{n}e_{β}^{∗} ≠ 0 for every n ≥ 0,

V_{β}^{+} = {v ∈ ℓ_{+}^{∞} : σ̄_{+}^{n}v ⪯ e_{β}^{∗} for every n ≥ 0}.

(5.3)

Here we are interested in the two-sided beta-shift space. We write v^{+} = (v_{1}, v_{2}, …) for every v = (v_{n}) ∈ ℓ^{∞}(ℤ, ℤ) and set

V_{β} = {v ∈ ℓ^{∞}(ℤ, ℤ) : (σ̄^{n}v)^{+} ∈ V_{β}^{+} for every n ∈ ℤ}

= {v ∈ ℓ^{∞}(ℤ, ℤ) : (σ̄^{n}v)^{+} ⪯ e_{β}^{∗} for every n ∈ ℤ}. (5.4)

For every v ∈ ℓ^{∞}(ℤ, ℤ) with v_{−n} = 0 for all sufficiently large n ≥ 0 we define the evaluation η_{β}(v) ∈ ℝ by

η_{β}(v) = ∑_{n∈ℤ}v_{n}β^{−n}.  (5.5)

If we view V_{β}^{+} as the subset {v ∈ V_{β} : v_{n} = 0 for n ≤ 0}, then the evaluation defines a continuous, surjective, at most two-to-one map η_{β}: V_{β}^{+} ⟶ [0, 1] with

e_{β}(η_{β}(v)) = v  (5.6)

for all v in the complement of a countable subset of V_{β}^{+} (cf. [14] and (5.1)): the only possible exceptions to (5.6) are points satisfying (σ̄^{k}v)^{+} = e_{β}^{∗} for some k > 0 (cf.(5.3)).

The following elementary observations follow directly from (5.3)–(5.4):

Proposition 5.1. Let β > 1, and let V_{β} ⊂ ℓ^{∞}(ℤ, ℤ) be the two-sided betashift space defined in (5.4).

(1) If v, w ∈ V_{β} satisfy that w^{+} ≺ v^{+} in the notation of (5.3)–(5.4), then the point v′ with

v′_{n} = \begin{cases}v_{n} & if n ≤ 0, \\ w_{n} & if n > 0, \end{cases}

lies in V_{β};

(2) If β is algebraic with minimal polynomial f ∈ R_{1}, and if v, w ∈ V_{β} satisfy that v_{n} = w_{n} for every n < 0, v_{0} > w_{0} and w − v ∈ f(σ̄)(ℓ^{∞}(ℤ, ℤ)), then v_{0} = w_{0} + 1, v^{+} = 0 and w^{+} = e_{β}^{∗}. It follows that

(v + f(σ̄)(ℓ^{1}(ℤ, ℤ))) ∩ V_{β} = {v}

for every v ∈ V_{β}.

(3) The homoclinic equivalence relation Δσ̄(V_{β}) (cf. Definition 4.5) is topologically transitive on V_{β}.

Proof. In order to prove (1) we note that (v′_{k+1}, v′_{k+2}, …) ≺ (v_{k+1}, v_{k+2}, …) ⪯ e_{β}^{∗} whenever k < 0, and (v′_{k+1}, v′_{k+2}, …) = (w_{k+1}, w_{k+2}, …) ⪯ e_{β}^{∗} otherwise. According to (5.4) this implies that v′ ∈ V_{β}.

If β is algebraic with minimal polynomial f, and if

w ∈ (v + f(σ̄)(ℓ^{∞}(ℤ, ℤ))) ∩ V_{β}

and v_{n} = w_{n} for all n < 0, then

η_{β}(w_{0}, w_{1}, …) = η_{β}(v_{0}, v_{1}, …),

and (5.3) and (5.6) imply (2).

In order to verify (3), we denote by 0 ∈ V_{β} the two-sided infinite sequence of zeros. For every v ∈ V_{β} and n ∈ ℕ the point v′ defined by

v′_{k} = \begin{cases}0 & if k < −n \\ v_{n} & if −n ≤ k ≤ n \\ 0 & if n < k \end{cases}

again lies in V_{β}, due to the lexicographic definition of the beta-shift in (5.3). It is also clearly homoclinic to 0. This shows that the homoclinic equivalence class of 0 is dense in V_{β}. □

Beta-shifts are in general not sofic (in fact, they are sofic if and only if the point e_{β}^{∗} in (5.3) is eventually periodic which implies, in turn, that β is algebraic — cf. e.g. [3]). However, even if V_{β} is not sofic, i.e. cannot be obtained by relabelling the letters of a shift of finite type with finite alphabet, it always has a nice description in terms of a certain shift of finite type Σ_{β} with a countable alphabet. This infinite state shift of finite type has additional nice properties which make it a useful tool in the study of beta-shifts.

We now present the construction in [9, 26] of this shift of finite type due to Hofbauer and Takahashi and its relation to the beta-shift (note that there is a gap in [26]; see [9] for details).

For any pair of points v, v′ ∈ V_{β}^{+} with v ≺ v′, let [v, v′] ⊂ V_{β}^{+} be the set of points which lie between v and v′ in the lexicographic order, and let 0_{+} denote the one-sided sequence of zeros.

Let A = {0, …, ⌈β − 1⌉}. The alphabet (or state space) Ā of Σ_{β} is given by Ā = Ā′ ∪ Ā′′ where

\begin{aligned}Ā′ & = \{(a, [0_{+}, e_{β}^{∗}]) : a = 0, …, ⌈β − 2⌉\} \\ Ā′′ & = \{((e_{β}^{∗})_{k}, [0_{+}, σ̄_{+}^{k}e_{β}^{∗}]) : k = 1, 2, … \}. \end{aligned}  (5.7)

Note that Ā is finite if and only if e_{β}^{∗} is eventually periodic.

The allowed transition in Σ_{β} are defined as follows. Each state ā ∈ Ā′ can be followed by any other state in Ā′ as well as by the state (⌈β − 1⌉, [0_{+}, σ̄_{+}e_{β}^{∗}]). Each state ā = ((e_{β}^{∗})_{k}, [0_{+}, σ̄_{+}^{k}e_{β}^{∗}]) ∈ Ā′′ can be followed by either ā′ = (a, [0, e_{β}^{∗}]) for a < (e_{β}^{∗})_{k+1}, or by ((e_{β}^{∗})_{k+1}, [0_{+}, σ̄_{+}^{k+1}e_{β}^{∗}]). We denote by P = (P (ā, ā′), ā, ā′ ∈ Ā) the corresponding transition matrix, i.e. P (ā, ā′) = 1 if and only if ā can be followed by ā′.

Let ϕ: Ā ⟶ A be the projection onto the first coordinate, and let ϕ be the corresponding map from Ā^{ℤ} to A^{ℤ}. One can show quite easily that ϕ(Σ_{β}) ⊂ V_{β}. In general, ϕ|_{Σ_{β}} need not be surjective. What is true (see [9]) is that the complement N of ϕ(Σ_{β}) is a shift invariant subset of V_{β} with the property that any measure supported on it has zero entropy.

This construction is used in particular to show that V_{β} has a unique σ̄- invariant measure μ_{β} of maximal entropy with entropy log β (cf. [9]).

Theorem 5.2 ([9]). The transition matrix P = (P (a, a′), a, a′ ∈ Ā) of Σ_{β} has maximal eigenvalue β and unique positive left and right eigenvectors x = (x(a), a ∈ Ā), y = (y(a), a ∈ Ā) with xP = βx, P y = βy and ∑_{a∈Ā} x(a) = ∑_{a∈Ā} y(a) = 1.

Let μ̄_{P} be the Markov measure on Σ_{P} defined by

μ̄_{P} ([a_{m_{1}}, …, a_{m_{2}}]) = \frac{β^{−(m_{2}−m_{1})}y(a_{m_{2}})}{y(a_{m_{1}})}  (5.8)

for every cylinder set

[a_{m_{1}}, …, a_{m_{2}}] = {y ∈ Σ_{P} : y_{n} = a_{n} for n = m_{1}, …, m_{2}}.

Then the restriction of ϕ to Σ_{P} is injective μ̄_{P} -a.e., and ϕ_{∗}μ̄_{P} = μ_{β}.

The beta-shift is known to be sofic for Pisot numbers as well as for Salem numbers of degree four. For general Salem numbers β it is not known whether V_{β} has to be sofic (cf. [2], [5]–[7] and [17]).

6. The beta-shift and symbolic embeddings for Salem numbers

We start this section with a brief review of the case where β is a Pisot number, and where the β-shift is a sofic model of the corresponding hyperbolic toral automorphism.

Proposition 6.1 ([21]). Let β > 1 be a Pisot number, f ∈ R_{1} its minimal polynomial of degree m, say, and let α = α_{R_{1}/(f)} be the expansive automorphism of the compact abelian group X = X_{R_{1}/(f)} described in (2.6) (if β is a Pisot unit then X ≅ 𝕋^{m}). Then the restriction of the equivariant map ξ: ℓ^{∞}(ℤ, ℤ) ⟶ X in (3.6) to the two-sided beta-shift V_{β} ⊂ ℓ^{∞}(ℤ, ℤ) is surjective and finite-to-one. In particular, if ν is a shift-invariant probability measure on V_{β}, then the measure μ = ξ_{∗}ν on X is α-invariant and has the same entropy as ν. Furthermore, every α-invariant probability μ on X can be obtained in this manner.

The restriction of ξ to V_{β} in Proposition 6.1 is conjectured to be almost one-to-one, although this has only been proved in some examples (cf. [21]– [23]). For earlier special cases of Proposition 6.1 we refer to [24].

Proposition 6.1 describes the close connection between the two-sided betashift of a Pisot unit β > 1 and the toral automorphism defined by the companion matrix of the minimal polynomial of β. One of the principal motivations of this paper was the question whether there exists an analogous result for Salem numbers.

The following discussion shows that, although Proposition 6.1 does not hold in this case, there does exist a connection between two-sided betashifts of Salem numbers and the nonhyperbolic ergodic toral automorphisms defined by the companion matrices of their minimal polynomials. However, this connection is much more complicated and tenuous than in the Pisot case.

For the remainder of this section we restrict ourselves to Salem numbers, their minimal polynomials and their companion matrices. Assume therefore that β is a Salem number with minimal polynomial f ∈ R_{1} of (even) degree m, say, and let α = α_{R_{1}/(f)} be the ergodic and nonexpansive automorphism of X = X_{R_{1}/(f)} defined by (2.6)–(2.7), which is algebraically conjugate to the companion matrix M_{f} in (3.4), acting on 𝕋^{m}. The corresponding twosided beta-shift will be denoted by V_{β} ⊂ ℓ^{∞}(ℤ, ℤ), and we write μ_{β} for the unique shift-invariant measure of maximal entropy on V_{β}.

Since the homoclinic equivalence relation Δσ̄(V_{β}) is topologically transitive on V_{β} by Proposition 5.1 (3), Proposition 4.6 shows that a simple symbolic description as in Proposition 6.1 is not possible in this case. A partial analogue to Proposition 6.1 is presented in Theorem 6.4 below.

We start our discussion by using the information in Theorem 5.2 to prove the following:

Proposition 6.2. The measure μ_{β} on V_{β} ⊂ ℓ^{∞}(ℤ, ℤ) is not weakly dbounded.

Proof. Assume in contradiction that μ_{β} is weakly d-bounded, and let b: V_{β} ⟶ W_{f}^{(0)} be the map in (4.19). We lift d and b to Σ_{β} by setting

d̄(n, v̄) = d(n, ϕ(v)),  b̄(v̄) = b(ϕ(v̄))

for every v̄ ∈ Σ_{β}.

For any a < b ∈ ℤ ∪ {±∞}, let C̄_{a}^{b} denote the sigma algebra of Borel subsets of Σ_{β} that depend only on coordinates a through b, and set D̄_{n} = C̄_{−∞}^{−1}∨C̄_{n}^{∞}. It follows from (5.8) that for almost every v̄ ∈ Σ_{β}, the conditional measure μD̄_{n} ; v̄ on the atom [v̄]D̄_{n} is the normalized counting measure (note that |[v̄]D̄_{n} | < ∞).

We also note that, for any v̄ ∈ Σ_{β},

the map v̄′ ↦ d̄(n, v̄′) is injective on [v̄]D̄_{n}.  (6.1)

Indeed, if v̄′, v̄′′ ∈ [v̄]D̄_{n} satisfy d̄(n, v̄′) = d̄(n, v̄′′) then, by (4.8),

∑_{i=0}^{n−1} ϕ(v̄′_{i})σ̄^{n−i}w^{Δ_{0}} = ∑_{i=0}^{n−1} ϕ(v̄′′_{i})σ̄^{n−i}w^{Δ_{0}},

implying that

ϕ(v̄′) ∈ ϕ(v̄′′) + f(σ̄)(ℓ^{1}(ℤ, ℤ)).

Since ϕ(v̄′), ϕ(v̄′′) ∈ V_{β} it follows from Proposition 5.1 (2) that

ϕ(v̄′) = ϕ(v̄′′).  (6.2)

However, since w̄_{i−1} and ϕ(w̄_{i}) determine w̄_{i} for any w̄ ∈ Σ_{β}, and since v̄′, v̄′′ ∈ [v̄]D̄_{n}, it follows that v̄′ = v̄′′, which proves (6.1). Note that if (6.1) holds for v̄, then it also holds for every w̄ ∈ [v̄]C̄_{−1}^{n}.

Pick n ∈ ℕ and v̄^{0} ∈ Σ_{β} so that v̄_{−1}^{0} ∈ Ā′, v̄_{n}^{0} ∈ Ā′ and (ϕ(v̄^{0})_{0}, …, ϕ(v̄_{n−1}^{0})) ≠ (0, …, 0) (cf. (5.7)). Then the lexicographic characterization of V_{β} in (5.4) and the definition of Σ_{β} together imply that

|[v̄]D̄_{n} | > 1 for every v̄ ∈ [v̄^{0}]C̄_{−1}^{n}.  (6.3)

Put

δ = min (∥d̄(n, v̄′) − d̄(n, v̄′′)∥_{∞} : v̄′, v̄′′ ∈ [v̄^{0}]D̄_{n}) .  (6.4)

Note that d̄(n, ·) depends only on the coordinates 0, …, n−1, i.e. that d̄(n, ·) is C̄_{0}^{n−1} measurable, and that the expression on the right hand side of (6.4) will therefore not change if we replace v̄^{0} by any other point in [v̄^{0}]C̄_{−1}^{n}.

Take k ∈ ℕ to be large (to be determined later) and let b̄_{k} be the conditional expectation Eμ̄_{P} (b̄|C̄_{−k}^{k}). We set

Z̄ = {v̄ : ∥b̄(v̄) − b̄_{k}(v̄)∥ < δ/20}

and choose k so that

μ̄_{P} (Z̄) > 1 − μ̄_{P} ([v̄^{0}]C̄_{−1}^{n})/10.  (6.5)

Let Z̄′ = σ̄^{−2k}Z̄ ∩ σ^{2k+n}Z̄. By definition of Z̄,

∥d̄(4k + n, σ̄^{−2k}v̄) − d̄(4k + n, σ̄^{−2k}v̄′)∥_{∞}
≤ ∥b̄(σ̄^{−2k}v̄) − b̄(σ̄^{−2k}v̄′)∥_{∞} + ∥b̄(σ̄^{2k+n}v̄) − b̄(σ̄^{2k+n}v̄′′)∥_{∞}

≤ δ/5 + ∥b̄_{k}(σ̄^{−2k}v̄) − b̄_{k}(σ̄^{−2k}v̄′)∥_{∞} + ∥b̄_{k}(σ̄^{2k+n}v̄) − b̄_{k}(σ̄^{2k+n}v̄′)∥_{∞}

= δ/5.

for any v̄ ∈ Z̄′ and v̄′ ∈ [v̄]D̄_{n} ∩ Z̄′. In view of (6.4) and the remark following it this shows that, for any v̄ ∈ [v̄^{0}]C̄_{−1}^{n}, at most one of the points in [v̄]D̄_{n}

is in Z̄′. Since μ̄D̄_{n} ; v̄ is the counting measure on at least two points for every v̄ ∈ [v̄^{0}]C̄_{−1}^{n} by (6.3),

μ̄D̄_{n} ; v̄(Z̄′) < \frac{1}{2}

for any v̄ ∈ [v̄^{0}]C̄_{−1}^{n}, and by integrating over [v̄^{0}]C̄_{−1}^{n} we get that

μ̄_{P} (Z̄′) < 1 − μ̄_{P} ([v̄^{0}]C̄_{−1}^{n})/2.

This violates our choice of k in (6.5), and the resulting contradiction proves the proposition. □

Definition 6.3. Let Z_{1}, Z_{2} be standard Borel spaces and ν a probability measure on Z_{1}. A Borel map g: Z_{1} ⟶ Z_{2} is countable-to-one ν-a.e. if there are Borel sets Z′_{1} ⊂ Z_{1}, Z′_{2} ⊂ Z_{2} with ν(Z′_{1}) = g_{∗}ν(Z′_{2}) = 1 so that g^{−1}(z) ∩ Z′_{1} is countable for every z ∈ Z′_{2}.

It is an easy exercise to see that entropy is preserved under almost everywhere countable-to-one factor maps.

Theorem 6.4. Let β > 1 be a Salem number of degree m, say, f ∈ R_{1} its minimal polynomial, and let α = α_{R_{1}/(f)} be the ergodic and nonexpansive automorphism of X = X_{R_{1}/(f)} ≅ 𝕋^{m} defined in (2.6)–(2.7).

Suppose that ν is a weakly d-bounded σ̄-invariant probability measure on the two-sided beta-shift V_{β}, and that ξ_{b}^{∗}: V_{β} ⟶ X is the ν-a.e. equivariant Borel map defined in (4.20). Then ξ_{b}^{∗} is countable-to-one ν-a.e., and the αinvariant probability measure μ = (ξ_{b}^{∗})_{∗}ν on X is singular with respect to Haar measure and satisfies that h_{ν}(σ̄) = h_{μ}(α).

For the proof of Theorem 6.4 we need several lemmas. The hypotheses of these lemmas are those of the theorem.

We call two points v, v′ ∈ V_{β} equivalent (in symbols: v ∼ v′) if v − v′ ∈ f(σ̄)(ℓ^{∗}(ℤ, ℤ)) or, equivalently, if ξ^{∗}(v) − ξ^{∗}(v′) ∈ X^{(0)} (cf. (4.6)). Denote by

R = {(v, v′) : v ∼ v′} ⊂ V_{β} × V_{β}  (6.6)

the resulting equivalence relation, and write

R(v) = {v′ ∈ V_{β} : v ∼ v′}  (6.7)

for the equivalence class of every v ∈ V_{β}.

Lemma 6.5. The set R ⊂ V_{β} × V_{β} is Borel and σ̄ × σ̄-invariant.

Proof. For every r > 0, the sets

B_{r}^{∗} = \{v ∈ ℓ^{∗}(ℤ, ℤ) : sup_{n∈ℤ} \frac{|v_{n}|}{|n| + 1} ≤ r\}

and C_{r} = f(σ̄)(B_{r}^{∗}) ⊂ ℓ^{∗}(ℤ, ℤ) are compact, and the map p: C_{r} × V_{β} ⟶ ℓ^{∗}(ℤ, ℤ) × V_{β}, given by p(v′, v) = (v′ + v, v), is continuous. Hence C̃_{r} = p(C_{r} × V_{β}) ∩ (V_{β} × V_{β}) is compact and R = ⋃_{r>0} C̃_{r} is Borel. The σ̄ × σ̄- invariance of R is obvious. □

For every subset F ⊂ ℤ we write π_{F} : ℓ^{∞}(ℤ, ℤ) ⟶ ℤ^{F} for the projection onto the coordinates in F.

Lemma 6.6. Let Y ⊂ V_{β} be a shift-invariant Borel set with ν(Y) = 1 such that (4.19) holds for every v ∈ Y, and let

Y (M) = {y ∈ Y : ∥b(y)∥_{∞} ≤ M},
L(M) = {y ∈ ℓ^{∗}(ℤ, ℤ) : ∥y − ξ̄^{∗} ∘ f(σ̄)(y)∥_{∞} ≤ M}
R(M, w) = (w + f(σ̄)(L(M))) ∩ V_{β} ⊂ R(w)

for every M ≥ 1 and w ∈ V_{β} (cf. (4.6)). Then there exists a constant M_{1} > 0 such that

\begin{aligned}|π_{\{0,...,n\}}(R(K, w) & ∩ Y (M) ∩ σ^{−n}(Y (M)))| \\ & ≤ (2M_{1}β + 4M + 2K + 1)^{2m+2}, \\ |π_{\{−n,...,0\}}(R(K, w) & ∩ Y (M) ∩ σ̄^{n}(Y (M)))| \\ & ≤ (2M_{1}β + 4M + 2K + 1)^{2m+2}, \end{aligned}  (6.8)

for every K, M ≥ 1, w ∈ V_{β} and n ≥ 1.

Proof. By (4.4) there exists a constant M_{1} > 0 such that

max_{j=0,...,m} |ξ̄^{∗}(w)_{j}| ≤ M_{1} · ∥w∥_{∞}

for every w ∈ ℓ^{∞}(ℤ, ℤ). As (σ̄^{∗})^{n} ∘ ξ̄^{∗}(w) = ξ̄^{∗} ∘ σ̄^{n}(w) + b(σ̄^{n}w) − σ̄^{n}b(w) for every n ∈ ℤ ,

max_{j=0,...,m} |ξ̄^{∗}(w)_{n+j}| ≤ M_{1}β + 2M

for every M ≥ 1, n ∈ ℤ and w ∈ Y (M) ∩ σ̄^{−n}(Y (M)). We fix w ∈ V_{β} and obtain that, for every v ∈ R(K, w) ∩ Y (M) ∩ σ̄^{−n}(Y (M))

max_{j=0,...,m} |ξ̄^{∗}(v)_{j}| ≤ M_{1}β, max_{j=0,...,m} |ξ̄^{∗}(v)_{n+j}| ≤ M_{1}β + 2M,

and that there exists a unique y ∈ ℓ^{∗}(ℤ, ℤ) with v = w + f(σ̄^{∗})(y) and ∥y − ξ̄^{∗} ∘ f(σ̄^{∗})(y)∥_{∞} ≤ K.

If v′ is a second element in R(K, w) ∩ Y (M) ∩ σ̄^{−n}(Y (M)) with v′ = w + f(σ̄^{∗})(y′) for some y′ ∈ ℓ^{∗}(ℤ, ℤ), then ∥y′ − ξ̄^{∗} ∘ f(σ̄^{∗})(y′)∥_{∞} ≤ K, and hence

max_{j=0,...,m} |y_{j} −y′_{j}| ≤ 2M_{1}β+2K and max_{j=0,...,m} |y_{n+j} −y′_{n+j}| ≤ 2M_{1}β+4M +2K.

For every L > 0 we set

B(n, L) = {w ∈ ℓ^{∗}(ℤ, ℤ) : |w|_{j} ≤ L
for 0 ≤ j ≤ m and n ≤ j ≤ n + m},

If the first inequality in (6.8) does not hold for some w ∈ V_{β} and n > 0, then we can find elements y, z ∈ B(n, 2M_{1}β + 4M + 2K + 1) with the following properties:

(y_{0}, …, y_{m}) = (z_{0}, …, z_{m}),
(y_{n}, …, y_{n+m}) = (z_{n}, …, z_{n+m}),
(y_{m+1}, …, y_{n−1}) ≠ (z_{m+1}, …, z_{n−1}).

so that ȳ = w + f(σ̄^{∗})(y), z̄ = w + f(σ̄^{∗})(z) are both in V_{β}. Note that these two points satisfy ȳ_{0} = z̄_{0}, ȳ_{n} = z̄_{n} and (ȳ_{1}, …, ȳ_{n−1}) ≠ (z̄_{1}, …, z̄_{n−1}).

Suppose, without loss in generality, that

(ȳ_{1}, …, ȳ_{n−1}) ≺ (z̄_{1}, …, z̄_{n−1})

and hence

(ȳ_{1}, ȳ_{2}, …) ≺ (z̄_{1}, z̄_{2}, …)

in the lexicographic order. We set

y′_{j} = \begin{cases}z_{j} & if j ≤ 0, \\ y_{j} & if j > 0, \end{cases}

and put z′ = z. Then ȳ′ = w + f(σ̄)(y′) is of the form

ȳ′_{j} = \begin{cases}z̄_{j} & if j ≤ 0, \\ ȳ_{j} & if j > 0, \end{cases}

and ȳ′ ∈ V_{β} by Proposition 5.1 (1). Put z̄′ = w + f(σ̄^{∗})(z′) = z̄, remember that ȳ′_{j} = z̄′_{j} for j ≤ 0 and for j = n, and assume for the moment that

(ȳ′_{n+1}, ȳ′_{n+2}, …) ≺ (z̄′_{n+1}, z̄′_{n+2}, …)

in the lexicographic order (if this is not the case we have to interchange the roles of y′ and z′ below). Let

z′′_{j} = \begin{cases}z′_{j} & if j ≤ n, \\ y′_{j} & if j > n, \end{cases}

and set y′′ = y′ and ȳ′′ = ȳ′. The point z̄′′ = w + f(σ̄^{∗})(z′′) is of the form

z̄′′_{j} = \begin{cases}z̄′_{j} & if j ≤ n, \\ ȳ′_{j} & if j > n, \end{cases}

and lies in V_{β} by Proposition 5.1 (1). By construction, y′′_{j} = z′′_{j} for j ≤ m and j ≥ n, and hence v′′ = y′′ −z′′ ∈ ℓ^{1}(ℤ, ℤ). Since z̄′′ and ȳ′′ = z̄′′ +f(σ̄)v′′ lie in V_{β} we obtain a contradiction to Proposition 5.1 (2). This proves the first inequality in (6.8), and the proof of the second one is analogous. □

Lemma 6.7. For ν-a.e. w ∈ Y, R(w) ∩ Y is countable, and the map ξ_{b}^{∗}: V_{β} ⟶ X is countable-to-one ν-a.e.

Proof. In the notation of Lemma 6.6 we set Y′(1) = Y (1) and Y′(M) = Y (M) ∖ Y (M − 1) for every M ≥ 2, and we define a map q: Y ⟶ ℝ by setting q(w) = 2^{−M} if w ∈ Y′(M), M ≥ 1. We fix an everywhere positive, Borel measurable and σ̄-invariant version p = E_{ν}(q|Sσ̄ ; Y) of the conditional expectation of q, given the sigma-algebra Sσ̄ ; Y of σ̄-invariant Borel subsets of Y. After decreasing Y by a σ̄-invariant ν-null set, if necessary, we also assume that

lim_{n→∞} \frac{1}{n} · ∑_{j=1}^{n} q(σ̄^{n}w) = lim_{n→∞} \frac{1}{n} · ∑_{j=1}^{n} q(σ̄^{−n}w) = p(w)

for every w ∈ Y.

We claim that there exists a C > 0 such that

∑_{v∈R(K,w)∩Y} q(v)p(v)^{2} = sup_{F ⊂R(K,w)∩Y F is finite} ∑_{v∈F} q(v)p(v)^{2} ≤ C  (6.9)

for every w ∈ Y and K ≥ 1.

Indeed, if F ⊂ R(K, w)∩Y is finite, then there exists an N ≥ 1 such that v_{n} = v′_{n} = w_{n} whenever v, v′ ∈ F and |n| ≥ N. It follows that

∑_{v∈F} q(v)p(v)^{2} = lim_{n→∞} ∑_{v∈F} q(v) · \frac{1}{n^{2}} · (∑_{j=1}^{n} q(σ̄^{−j}v)) · (∑_{j′=1}^{n} q(σ̄^{j′}v))

= lim_{n→∞} \frac{1}{n^{2}} · ∑_{j=1}^{n} ∑_{j′=1}^{n} ∑_{M≥1} ∑_{M′≥1} ∑_{M′′≥1} 2^{−M−M′−M′′}

· |π_{{−j,...,j′}}(F ∩ σ̄^{j}(Y′(M)) ∩ Y′(M′′) ∩ σ̄^{−j′}(Y′(M′)))|

= lim_{n→∞} \frac{1}{n^{2}} · ∑_{j=N+1}^{N+n} ∑_{j′=N+1}^{N+n} ∑_{M≥1} ∑_{M′≥1} ∑_{M′′≥1} 2^{−M−M′−M′′}

· |π_{{−j,...,j′}}(F ∩ σ̄^{j}(Y′(M)) ∩ Y′(M′′) ∩ σ̄^{−j′}(Y′(M′)))|

≤ lim_{n→∞} \frac{1}{n^{2}} · ∑_{j=N+1}^{N+n} ∑_{j′=N+1}^{N+n} ∑_{M≥1} ∑_{M′≥1} ∑_{M′′≥1} 2^{−M−M′−M′′}

· |π_{{−j,...,0}}(R(K, w) ∩ σ̄^{j}(Y′(M)) ∩ Y′(M′′))|

· |π_{{0,...,j′}}(R(K, w) ∩ σ̄^{−j′}(Y′(M′)) ∩ Y′(M′′))|

≤ ∑_{M≥1} ∑_{M′≥1} ∑_{M′′≥1} 2^{−M−M′−M′′}

· [(2M_{1}β + 2K + 4 max {M, M′′} + 1)^{2m+2}]

· [(2M_{1}β + 2K + 4 max {M′, M′′} + 1)^{2m+2}] = C < ∞

by (6.8), which proves (6.9).

Since the maps p, q: Y ⟶ ℝ are everywhere positive, (6.9) implies that the sets R(K, w) ∩ Y and R(w) ∩ Y = ⋃_{K≥1} R(K, w) ∩ Y are countable for every w ∈ Y, and that the equivariant Borel map ξ_{b}^{∗}: Y ⟶ X in (4.20) is therefore countable-to-one. □

Proof of Theorem 6.4. By Lemma 6.7 there exists a shift-invariant Borel set Y ⊂ V_{β} with ν(Y) = 1 such that the Borel map ξ_{b}^{∗}: Y ⟶ X in (4.20) is countable-to-one. Since countable-to-one factor maps do not decrease entropy, h_{ν}(σ̄) = h_{μ}(α). Furthermore, the Borel set Z = ξ_{b}^{∗}(Y) ⊂ X is αinvariant with μ(Z) = 1 and intersects each coset of X^{(0)} in a countable set. Hence λ_{X}(Z) = 0, which proves that λ_{X} and μ are mutually singular. □

7. Some examples of invariant measures in the Salem case

Theorem 7.1. Let β > 1 be a Salem number, and let V_{β} ⊂ ℓ^{∞}(ℤ, ℤ) be the corresponding two-sided beta-shift space. For every ε > 0 there exists a dbounded shift-invariant probability measure ν on V_{β} with h_{ν}(σ_{β}) > log β − ε, where σ_{β} = σ̄_{V_{β}} is the beta-shift.

Proof. As in the proof of Theorem 4.13 we choose an enumeration Ω_{f}^{(0)} = {ω_{1}, …, ω_{m_{0}}} of Ω_{f}^{(0)}, write W_{f}^{(0)} = ℂ ⊗_{ℝ} W_{f}^{(0)} for the complexification of W_{f}^{(0)}, and use the basis {w(ω_{i}) : i = 1, …, m_{0}} in (2.15) to identify W_{f}^{(0)} with ℂ^{m_{0}}. Let

Γ_{β} = \overline{\{(ω_{1}^{n}, …, ω_{m_{0}}^{n}) : n ∈ ℤ\}}.

Put V = V_{β} × Γ_{β} (cf. (4.24)) and define a map S_{β}: V ⟶ V by S_{β}(v, γ) = (σ̄v, M_{ω}γ) for every v ∈ V_{β} and γ = (γ_{1}, …, γ_{m_{0}}) ∈ Γ_{β} ⊂ ℂ^{m_{0}}, where M_{ω} is defined in (4.25) and (4.26).

Let λ be the Haar (= normalized Lebesgue) measure on Γ_{β}. Since the unique shift-invariant measure of maximal entropy μ_{β} on V_{β} is mixing (cf. [9]), the measure μ_{β} ×λ on V is ergodic under T_{β}. The map F_{β}: V ⟶ ℂ^{m_{0}}, given by F_{β}(v, γ)_{i} = γ_{i}v_{0} for every v = (v_{n}) ∈ V_{β}, γ = (γ_{1}, …, γ_{m_{0}}) ∈ Γ_{β} and i = 1, …, m_{0}, satisfies that ∫ F_{β} d(μ_{β} ×λ) = 0, and the ergodic theorem implies that

lim_{K→∞}∥\frac{1}{K} ∑_{k=0}^{K−1} F_{β}(T_{β}^{k}(v, γ))∥_{∞} = lim_{K→∞}|\frac{1}{K} ∑_{k=0}^{K−1} ω_{i}^{k}v_{k}| = 0 (μ_{β} × λ)-a.e.

for i = 1, …, m_{0}, where ∥ · ∥_{∞} is the maximum norm on ℂ^{m_{0}}. Hence

lim_{K→∞} ∥d(K, v)/K∥_{∞} = 0 μ_{β}-a.e.

We fix a positive integer J and choose K > 0 sufficiently large so that μ_{β}(B_{K,J}) > 1 − 1/J, where

B_{K,J} = {v ∈ V_{β} : ∥d(k, v)∥_{∞} ≤ K for k = 0, …, KJ}.

Note that the set B_{K,J} is a union of cylinder sets which depend only on the coordinates 0, …, KJ − 1.

Since M_{ω} acts minimally on Γ_{β}, there exists an L > 0 with the following property: for every pair v, w ∈ ℂ^{m_{0}} of vectors with ∥v∥_{∞} ≤ 1 and ∥w∥_{∞} ≤ 1 there exists an l ∈ {0, …, L − 1} with ∥M_{ω}^{l}v + w∥_{∞} ≤ 1.

Let v ∈ V_{β}. By inserting zero coordinates in an appropriate manner we modify v to a point v^{∗} ∈ V_{β} with v_{n}^{∗} = v_{n} for n < 0 such that ∥d(m, v^{∗})∥_{∞} < 4K for every m ≥ 0.

In order to describe this modification we proceed by induction and assume that v = v^{(0)} ∈ V_{β}. If v ∈ B_{K,J} we put v^{(1)} = v and v^{(1)} = d(JK, v^{(0)}).

If v ∉ B_{J,K} we use our choice of L to find an integer l_{1} ∈ {0, …, L − 1} such that the point v(1), given by

v(1)_{n} = \begin{cases}v_{n} & if n ≤ K − 1, \\ 0 & if n = K, …, K + l_{1} − 1, \\ v_{n−l_{1}} & if n ≥ K + l_{1}, \end{cases}

which satisfies that ∥d(2K+l_{1}, v(1))∥_{∞} ≤ K. Next we choose l_{2} ∈ {0, …, L− 1} such that the point v(2) with

v(2)_{n} = \begin{cases}v(1)_{n} & if n ≤ 2K + l_{1} − 1, \\ 0 & if n = 2K + l_{1}, …, 2K + l_{1} + l_{2} − 1, \\ v_{n−l_{1}−l_{2}} & if n ≥ 2K + l_{1} + l_{2}, \end{cases}

satisfies that ∥d(3K + l_{1} + l_{2}, v(2))∥_{∞} ≤ K. By continuing in this manner we eventually obtain integers l_{1}, …, l_{J−1} ∈ {0, …, L−1} and a point v^{(1)} = v(J − 1) ∈ V_{β} (cf. Proposition 5.1 (1)) with

v_{n}^{(1)} = {v_{n}  if n ≤ K − 1, 0  if n = K, …, K + l_{1} − 1, ⋮ 0  if n = (J − 1)K + l_{1} + ⋯ + l_{J−2}, …, (J − 1)K + l_{1} + ⋯ + l_{J−1} − 1, v_{n−l_{1}−···−l_{J−1}} if n ≥ (J − 1)K + l_{1} + ⋯ + l_{J−1},

satisfies that ∥d(JK+l_{1}+⋯+l_{J−1}, v^{(1)})∥_{∞} ≤ K. We set l^{(1)} = l_{1}+⋯+l_{J−1}, v^{(1)} = d(JK + l^{(1)}, v^{(1)}) and note that σ̄^{JK+l^{(1)}}v^{(1)} ∈ B_{K,J} if and only if σ̄^{JK}v ∈ B_{K,J}, and that

∥d(j, v^{(1)})∥_{∞} ≤ 2K

for j = 0, …, JK + l^{(1)}.

We repeat this process with v replaced by w = σ̄^{JK+l^{(1)}}v^{(1)} and obtain an integer l^{(2)} ∈ {0, …, J(L − 1)} and a point w′ ∈ V_{β} with the following properties.

(i) ∥d(JK + l^{(2)}, w′)∥_{∞} ≤ K and ∥d(j, w′)∥_{∞} ≤ 2K for j = 0, …, JK + l^{(2)},

(ii) w′_{n} = w_{n} for n < 0 and w′_{n+l^{(2)}} = w_{n} for n ≥ JK, (iii) w′ is obtained from w by inserting l^{(2)} ≤ (J − 1)(L − 1) zeros among

the coordinates w_{0}, …, w_{JK−1}, and l^{(2)} = 0 if and only if σ̄^{JK}(v) ∈ B_{K,J}.

Next we set w = d(JK + l^{(2)}, w′), choose a j^{(1)} ∈ {0, …, L − 1} with ∥M_{ω}^{j^{(1)}}v^{(1)} + w∥_{∞} ≤ K, and define v^{(2)} ∈ V_{β} by

v_{n}^{(2)} = \begin{cases}v_{n}^{(1)} & if n < JK + l^{(1)}, \\ 0 & if n = JK + l^{(1)}, …, JK + l^{(1)} + j^{(1)} − 1 \\ w′_{n−JK−l^{(1)}−j^{(1)}} & if n ≥ JK + l^{(1)} + j^{(1)}. \end{cases}

The point v^{(2)} lies in V_{β} by Proposition 5.1 (1) and has the following properties.

(i’) ∥d(2JK + l^{(1)} + j^{(1)} + l^{(2)}, v^{(2)})∥_{∞} ≤ K and ∥d(j, v^{(2)})∥_{∞} ≤ 2K for

j = 0, …, 2JK + l^{(1)} + j^{(1)} + l^{(2)}, (ii’) v_{n}^{(2)} = v_{n} for n < 0 and v_{n+l^{(1)}+j^{(1)}+l^{(2)}}^{(2)} = v_{n} for n ≥ 2JK, (iii’) v^{(2)} is obtained from v^{(1)} by inserting l^{(2)} ≤ J(L − 1) zeros among

the coordinates v_{JK+l^{(1)}}, …, v_{2JK−1+l^{(1)}}, and l^{(2)} = 0 if and only if σ̄^{JK}v ∈ B_{K,J} (or, equivalently, if and only if σ̄^{JK+l^{(1)}}v^{(1)} ∈ B_{K,J}).

By repeating this process we obtain sequences (v^{(m)}, m ≥ 1) in V_{β} and (l^{(m)}, m ≥ 1) and (j^{(m)}, m ≥ 1) of positive integers satisfying the following conditions for every m ≥ 1.

(1) 0 ≤ l^{(m)} ≤ J(L − 1) and 0 ≤ j^{(m)} ≤ L − 1,

(2) If L^{(m)} = ∑_{i=1}^{m} l^{(i)}, J^{(m)} = ∑_{i=1}^{m−1} j^{(i)} and L^{(i)} = J^{(i)} = 0 for i ≤ 0,

then

L^{(m)} ≤ J(L − 1) · ∑_{i=0}^{m−1} 1_{V_{β}∖B_{K,J}}(σ̄^{iJK}v),

where 1_{S} denotes the indicator function os a set S ⊂ V_{β}, and

∥d(mJK + L^{(m)} + J^{(m−1)}, v^{(m)})∥_{∞} ≤ K,

(3) ∥d(j, v^{(m)})∥_{∞} ≤ 2K for j = 0, …, mJK + L^{(m)} + J^{(m−1)},

(4) v_{n}^{(m)} = \begin{cases}v_{n}^{(m−1)} & if n < (m − 1)JK + L^{(m−1)} + J^{(m−2)}, \\ v_{n−L^{(m)}−J^{(m−1)}} & if n ≥ mJK + L^{(m)} + J^{(m)}. \end{cases}

From the conditions (3)–(4) above we see that the sequence (v^{(m)}, m ≥ 1) converges to an element v^{∗} ∈ V_{β} with

∥d(j, σ̄^{j′}v^{∗})∥_{∞} ≤ 4K

for every j, j′ ≥ 0.

If m is sufficiently large, then the set

C_{m} = {v ∈ V_{β} : ∑_{i=0}^{m−1} 1_{V_{β}∖B_{K,J}}(σ̄^{iJK}v) ≤ 2m/J}

has μ_{β}-measure > 1 − 1/J.

So far we have kept J and K fixed, but now we begin to vary them. If

P (m) = π_{{0,...,mJK−1}}(C_{m})

is the projection of the set C_{m} onto the coordinates 0, …, mJK − 1, then the Shannon-McMillan-Breiman theorem applied to μ_{β} (cf. [15]) implies that the cardinality of P (m) satisfies that

lim_{J→∞} \frac{1}{mJK} log |P (m)| = log β,

since h_{μ_{β}}(σ̄_{β}) = log β (note that K depends on J and tends to infinity as J → ∞). We fix ε > 0 and choose J (and hence K) sufficiently large so that P (m) > (β − ε)^{mJK} for all sufficiently large m. For every v ∈ C_{m}, the number of zero coordinates inserted among the coordinates v_{0}, …, v_{mJK−1} in the transition from v to v^{∗} is less than m · (L − 1) + 2m · (L − 1) · K/J, so that

|π_{{0,...,mJK−1}}({v^{∗} : v ∈ C_{m}})| ≥ |π_{{0,...,m·(JK−L−2·(L−1)·K/J)}}(C_{m})|.

This shows that, for sufficiently large K, the topological entropy of the closed, σ̄-invariant subset

{v ∈ V_{β} : ∥d(j, σ̄^{j′}v^{∗})∥_{∞} ≤ 4K for every j ≥ 0 and j′ ∈ ℤ}

is arbitrarily close to log β, and the variational principle (cf. [30]) guarantees that we can find σ̄-invariant and ergodic probability measures ν on V_{β} with entropy arbitrarily close to log β. □

Acknowledgment

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 Departments of the University of Washington, Stanford University, the Newton Institute, Cambridge and the ETH Zürich for hospitality during parts of this work. E.L. is a Clay Research Fellow and is grateful for this generous support from the Clay Mathematics Institute. E.L. would also like to thank Rick Kenyon for an interesting and helpful discussion on these and related topics.

References

[1] R.L. Adler and B. Weiss, Similarity of automorphisms of the torus, Mem. Amer. Math. Soc. 98 (1970).

[2] A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris Sér. I Math. 285 (1977), 419–421.

[3] F. Blanchard, Beta-expansion and symbolic dynamics, Theoret. Comput. Sci. 65 (1989), 131–141.

[4] R. Bowen, Markov partitions for axiom A diffeomorphisms, Amer. J. Math. 92 (1970), 725–747.

[5] D. Boyd, Salem numbers of degree four have periodic beta-expansions, in: Théorie des nombres (Conference Proceedings, Banff 1988), de Gruyter, Berlin-New York, 1990, 57–64.

[6] D.W. Boyd, On the beta expansion for Salem numbers of degree 6, Math. Comp. 65 (1996), 861–875.

[7] D.W. Boyd, The beta expansion for Salem numbers, Canad. Math. Soc. Conf. Proc. 20 (1997), 118–130.

[8] M. Einsiedler and K. Schmidt, Irreducibility, homoclinic points and adjoint actions of algebraic Z^{d}-actions of rank one, in: Nonlinear Phenomena and Complex Systems, ed. A. Maass, S. Martinez and J. San Martin, Kluwer Academic Publishers, Dordrecht, 2002, 95–124.

[9] F. Hofbauer, β-shifts have unique maximal measure, Monatsh. Math. 85 (1978), 189– 198.

[10] R. Kenyon and A. Vershik, Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic Theory Dynam. Systems 18 (1998), no. 2, 357–372.

[11] D. Lind and B. Marcus, Symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995.

[12] D. Lind and K. Schmidt, Homoclinic points of algebraic Z^{d}-actions, J. Amer. Math. Soc. 12 (1999), 953–980.

[13] E. Lindenstrauss and K. Schmidt, Invariant measures of nonexpansive group automorphisms, in preparation.

[14] W. Parry, On the β-expansions of real numbers, Acta Math. Hungar. 11 (1960), 401– 416.

[15] W. Parry, Topics in ergodic theory, Cambridge University Press, Cambridge, 1981.

[16] A. Rényi, Representations of real numbers and their ergodic properties, Acta Math. Hungar. 8 (1957), 477–493.

[17] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269–278.

[18] K. Schmidt, Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group actions, Ergod. Th. & Dynam. Sys. 1 (1981), 223–236.

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

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

[21] K. Schmidt, Algebraic coding of expansive group automorphisms and two-sided betashifts, Monatsh. Math. 129 (2000), 37–61.

[22] N. Sidorov, Bijective and general arithmetic codings for Pisot toral automorphisms, J. Dynam. Cont. Sys. 7 (20002001), 447–472.

[23] N. Sidorov, An arithmetic group associated with a Pisot unit, and its symbolicdynamical representation, Acta Arith. 101 (2002), 199–213.

[24] N. Sidorov and A. Vershik, Bijective arithmetic codings of the 2-torus, and binary quadratic forms, J. Dynam. Cont. Sys. 4 (1998), 365–400.

[25] Ya.G. Sinai, Markov partitions and Y -diffeomorphisms, Functional Anal. Appl. 2 (1986), 64–89.

[26] Y. Takahashi, Isomorphisms of β-automorphisms to Markov automorphisms, Osaka J. Math. 10 (1973), 175–184.

[27] A. Vershik, The fibadic expansion of real numbers and adic transformations, Preprint, Mittag-Leffler Institute, 1991/92.

[28] A. Vershik, Arithmetic isomorphism of hyperbolic toral automorphisms and sofic systems, Functional Anal. Appl. 26 (1992), 170–173.

[29] A.M. Vershik, Locally transversal symbolic dynamics, St. Petersburg Math.J. 6 (1995), 529–540.

[30] P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer Verlag, Berlin-Heidelberg-New York, 1982.

[31] B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math. 77 (1973), 462–

Elon Lindenstrauss: Department of Mathematics, Princeton University, Princeton, NJ 08540, USA

E-mail address: elonl@math.princeton.edu

Klaus Schmidt: Mathematics Institute, University of Vienna, Nordbergstraße 15, 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