PRODUCTS OF NON-STATIONARY RANDOM MATRICES

AND MULTIPERIODIC EQUATIONS OF SEVERAL SCALING FACTORS

AI-HUA FAN, BENOÎT SAUSSOL, AND JÖRG SCHMELING

Abstract. Let β > 1 be a real number and M : ℝ → GL(ℂ^{d}) be a uniformly almost periodic matrix-valued function. We study the asymptotic behavior of the product

P_{n}(x) = M(β^{n−1}x) ⋯ M(βx)M(x).

Under some condition we prove a theorem of Furstenberg-Kesten type for such products of non-stationary random matrices. Theorems of Kingman and Oseledec type are also proved. The obtained results are applied to multiplicative functions defined by commensurable scaling factors. We get a positive answer to a Strichartz conjecture on the asymptotic behavior of such multiperiodic functions. The case where β is a Pisot– Vijayaraghavan number is well studied.

1. Introduction

Kingman’s subadditive ergodic heorem was originally proved in 1968 [Ki1, Ki2]. A more recent proof was given by Katznelson and Weiss in 1982 [KW]. It is one of the most important results in ergodic theory. In this paper we consider the following set-up which resembles a dynamical system without invariant measure and try to get results similar to Kingman’s Theorem. Let β > 1 be a positive real number. Let {f_{n}} be a sequence of uniformly almost periodic functions (i.e. in the sense of Bohr, see Section 2.1) defined on the real line ℝ. Suppose the following subadditivity condition is fulfilled

f_{n+m}(x) ≤ f_{n}(x) + f_{m}(β^{n}x) for a.e. x and all n, m.

where a.e. refers to the Lebesgue measure. We would like to study the almost everywhere convergence of n^{−1}f_{n}(x). Kingman’s Theorem applies in the special case where β > 1 is an integer and the f_{n}’s are periodic. The typical case in our mind is

f_{n}(x) = log ∥M(β^{n−1}x) ⋯ M(βx)M(x)∥  (1.1)

Date: November 8, 2002. 2000 Mathematics Subject Classification. Primary:28A80,42A38. Key words and phrases. Kingman’s ergodic Theorem,Random matrix product, Multi-

periodic functions, PV–numbers.

Acknowledgement. This work was partially done during the first author’s visit to Lund University, Sweden and the last author’s visit to WIPM of the academy of sciences of China and Wuhan University, China.

where M : ℝ → GL(ℂ^{d}) is a matrix-valued uniformly almost periodic function. We will prove that the limit lim_{n→∞} n^{−1}f_{n}(x) exists almost everywhere (a.e. for short) with respect to the Lebesgue measure under the condition that the n^{−1}f_{n}(x) have joint periods (see Theorem 2.5). As a consequence, an Oseledec type theorem is proved for the matrix products involved in (1.1) (see Theorem 2.9). It is proved that the condition on the existence of joint periods is satisfied when β is a PV-number (see Section 3).

Our consideration is partially motivated by the study of multiperiodic functions, already investigated by Strichartz et al. [JRS], Fan and Lau [FL], and Fan [F]. By a Multiperiodic function of one real variable we mean any function F : ℝ → ℝ which is a solution of a functional equation of the following form

F (ξ) = f_{1} (\frac{ξ}{ρ_{1}} ) F (\frac{ξ}{ρ_{1}} ) + ⋯ + f_{d} (\frac{ξ}{ρ_{d}} ) F (\frac{ξ}{ρ_{d}} )

where d ≥ 1 is an integer; ρ_{1} > 1, …, ρ_{d} > 1 are d real numbers, called scaling factors; f_{1}, …, f_{d} are d complex valued functions defined on the real line, called determining functions. The equation will be called a multiperiodic equation.

We will assume that the determining functions f_{j} are periodic or almost periodic in the sense of Bohr, as is the case in most applications. We will also assume that the scaling factors ρ_{j} are commensurable in the sense that ρ_{j} are powers of some real number β > 1. Without lost of generality, we assume that ρ_{j} = β^{j} for 1 ≤ j ≤ d. Then the multiperiodic equation becomes

F (ξ) = f_{1} (\frac{ξ}{β} ) F (\frac{ξ}{β} ) + ⋯ + f_{d} (\frac{ξ}{β^{d}} ) F (\frac{ξ}{β^{d}} )  (1.2)

As far as we know, there is few work done for the non–commensurable case which is much more difficult.

In the literature, the case where d = 1 and β = 2 (or an arbitrary integer) has been studied, especially in the theory of wavelets [D]. In fact, the scaling function φ of a wavelet satisfies a scaling equation

φ(x) = ∑ a_{n}φ(2x − n).

The Fourier transform of φ satisfies a multiperiodic equation of the form (1.2) with only one scaling factor β = 2 and only one determining function f_{1}(x) = f(x) = \frac{1}{2} ∑_{n} a_{n}e^{inx}.

The scaling functions in wavelets constitute a class of functions sharing a kind of similarity. More generally, multiperiodic functions arise as Fourier transforms of self-similar objects such as Bernoulli convolution measures (d = 1, β > 1 being a real number and f being a trigonometric polynomial), inhomogeneous Cantor measures (d may be greater than 1) or more general self-similar measures produced by iterated function systems (see [S]).

In the case of one scaling factor (i.e. d = 1, then we write f_{1}(x) = f(x)), the existence of the solution of the multiperiodic equation (1.2) is simple and is assured by the consistency condition f(0) = 1 and a regularity condition, say f is Lipschitz continuous. Actually the solution can be written as an infinite product

F (x) = ∏_{n=1}^{∞} f (\frac{x}{β^{n}} ) .

For the existence of the general equation (1.2), we have

Theorem A. Let d ≥ 1. Suppose that the determining functions f_{1}, ⋯, f_{d} are Lipschitz continuous, and satisfy the consistency condition

f_{1}(0) + ⋯ + f_{d}(0) = 1.

Suppose furthermore that f_{1}(0), …, f_{d}(0) ∈ [0, +∞). Then equation (1.2) admits a unique continuous solution F such that F (0) = 1.

The proof of this theorem is postponed to Section 4.2.

Our study of equation (1.2) is converted to that of vector valued equations of the form

G(x) = M (\frac{x}{β} ) G (\frac{x}{β} )  (1.3)

where M : ℝ → M_{d×d}(ℂ) is a matrix valued determining function and G : ℝ → ℝ^{d} is a vector valued unknown function. Matrix products will be involved in the study of equation (1.3), which produces some difficulties. However, equation (1.3) is a simple recursive relation because it contains only one scaling factor. Equation (1.2) is equivalent to equation (1.3) with M(x) and G(x) equal respectively to

( \begin{matrix}f_{1}(x) & f_{2}(\frac{x}{β}) & ⋯ & f_{d−1}(\frac{x}{β^{d−2}} ) & f_{d}(\frac{x}{β^{d−1}} ) \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & ... & ... & ⋮ & ⋮ \\ ⋮ & ... & & 0 & 0 \\ 0 & ⋯ & 0 & 1 & 0 \end{matrix} ) and ( F (x) ; F (\frac{x}{β}) ; ⋮ ; F (\frac{x}{β^{d−1}} ) ) (1.4)

We would like to know the asymptotic behavior at infinity of the solution G. This is a natural question because G often represents Fourier transform of a function (a measure or a distribution) and the asymptotic behavior at infinity describes quantitatively the regularity of the solution. Unfortunately, there is no closed form formula for G in general and the behavior of G is rather complicated, as is shown by the Fourier transform of the Cantor measure (d = 1, β = 3 and f(ξ) = cos ξ).

Following [JRS], we will study the pointwise asymptotic behaviors of

h_{n}(x) := \frac{1}{n} log |F (β^{n}x)|

as n → ∞. We will prove that, under some conditions, the limit lim_{n→∞} h_{n}(x) exists and is equal to a constant almost everywhere with respect to Lebesgue measure. This answers partially a questions in [JRS]. More precisely, we have the following results, whose proofs are postponed to Section 4.3.

Theorem B. Suppose that the conditions in Theorem A are satisfied. Furthermore, suppose that the determining functions f_{1}, ⋯, f_{d} are either identically zero or strictly positive and 1-periodic, and that β > 1 is a Pisot number. Let F be the solution of the equation (1.2). Then there is a constant L such that

lim_{n→∞} \frac{1}{n} log F (β^{n}x) = L a.e.

The constant L in the theorem is the leading Liapunov exponent of the matrix M(x) above, defined by

L(M) = inf_{n≥1} \frac{1}{n}𝕄 log ∥M(β^{n−1}x)M(β^{n−2}x) ⋯ M(βx)M(x)∥.

where 𝕄f denotes the Bohr mean of an almost periodic function f (see Section 2.1 below).

Remark 1.1. Setting β^{n}x = y in the above theorem gives a growth rate F (y) ∼ y^{L/ log β}.

Theorem C. Suppose that the conditions in Theorem A are satisfied. Furthermore, suppose that the determining functions f_{1}, ⋯, f_{d} are 1-periodic, Lipschitz, and that β > 1 is a Pisot number with maximal conjugate of modulus ρ. Let F be the solution of the equation (1.2). If

sup_{x} \frac{(1 + |f_{1}(x)| + ⋯ + |f_{d−1}(x/β^{d−2})|)(|f_{1}(x)| + ⋯ + |f_{d}(x/β^{d−1})|)}{|f_{d}(x/β^{d−1})|} < ρ^{−1}

then there is a constant λ ∈ ℝ such that

lim_{n→∞} \frac{1}{n} log ∑_{j=0}^{d−1} |F (β^{n−j}x)| = λ a.e.

In this case we do not know if the constant λ equals the leading Lyapunov exponent of the matrix.

Theorem A will be proved in Section 4.2 as a special case of a more general result (Theorem 4.1), Theorem B and Theorem C in Section 4.3. Both Theorem B and Theorem C are consequences of our Kingman’s Theorem (Theorem 2.5) and Oseledec’s Theorem (Theorem 2.9) which are discussed in Section 2. In Section 3, we prove that the joint period condition required in both Kingman’s Theorem and Oseledec’ Theorem is satisfied when β > 1 is a Pisot number.

2. Kingman’s Theorem and Oseledec’ Theorem

2.1. Total Bohr ergodicity and joint ϵ-period.

Let us first recall the definition of uniformly almost periodic functions and some of their properties (see [Bo]). Next we will introduce the notions of total Bohr ergodicity and a joint ϵ-period.

Let f be a real or complex valued function defined on the real line. A number τ is called a translation number of f belonging to ϵ ≥ 0 (or an ϵ-period) if

sup_{x∈ℝ}|f(x + τ) − f(x)| ≤ ϵ.

We say that f is a uniformly almost periodic (u.a.p.) function if it is continuous and if for any ϵ > 0 the set of its translation numbers belonging to ϵ is relatively dense (i.e. there exists a number ℓ > 0 such that any interval of length ℓ contains at least one such translation number). H. Bohr proved that the space of all u.a.p. functions is a closed sub-algebra of the Banach algebra C_{b}(ℝ) of bounded continuous functions equipped with the uniform norm and that it is the closure of the space of all (generalized) trigonometric polynomials of the form

∑_{finite} A_{n}e^{iΛ_{n}x} (A_{n} ∈ ℂ, Λ_{n} ∈ ℝ).

For any u.a.p. function f, as is proved by Bohr, the following limit exists

𝕄f = lim_{T →∞} \frac{1}{T} ∫_{0}^{T} f(x)dx.

It is called the Bohr mean of f. For any locally integrable not necessarily u.a.p. f, we define 𝕄f as the limsup instead of the limit.

Definition 2.1. A sequence of real numbers (u_{n})_{n≥0} is said to be totally Bohr ergodic if for any arithmetic subsequence (u_{am+b})_{m≥0} (a ≥ 1, b ≥ 0 being fixed) and for any real p > 0, the sequence (u_{am+b}x)_{m≥0} is uniformly distributed (modulo p) for almost every x ∈ ℝ with respect to the Lebesgue measure.

The following is the main property of totally Bohr ergodic sequences that we will use.

Lemma 2.2. Suppose that (u_{n})_{n≥0} is a totally Bohr ergodic sequence. Then for any u.a.p. function f and any integers a ≥ 1, b ≥ 0, we have

lim_{N→∞} \frac{1}{N} ∑_{n=0}^{N} f(u_{an+b}x) = 𝕄f a.e.

Proof. It is a consequence of the fact that u_{an+b}x is uniformly distributed (modulo p) for almost all x, for any real p > 0, the fact that f can be uniformly approximated by trigonometric polynomials and the Weyl criterion. □

Remark 2.3. Suppose that (u_{n})_{n≥0} is such that inf_{n≠m} |u_{n} − u_{m}| > 0, then the sequence (u_{n}x) is uniformly distributed for almost every point x [KL]. Consequently, the sequence (u_{n}) is totally Bohr ergodic. A more special case is u_{n} = β^{n} with β > 1. This is the most interesting case for us. On the other hand, no bounded sequence can be totally Bohr ergodic.

Definition 2.4. Let (F_{n})_{n≥0} be a sequence of u.a.p. functions. Let ϵ > 0 and N ∈ ℕ. A real number τ is called a joint ϵ-translation number for (F_{n})_{n≥N} if

sup_{n≥N} sup_{x∈ℝ}|F_{n}(x + τ) − F_{n}(x)| ≤ ϵ.

If for any ϵ > 0 there exists N(ϵ) such that such joint ϵ-translation numbers for (F_{n})_{n≥N(ϵ)} are relatively dense, we say that (F_{n})_{n≥0} has joint periods.

2.2. Kingman’s Theorem.

Following ideas of Katznelson and Weiss [KW] we prove the following version of Kingman’s Theorem. The difficulty in our case is that we have to deal with an infinite measure space. We are also dealing with non–stationary sequences.

Theorem 2.5. Let (u_{n})_{n≥0} be a totally Bohr ergodic sequence of real numbers and (f_{n})_{n≥0} be a sequence of uniformly almost periodic functions. Suppose

(i) The sequence (n^{−1}f_{n}) has joint periods.

(ii) The following subadditivity is fulfilled

f_{n+m}(x) ≤ f_{n}(x) + f_{m}(u_{n}x) for a.e. x and all n, m.

(iii) For any n ≥ 1

sup_{m} (f_{m}(u_{n}x) − f_{m}(x)) < ∞ for a.e. x.  (2.1)

Then the following limit exists and is a constant

lim_{n→∞} \frac{1}{n}f_{n}(x) = inf_{n} \frac{1}{n}𝕄f_{n} for a.e. x.

Proof. The proof is a modification of Katznelson-Weiss’ proof [KW]. Without loss of generality we assume that u_{0} = 1.

Let γ = inf_{n} \frac{1}{n}𝕄f_{n}. Let us put

f^{−}(x) = lim inf_{n→∞} \frac{1}{n}f_{n}(x), f^{+}(x) = lim sup_{n→∞} \frac{1}{n}f_{n}(x).

We remark that the subadditivity implies that f^{±}(x) ≤ f^{±}(u_{n}x) for all n ∈ ℕ and a.e. x ∈ ℝ. In a finite measure space this would imply the invariance a.e. In our case it is the boundedness (2.1) which implies that f^{±}(u_{n}x) ≤ f^{±}(x) a.e., what makes the functions f^{±} invariant in the sense that f^{±}(x) = f^{±}(u_{n}x) a.e. (∀n).

The first part of the proof, i.e. f^{+} ≤ γ a.e., is simple. We just exploit the fact that any (infinite) arithmetical subsequence of (u_{n}x) is Bohr-uniform distributed. This provides us with a kind of ergodic theorem. In fact, fix an integer N. For any integer n write n = mN + r with 0 ≤ r < N. We have

f_{n}(x) ≤ ∑_{k=0}^{m−1} f_{N}(u_{kN}x) + f_{r}(u_{mN}x).

The N functions f_{r} (r = 0, 1, ⋯, N − 1) being bounded, by Lemma 2.2, this readily implies by the Bohr-uniform distribution of (u_{kN}x)_{k≥0} that

f^{+}(x) ≤ lim_{m→∞} \frac{1}{mN + r} ∑_{k=0}^{m−1} f_{N}(u_{kN}(x)) = \frac{1}{N} 𝕄f_{N}

for a.e. x. Hence f^{+}(x) ≤ γ for a.e. x.

Next, we want to prove that f^{−}(x) ≥ γ for a.e. x. For this we assume that γ > −∞, otherwise it is trivially true. Adding to each f_{n} the constant value −n∥f_{1}∥ creates a new subadditive sequence f̃_{n} := f_{n} − n∥f_{1}∥ with f̃_{n} ≤ 0 (n ≥ 1), f^{−} = f̃^{−} + ∥f_{1}∥ and γ_{f} = γf̃ + ∥f_{1}∥. So, we may assume that f_{n} ≤ 0 (n ≥ 1). We can furthermore set f_{1} = 0 (observe that this will not affect the subbadditivity condition since f_{n} ≤ 0). Then for any Δ > 0 we truncate the function f_{n} in the way

f_{n,Δ} = max(f_{n}, −nΔ).

Note that the sequence f_{n,Δ} fulfills the assumptions of the theorem. Note that in this case γ_{Δ} ≥ −Δ and also f_{Δ}^{−}(x) ≥ −Δ for all x. It is clear that

f_{Δ}^{±}(x) = max(f^{±}(x), −Δ).

Assume that we proved the theorem for the sequence f_{n,Δ} for any Δ. Then we claim that f_{Δ}^{−}(x) ↘ f^{−}(x) for a.e. x as Δ goes to ∞. In fact, if f_{Δ}^{−}(x) > −Δ for some Δ then f_{Δ}^{−}(x) = f^{−}(x) and γ_{Δ} = γ_{f} > −∞. On the other hand if f_{Δ}^{−}(x) = −Δ for all Δ then obviously f^{−}(x) = −∞ and γ_{Δ} = γ_{f} = −∞. This proves the theorem for the sequence f_{n}.

now on we assume that f_{1} = 0, f_{n} ≤ 0 and f_{n} is truncated and we skip the subscript Δ. Let ϵ > 0. By the hypothesis (i) on the joint periods, there is an integer N(ϵ) such that the joint ϵ-translation numbers are relatively dense. For these numbers τ we have

|\frac{f_{n}(x + τ)}{n} − \frac{f_{n}(x)}{n}| ≤ ϵ  (∀x ∈ ℝ, ∀n ≥ N(ϵ))  (2.2)

Notice that there is no loss of generality to suppose that N(ϵ) increases as ϵ decreases to 0. We define

n_{ϵ}(x) = min \{n ≥ N(ϵ): \frac{1}{n}f_{n}(x) ≤ f^{−}(x) + ϵ\}

Let A_{K}^{ϵ} = {x: n_{ϵ}(x) > K}. Notice that if ϵ′ < ϵ′′, we have N(ϵ′) ≥ N(ϵ′′) and n_{ϵ′}(x) ≥ n_{ϵ′′}(x), so that A_{K}^{ϵ′′} ⊂ A_{K}^{ϵ′} for K > N(ϵ′).

We claim that Lemma 2.6. For any ϵ > 0, we have

lim_{K→∞} 𝕄̂(A_{K}^{ϵ}) = 0

where 𝕄̂A = 𝕄1_{A} denotes the Bohr mean of the characteristic function of the set A (defined if necessary with the limsup).

In order to prove this Lemma 2.6 we need the following lemma which says that A_{K}^{ϵ} is to some extent periodic. Lemma 2.7. For any joint ϵ-translation number τ of (f_{n}/n)_{n≥N(ϵ)}, we have A_{K}^{2ϵ} + τ ⊂ A_{K}^{ϵ}.

Let us first prove Lemma 2.7. Suppose x ∈ A_{K}^{2ϵ} + τ, i.e. x − τ ∈ A_{K}^{2ϵ}, then

\frac{f_{n}(x − τ)}{n} > f^{−}(x − τ) + 2ϵ (N(2ϵ) ≤ n ≤ K).

This, together with the fact that τ is a joint ϵ-translation number for all f_{n}/n with n ≥ N(ϵ)(≥ N(2ϵ)) (see (2.2)), implies

\frac{f_{n}(x)}{n} ≥ \frac{f_{n}(x − τ)}{n} − ϵ > f^{−}(x − τ) + ϵ (N(ϵ) ≤ n ≤ K).

That means x ∈ A_{K}^{ϵ}. Thus we have finished the proof of Lemma 2.7.

Now let us prove Lemma 2.6. Since joint \frac{ϵ}{2}-translation numbers are relatively dense there exists L = L(\frac{ϵ}{2}) > 0 such that any interval of length L contains such a joint \frac{ϵ}{2}-translation number. Since ∩_{K}AK^{\frac{ϵ}{2}} = ∅, for any η > 0 there exists K_{0} > 0 such that

|AK^{\frac{ϵ}{2}} ⋂[−L, L]| < Lη  (∀K ≥ K_{0})  (2.3)

where | · | denotes the Lebesgue measure (see the definition of n_{ϵ}(x)). We claim that 𝕄̂(A_{K}^{ϵ}) ≤ η (∀K ≥ K_{0}). Otherwise 𝕄̂(A_{K}^{ϵ}) > η for some K ≥ K_{0}. Then by the definition of 𝕄̂(A_{K}^{ϵ}) there exists x_{0} ∈ ℝ such that

∫_{x_{0}}^{x_{0}+L} χ_{A_{K}^{ϵ}}(x)dx ≥ Lη.

Take a joint \frac{ϵ}{2}-translation number τ ∈ [−x_{0} − L, −x_{0}], i.e. −L ≤ x_{0} + τ ≤ 0. Then by Lemma 2.7, we have

|AK^{\frac{ϵ}{2}} ⋂[−L, L]| ≥ ∫_{x_{0}+τ}^{x_{0}+τ+L} χ_{A \frac{ϵ}{2} ; K}(x)dx
= ∫_{x_{0}}^{x_{0}+L} χ_{A \frac{ϵ}{2} ; K}(y + τ)dy
≥ ∫_{x_{0}}^{x_{0}+L} χ_{A_{K}^{ϵ}}(y)dy

For the first inequality we have used the fact that [x_{0}+τ, x_{0}+τ +L] ⊂ [−L, L] and for the last inequality we have used Lemma 2.7. What we have deduced contradicts (2.3). Thus Lemma 2.6 is proved.

We continue our proof of Theorem 2.5. Let S := ∥f^{−}∥_{∞} < ∞. Let K be such that 𝕄̂(A_{K}^{ϵ}) ≤ ϵ/S. We define first

g(x) = \begin{cases}f^{−}(x) & if x ∉ A_{K}^{ϵ} \\ 0 & if x ∈ A_{K}^{ϵ} \end{cases} and m(x) = \begin{cases}n^{ϵ}(x) & if x ∉ A_{K}^{ϵ} \\ 1 & if x ∈ A_{K}^{ϵ} \end{cases}

Lemma 2.6 implies that

𝕄g ≤ 𝕄f^{−} + ϵ a.e..  (2.4)

Moreover by the invariance of f^{−} we have (remember that u_{0} = 1)

g(x) ≤ g(u_{k}x) for a.e. x and all 0 ≤ k ≤ m(x) − 1.  (2.5)

Then we have

f_{m(x)}(x) ≤ (g(x) + ϵ)m(x) ≤ ∑_{k=0}^{m(x)−1} g(u_{k}x) + ϵm(x).  (2.6)

We define inductively m_{0}(x) = 0 and

m_{k}(x) = m_{k−1}(x) + m(u_{m_{k−1}(x)}x).

Now choose R > K and let k(x) be the maximal k for which m_{k}(x) ≤ R. Note that R−m_{k(x)}(x) < K. Now we get by the subadditivity and Equation (2.6)

f_{R}(x) ≤ ∑_{k=0}^{k(x)−1} f_{m(u_{m_{k}(x)}x)}(u_{m_{k}(x)}x) + \underbrace{f_{R−m_{k(x)}(x)}(u_{m_{k(x)}(x)}x)}_{≤0}

≤ ∑_{k=0}^{k(x)−1} ∑_{j=m_{k−1}(x)}^{m_{k}(x)} g(u_{j}x) + (m_{k}(x) − m_{k−1}(x))ϵ

≤ ∑_{j=0}^{m_{k(x)}(x)} g(u_{j}x) + m_{k(x)}(x)ϵ

≤ ∑_{j=0}^{R−1} g(u_{j}x) + Rϵ + KS.

Taking the Bohr mean, using that 𝕄g = 𝕄(g ∘ u_{j}) and dividing by R gives

\begin{aligned}\frac{1}{R}𝕄f_{R} & ≤ 𝕄g + ϵ + \frac{KS}{R} \\ & ≤ 𝕄f^{−} + 2ϵ + \frac{KS}{R} \end{aligned}

by Equation (2.4). Now we let R → ∞ and we get

γ ≤ 𝕄f^{−}.

We claim that f^{−} ≤ γ implies f^{−} = γ for a.e. x. Suppose this was not the case, then one could find ϵ > 0, δ > 0 and an interval J = (0, L) of length |J| = L = L_{ϵ} such that |A ∩ J| = δ > 0, where

A = {x: f^{−}(x) < γ − ϵ}

By the invariance of f^{−} we have u_{k}A = A for all k ∈ ℕ. Hence, for all k ∈ ℕ

\frac{1}{u_{k}L} ∫_{0}^{u_{k}L} χ_{A} dx = \frac{1}{L} ∫_{0}^{L} χ_{A} dx > \frac{δ}{L} > 0.

Since lim sup_{k} u_{k} = +∞ (see remark 2.3), we have 𝕄̂A > \frac{δ}{L}, and thus

\underline{𝕄}f^{−} < γ (1 − \frac{δ}{L}) + (γ − ϵ)\frac{δ}{L} < γ.

□

Remark 2.8. One can prove a similar theorem for more general sequences (u_{n}(x)). In this case it seems to be necessary to assume L^{1} Bohr-uniform distribution.

2.3. Oseledec’ Theorem.

Kingman’s Theorem implies the following Oseledec type theorem (see Ruelle [Ru]).

Theorem 2.9. Let β > 1 be a real number. Let M: ℝ → GL_{d}(ℂ) be a uniformly almost periodic function. Write

M_{x}^{n} = M(β^{n−1}x) ⋯ M(βx)M(x).

Suppose the q-exterior products \frac{1}{n} log ∥(M_{x}^{n})^{∧q}∥ have joint periods, for q = 1, …, d. Then there is Γ ⊂ ℝ with βΓ ⊂ Γ of full Lebesgue measure (in the sense that ℝ \ Γ has 0 measure) such that if x ∈ Γ then

a) lim_{n→∞}(M_{x}^{n∗}M_{x}^{n})^{\frac{1}{2n}} = Λ_{x} exists.

b) Let exp λ_{x}^{(1)} < ⋯ < exp λ_{x}^{(s)} be the eigenvalues of Λ_{x} (where s = s(x) and the λ_{x}^{(r)} are reals), and U_{x}^{(1)}, …, U_{x}^{(s)} the corresponding eigenspaces. Let m_{x}^{(r)} = dim U_{x}^{(r)}. We have λ_{βx}^{(r)} = λ_{x}^{(r)} and m_{βx}^{(r)} = m_{x}^{(r)} and

lim_{n→∞} \frac{1}{n} log ∥M_{x}^{n}v∥ = λ_{x}^{(r)} when v ∈ V_{x}^{(r)} \ V_{x}^{(r−1)}

for r = 0, …, s where V_{x}^{(0)} = {0} and V_{x}^{(r)} = U_{x}^{(1)} + ⋯ + U_{x}^{(r)}.

c) Moreover V_{x}^{(r)} depends measurably on x and M_{x}V_{x}^{(r)} = V_{βx}^{(r)}.

d) In addition the functions λ_{x}^{(r)} and m_{x}^{(r)} are constant a.e.

Proof. This theorem follows in the standard way from the a.e. convergence of

lim_{n→∞} \frac{1}{n} log ∥ (M_{x}^{n})^{∧q} ∥

for 1 ≤ q ≤ d, which in the classical case is insured by Kingman’s Theorem. So we only need to check that for the functions f_{n}^{(q)}(x) = log ∥ (M_{x}^{n})^{∧q} ∥ and the sequence u_{n} = β^{n} the assumptions of Theorem 2.5 hold.

First we note that M^{−1} is uniformly almost periodic because M is uniformly almost periodic and M^{−1}(x) ∈ GL_{d}(ℂ). Second we note that M^{∧q} and (M^{∧q})^{−1} are again uniformly almost periodic, since each entry is a rational function of the entries of M and M^{−1}, respectively. Hence,

sup_{x∈ℝ}∥M(x)^{∧q}∥ + sup_{x∈ℝ}∥(M^{∧q})^{−1}(x)∥ = W_{q} < ∞.

Subadditivity is obviously fulfilled since (M_{x}^{n+m})^{∧q} = (M_{β^{n}x}^{m})^{∧q}(M_{x}^{n})^{∧q}. Condition (2.1) follows from

∥(M_{βx}^{n})^{∧q}∥ = ∥M(β^{n}x)^{∧q}(M_{x}^{n})^{∧q}(M(x)^{∧q})^{−1}∥ ≤ ∥(M_{x}^{n})^{∧q}∥ + W_{q}.

Finally by Remark 2.3 the sequence β^{n} is totally Bohr ergodic, so Theorem 2.5 applies. Assertions (a), (b) and (c) follows from Proposition 1.3 (see also the proof of Theorem 1.6) in [Ru].

Now we prove d). By Kingman’s Theorem (Theorem 2.5), we have for almost all x

lim_{n→∞} \frac{1}{n} log ∥(M_{x}^{n})^{∧k}∥ = inf_{n≥1} \frac{1}{n}𝕄 log ∥(M_{x}^{n})^{∧k}∥  (1 ≤ k ≤ s). (2.1)

On the other hand, by the properties of exterior product, if we write k = ∑_{i=1}^{j−1} m_{x}^{(s−i)} + ℓ with 0 ≤ j < s and 0 ≤ ℓ ≤ m_{x}^{(s−j)} we have

lim_{n→∞} \frac{1}{n} log ∥(M_{x}^{n})^{∧k}∥ = m_{x}^{(s)}λ_{x}^{(s)} + ⋯ + m_{x}^{(s−j+1)}λ_{x}^{(s−j+1)} + ℓλ_{x}^{(s−j)}. (2.2)

We can solve λ_{x}^{(r)} (1 ≤ r ≤ s) from the system (2.1)-(2.2). The solution is independent of x since it depends only on the right hand side terms in (2.1). Consequently m_{x}^{(r)} is also independent of x. □

3. When β is a Pisot–Vijayaraghavan number

We restrict our attention to the special case where β > 1 is a Pisot– Vijayaraghavan (PV) number and f_{n} are defined by (1.1). We will prove that, under some extra condition, the sequence n^{−1}f_{n} has joint periods and the Kingman’s Theorem and the Oseledec’ Theorem apply. To do this, we need a distortion lemma and some properties of PV-numbers.

3.1. Distortion lemmas.

Lemma 3.1. Let M: ℝ → GL_{d}(ℂ) such that

D := sup_{x∈ℝ}∥M(x)∥ ∥M(x)^{−1}∥ < ∞  (3.3)

Let (x_{k}) and (y_{k}) be two sequences in ℝ and let θ_{k} = ∥M(x_{k}) − M(y_{k})∥. Then for all n ∈ ℕ and 0 ≠ v ∈ ℂ^{d} we have

\frac{|M(x_{1}) ⋯ M(x_{n−1})M(x_{n})v|}{|M(y_{1}) ⋯ M(y_{n−1})M(y_{n})v|} ≤ exp (C ∑_{k=1}^{n} D^{k}θ_{k}) .

where C = (sup_{x∈ℝ} ∥M(x)∥)^{−1}.

Proof. Let

Q_{1}(x) = 𝕀,  Q_{k}(x) = M(x_{1})M(x_{2}) ⋯ M(x_{k−1})  (1 < k ≤ n)
Q^{n}(x) = 𝕀,  Q^{k}(x) = M(x_{k+1})M(x_{k+2}) ⋯ M(x_{n}) (1 ≤ k < n).

We can write

\frac{|Q_{n}(x)v|}{|Q_{n}(y)v|} = ∏_{k=1}^{n} \frac{|Q_{k}(y)M(x_{k})Q^{k}(x)v|}{|Q_{k}(y)M(y_{k})Q^{k}(x)v|}.  (3.4)

Setting E_{k} = M(x_{k}) · M^{−1}(y_{k}) − 𝕀 we get the following estimate for the numerator of the general term in the above product

|Q_{k}(y)M(x_{k})Q^{k}(x)v| = |Q_{k}(x)(𝕀 + E_{k})M(y_{k})Q^{k}(x)v|
= |Q_{k}(x)(𝕀 + E_{k})Q_{k}(x)^{−1}Q_{k}(x)M(y_{k})Q^{k}(x)v|
= |(𝕀 + Ẽ_{k})Q_{k}(x)M(y_{k})Q^{k}(x)v|

where Ẽ_{k} = Q_{k}(x)E_{k}Q_{k}(x)^{−1}. It follows that

\frac{|Q_{k}(y)M(x_{k})Q^{k}(x)v|}{|Q_{k}(y)M(y_{k})Q^{k}(x)v|} ≤ ∥𝕀 + Ẽ_{k}∥.  (3.5)

It is obvious that

∥Ẽ_{k}∥ ≤ D^{k−1}∥E_{k}∥.  (3.6)

On the other hand

∥E_{k}∥ ≤ sup_{x} ∥M(x)^{−1}∥θ_{k}.  (3.7)

By combining (3.4), (3.5), (3.6) and (3.7), we obtain

\frac{|M(x_{1}) ⋯ M(x_{n−1})M(x_{n})v|}{|M(y_{1}) ⋯ M(y_{n−1})M(y_{n})v|} ≤ ∏_{k=0}^{n−1} (1 + CD^{k}θ_{k}) .

□

If M(x) is non-negative, the next lemma shows that condition (3.3) is not needed for positive vectors Lemma 3.2. Let M: ℝ → GL_{d}(ℝ) be such that the entries of M(x) are either identically zero or bounded from below by a positive number δ > 0 (independent of entries). Then for any sequences (x_{k}) and (y_{k}) in ℝ and for any non-negative vector v we have

\frac{|M(x_{1}) ⋯ M(x_{n−1})M(x_{n})v|}{|M(y_{1}) ⋯ M(y_{n−1})M(y_{n})v|} ≤ exp (\frac{1}{δ} ∑_{k=1}^{n} θ_{k}) ,

where θ_{k} = ∥M(x_{k})−M(y_{k})∥ and the norm |v| = ∑_{i=1}^{d} |v_{i}| on ℝ^{d} is chosen.

Proof. We may write

|M(x_{1}) ⋯ M(x_{n−1})M(x_{n})v|
= ∑_{i_{0},i_{1}··· ,i_{n}} M(x_{1})_{i_{0},i_{1}}M(x_{2})_{i_{1},i_{2}} ⋯ M(x_{n})_{i_{n−1},i_{n}}v_{i_{n}}

We have a similar expression for |M(y_{1}) ⋯ M(y_{n−1})M(y_{n})v|. Now compare the two expressions term by term. By the hypothesis, both quantities M(x_{1})_{i_{0},i_{1}} and M(y_{1})_{i_{0},i_{1}} are either zero or larger than δ. So, using the trivial inequality x/y ≤ e^{x/y−1} we have

M(x_{1})_{i_{0},i_{1}} ≤ M(y_{1})_{i_{0},i_{1}}e^{δ^{−1}θ_{k}}.

The same estimates hold for other pairs M(x_{k})_{i_{k−1},i_{k}} and M(y_{k})_{i_{k−1},i_{k}}. The desired inequality follows. □

3.2. Two properties of PV–numbers.

Let β > 1 be a PV–number of order r. We denote its conjugates by β′_{1}, ⋯, β′_{r−1}. Then for n ≥ 1, denote

F_{n} = β^{n} + β′_{1}^{n} + ⋯ + β′_{r−1}^{n}.

Lemma 3.3. The number F_{n} is an integer and we have

|β^{n} − F_{n}| ≤ (r − 1)ρ^{n} (∀n ≥ 1)

where ρ = max_{1≤j≤r−1} |β′_{j}| < 1.

Given any real number β > 1 (not necessarily integral), we can expand each number x ∈ [0, 1) in a canonical way into its β-expansion [Re] (see also [P, Bl]):

x = ∑_{n=1}^{∞} \frac{ϵ_{n}(x)}{β^{n}}

where (ϵ_{n}(x))_{n≥1} is a uniquely determined sequence in {0, 1, ⋯, [β]}^{ℕ}. We may also call (ϵ_{n}(x))_{n≥1} the β-expansion of x. We note that not all sequences in {0, 1, ⋯, [β]}^{ℕ} are β-expansions. Let D_{β} be the set of all possible βexpansions of numbers in [0, 1). A finite sequence ϵ = (ϵ_{1}, ⋯, ϵ_{n}) (of length n) in {0, 1, ⋯, [β]}^{n} is said to be admissible if it is the prefix of the βexpansion of some number x. For such an admissible sequence, we define

I(ϵ_{1}, ⋯, ϵ_{n}) = {x ∈ [0, 1) : ϵ_{1}(x) = ϵ_{1}, ⋯, ϵ_{n}(x) = ϵ_{n}}.

It is known that if D_{β} is endowed with the lexicographical order, the map which associates x to its β-expansion is strictly increasing. The set I(ϵ) = I(ϵ_{1}, ⋯, ϵ_{n}) is an interval, called a β-interval of level n. Its length is denoted by |I(ϵ)|.

Lemma 3.4. Suppose β > 1 is a PV–number. There is a constant C > 0 such that

C^{−1}β^{−n} ≤ |I(ϵ_{1}, ⋯, ϵ_{n})| ≤ Cβ^{−n}

for any integer n ≥ 1 and any β-interval I(ϵ_{1}, ⋯, ϵ_{n}).

See [F] for proofs of Lemma 3.3 and Lemma 3.4.

3.3. Existence of joint periods.

Definition 3.5. Let β > 1 be a positive real number and let M : ℝ → M_{d×d}(ℂ). If the entries of M are functions of the form f(β^{n}x) (n ∈ ℤ) where f is 1–periodic continuous, we say that M is β–adapted u.a.p.

Remark 3.6. The matrix M(x) defined by (1.4) associated to a multiperiodic function is β–adapted. Proposition 3.7. Let β > 1 be a PV–number and let M : ℝ → GL_{d}(ℂ) be β–adapted and α–Hölder continuous. Suppose that

Dρ^{α} < 1  (3.8)

where ρ is the maximal modulus of the conjugates of β and D is the same as in the distortion lemma (Lemma 3.1). Then for any 1 ≤ q ≤ d the sequence n^{−1}f_{n}^{(q)}(x) has joint periods, where

f_{n}^{(q)}(x) = log ∥(M(β^{n−1}x) ⋯ M(βx)M(x)))^{∧q}∥.

Proof. Since M is β–adapted, the entries of M(β^{k}x) are all of the form h_{i,j}(β^{ℓ_{i,j}}x) with 1–periodic function h_{i,j}(x) and integer ℓ_{i,j} ≥ 0 for sufficiently large k. So, if necessary, we consider log ∥(M(β^{n−1}x) ⋯ M(β^{k_{0}}x))^{∧q}∥ for some sufficiently large but fixed k_{0} ≥ 0.

Consider τ = β^{m}η_{m} + ⋯ + βη_{1} + η_{0} where m ≥ 1 and 0 ≤ η_{i} ≤ β are integers. We are going to show that all such τ are joint ϵ-translation numbers for n^{−1}f_{n}^{(n)}(x) with n ≥ N(ϵ), where N(ϵ) depending on ϵ is an integer to be determined.

By Lemma 3.3, we have

inf_{j∈ℤ}|β^{k}τ − j| ≤ C′ρ^{k}

for all k and some constant C′ independent of k and τ. For k ≥ 0 each entry of M^{∧q}(β^{k+k_{0}}x) is a degree q polynomial in d^{2} variables of the form h(β^{ℓ+k}x), with h α-Hölder, 1–periodic, and ℓ ≥ 0. Notice that we have h(β^{ℓ+k}(x + τ)) = h(β^{ℓ+k}x) + O(ρ^{kα}), hence

∥M(β^{k+k_{0}}(x + τ))^{∧q} − M(β^{k+k_{0}}(x))^{∧q}∥ = C_{q}ρ^{kα}

for some constant C_{q}. By the distortion Lemma 3.1 and the above estimate we have

|f_{n}^{(q)}(x + τ) − f_{n}^{(q)}(x)|
= |log \frac{∥(M(β^{n−1}x + β^{n−1}τ) ⋯ M(βx + βτ)M(x + τ))^{∧q}∥}{∥(M(β^{n−1}x) ⋯ M(βx)M(x))^{∧q}∥}|

= C_{q}C′ ∑_{k=1}^{n} D^{k}ρ^{(k−k_{0})α} ≤ \frac{CC′Dρ^{−k_{0}}}{1 − Dρ^{α}} =: C

So, we may choose N(ϵ) = C/ϵ. In order to finish the proof, it suffices to notice that Lemma 3.4 implies that all these τ form a subset with bounded gap in ℝ. □ Proposition 3.8. Let β > 1 be a PV–number and let M : ℝ → GL_{d}(ℂ) be β–adapted and α–Hölder continuous. Suppose that the entries of M(x) are either identically zero or larger than a constant δ > 0. Then n^{−1}f_{n}(x) has joint periods, where

f_{n}^{(q)}(x) = log ∥(M(β^{n−1}x) ⋯ M(βx)M(x))^{∧q})∥.

Proof. The proof is the same as the last proposition. But we use Lemma 3.2 instead of Lemma 3.1. □

4. Multiperiodic functions

As we pointed out in the introduction and as we will see in Section 4.2, our scalar equation (1.2) can be converted to the vector equation (1.3). So, we first study the vector equation (1.3).

4.1. Equation G(x) = M(x/β)G(x/β).

Let M : ℝ → M_{d×d}(ℂ) be a matrix valued function. We consider the following vector valued equation

G(x) = M (\frac{x}{β} ) G (\frac{x}{β} )

where the unknown G : ℝ → ℂ^{d} is a vector valued function.

Theorem 4.1. Let β > 1 be a real number and M(x) be a complex matrix valued function. Suppose that M is Lipschitzian and that M(0) is non negative and has 1 as a simple eigenvalue with a corresponding strictly positive eigenvector v. Then there exists, up to a multiplicative constant, a unique continuous solution G(0) ≠ 0 of the equation G(x) = M(x/β)G(x/β). The solution can be defined by

G(x) = lim_{n→∞} M (\frac{x}{β} ) M (\frac{x}{β^{2}} ) ⋯ M (\frac{x}{β^{n}} ) v

where the convergence is uniform on every compact subset in ℝ.

Proof. Write v = (v_{1}, ⋯, v_{d})^{t}. We introduce the following norm for ℂ^{d}

∥z∥ = max_{1≤j≤d} \frac{|z_{j}|}{v_{j}} (z = (z_{j})_{1≤j≤d} ∈ ℂ^{d}).

Then a matrix A = (a_{i,j}) ∈ M_{d×d}(ℂ), considered as an operator on the normed space (ℂ^{d}, ∥ · ∥), admits its operator norm

∥A∥ = max_{1≤i≤d} \frac{1}{v_{i}} ∑_{j=1}^{d} |a_{i,j}|v_{j}.

Notice that ∥M(0)∥ = 1 because M(0)v = v.

Since the eigenvalue 1 of M(0) is simple (and isolated), and M(x) is Lipschitz continuous, by the perturbation theory of matrices, there is a neighborhood of 0, say [−δ, δ] (δ > 0), such that for any x ∈ [−δ, δ], M(x) has a simple eigenvalue λ(x) and a corresponding eigenvector v(x) satisfying

|λ(x) − 1| ≤ C|x|,  ∥v(x) − v∥ ≤ C|x|  (x ∈ [−δ, δ])  (4.1)

for some constant C > 0. We claim that the limit

G(x) = lim_{n→∞} M (\frac{x}{β} ) M (\frac{x}{β^{2}} ) ⋯ M (\frac{x}{β^{n}} ) v  (4.2)

exists (uniformly on any compact set). It is clear that the limit function is a solution.

Denote

Q_{n}(x) = M (\frac{x}{β} ) M (\frac{x}{β^{2}} ) ⋯ M (\frac{x}{β^{n}} ) .

The proof of the existence of the limit in (4.2) is based on the following lemma.

Lemma 4.2. For any δ > 0, there exists a constant D > 0 such that for any n ≥ 1 and any x ∈ [−δ, δ] we have

∥Q_{n}(x)∥ ≤ D ∥Q_{n}(x)v − v∥ ≤ D|x|.

To get the boundedness of ∥Q_{n}(x)∥, it suffices to notice that

∥Q_{n}(x)∥ ≤ ∏_{j=1}^{n} f (\frac{x}{β^{j}} )

where the scalar function f(x) = ∥M(x)∥ is Lipschitzian and f(0) = 1 (we have used our choice of the norm of ℂ^{d}), and that the products converge uniformly on [−δ, δ] to a continuous function [FL]. Now we prove that

∥Q_{n}(x)v − Q_{n−1}(x)v∥ ≤ C′ \frac{|x|}{β^{n}}  (4.3)

where C′ > 0 is some constant. In fact, since M(x)v(x) = λ(x)v(x), we have

M (\frac{x}{β^{n}} ) v − v = M (\frac{x}{β^{n}} ) [v − v (\frac{x}{β^{n}} )] + [λ (\frac{x}{β^{n}} ) v (\frac{x}{β^{n}} ) − v] .

Multiplying both sides by Q_{n−1}(x), we get

∥Q_{n}(x)v − Q_{n−1}(x)v∥
≤ ∥Q_{n}(x) (v − v (\frac{x}{β^{n}} ))∥ + ∥Q_{n−1}(x) (v − λ (\frac{x}{β^{n}} ) v (\frac{x}{β^{n}} ))∥

Notice that

∥λ(x)v(x) − v∥ ≤ ∥λ(x) − 1∥ ∥v(x)∥ + ∥v(x) − v∥.

Using the last inequality, the estimates in (4.1) and that we have just proved ∥Q_{n}(x)∥ ≤ D, we obtain (4.3). Then for n > m

∥Q_{n}(x)v − Q_{m}(x)v∥ ≤ ∑_{k=m+1}^{n} ∥Q_{k}(x)v − Q_{k−1}(x)v∥ ≤ \frac{C′|x|}{β^{m−1}(β − 1)}.

That means Q_{n}(x)v is a Cauchy sequence in the space C([−δ, δ]) of continuous functions equipped with uniform norm. Since for any fixed integer n_{0}, we have

lim_{n→∞} Q_{n}(x)v = Q_{n_{0}}(x) · lim_{n→∞} Q_{n} (\frac{x}{β^{n_{0}}} ) v,

it follows that the uniform convergence of Q_{n}(x)v on [−δ, δ] implies its uniform convergence on any compact set.

The uniqueness of solution is easy. Let G ≠ 0 be a solution. First notice that G(0) is an eigenvector of M(0) associated to the simple eigenvalue 1. Hence we may assume that G(0) = v. By iterating the equation, we get

G(x) = Q_{n}(x)G (\frac{x}{β^{n}} ) = Q_{n}(x)v + Q_{n}(x) (G (\frac{x}{β^{n}} ) − v)

The last term converges to zero (uniformly on any compact set) because of ∥Q_{n}(x)∥ ≤ D. So, G(x) must be the limit of Q_{n}(x)v. □

Remark 4.3. In the theorem, neither the almost periodicity of M(x) nor the positivity of M(x) is required, but only the positivity of M(0). That 1 is an eigenvalue of M(0) is necessary for the equation (1.3) to have a solution G(x) such that G(0) ≠ 0.

Remark 4.4. The Lipschitz continuity is not really necessary. Hölder continuity or even Dini continuity is sufficient.

Remark 4.5. If the entries of M are (real) analytic, then the solution G is also analytic. Because, for any x_{0} ∈ ℝ, there is a disk on the complex plane centered at x_{0} on which Q_{n}(x)v (as functions of complex variable x) uniformly converges.

4.2. Existence of multiperiodic functions.

Here we give a proof of Theorem A based on Theorem 4.1.

Let M(x) be as in (1.4). It is easy to see that the characteristic polynomial of M(0) takes the form

P (u) = u^{d} − f_{1}(0)u^{d−1} − f_{2}(0)u^{d−2} − ⋯ − f_{d−1}(0)u − f_{d}(0)

The consistency condition implies that 1 is an eigenvalue of M(0). Notice that

P′(1) = f_{1}(0) + 2f_{2}(0) ⋯ + (d − 1)f_{d−1}(0) + df_{d}(0) > 0.

So, the eigenvalue 1 is simple. By Theorem 4.1, there is a unique solution of G(x) = M(x/β)G(x/β). Let

G(x) = (G_{1}(x), G_{2}(x), ⋯, G_{d}(x))^{t}

Then G_{1}(x) is a solution of (1.2). If F is a solution of (1.2). Let

G̃_{1}(x) = F (x), G̃_{2}(x) = F (x/β), ⋯, G̃_{d}(x) = F (x/β^{d−1}).

Then G̃ = (G̃_{1}, ⋯, G̃_{d})^{t} is a solution of G(x) = M(x/β)G(x/β). Thus the uniqueness of the solution of Equation (1.3) implies that of Equation (1.2).

4.3. Asymptotic behavior of multiperiodic functions.

Let us consider the asymptotic behavior of a multiperiodic function, or more generally the asymptotic behavior of a solution G of Equation (1.3) provided it exists (the existence may be guaranteed by Theorem 4.1).

Theorem 4.6. Let β > 1 be a PV–number whose maximal conjugate has modulus ρ. Let M : ℝ → GL_{d}(ℂ) be a β–adapted u.a.p. Hölder function of order α > 0. Suppose that G is a solution of G(x) = M(x/β)G(x/β). Suppose furthermore that one of the following conditions is satisfied

(i) Dρ^{α} < 1 where D = sup_{x∈ℝ} ∥M(x)∥∥M(x)^{−1}∥ (NB. β must be Pisot). (ii) The entries of M(x) are either identically zero or larger than a con-

stant δ > 0.

Then for a.e. x ∈ ℝ the limit

h(x) = lim_{n→∞} \frac{1}{n} log |G(β^{n}x)|

exists and is independent of x.

Proof. We first consider the case (i). By Proposition 3.7, Theorem 2.9 applies. Hence for a.e. x, if we denote by r(x) the integer such that the vector G(x) ∈ V_{x}^{(r)} \ V_{x}^{(r−1)} we get

lim_{n→∞} \frac{1}{n} log |G(β^{n}x)| = lim_{n→∞} \frac{1}{n} log |M_{x}^{n}G(x)| = λ_{x}^{(r(x))}.

But G(βx) = M(x)G(x) hence G(βx) ∈ V_{βx}^{(r)} \ V_{βx}^{(r−1)}, from what follows r(βx) = r(x), i.e. r is invariant. Hence constant a.e. because of the total Bohr ergodicity of the sequence β^{n}.

Case (ii). We use the notation of Proposition 3.8. Since n^{−1}f_{n} has joint periods, Theorem 2.5 applies (see the proof of Theorem 2.9 for details). Hence the following limit exists a.e.

lim_{n→∞} \frac{1}{n}f_{n}(x) = L,

where L = inf_{n} \frac{1}{n}𝕄(f_{n}). In view of G(β^{n}x) = M(β^{n−1}x) ⋯ M(x)G(x) the positivity of M and G gives

lim_{n→∞} \frac{1}{n} log |G(β^{n}x)| = L a.e.
□

Note that when G is the solution of equation (1.3) with M and G given by (1.4) we have

|G(x)| = |F (x)| + |F (x/β)| + ⋯ + |F (x/β^{d−1})|,  (4.4)

thus the asymptotic behavior of \frac{1}{n} log |G(β^{n}x)| and \frac{1}{n} log ∑_{j=0}^{d−1} |F (β^{n−j}x)| are the same. Thus Theorem C follows as an immediate corollary of Theorem 4.6. This partially answers a question in [JRS] (Conjecture 4.1., p. 263).

We prove now Theorem B. By the primitivity of M(0) and the hypothesis, there exists an ingeter τ ≥ 1 such that M̃(x) := M(x/β^{τ−1}) ⋯ M(x/β)M(x) has all its entries strictly positive (even larger than cδ^{τ} for some constant c > 0). Consider the equation

G(x) = M̃(x/β)G(x/β^{τ}).

We examine the first entries of both sides. We can find two constants 0 < c_{1} < c_{2} such that we get

c_{1}F (x) ≤ F (x/β^{τ+1}) + ⋯ + F (x/β^{τ+d}) ≤ c_{2}F (x).

Thus Theorem B follows from Theorem 4.6.

Theorem 4.7. Under the same conditions as Theorem 4.6, for any q ∈ ℝ^{+}, the following limit exists

lim_{n→∞} \frac{1}{n} log ∫_{0}^{1} ∥M(β^{n−1}x) ⋯ M(βx)M(x)∥^{q}dx.

Proof. Write

Z_{n} = ∫_{0}^{1} P_{n}(x)^{q}dx with P_{n}(x) = ∥M(β^{n−1}x) ⋯ M(βx)M(x)∥.

It suffices to show that there is a constant C > 0 such that

Z_{n+m} ≤ CZ_{n}Z_{m} (n ≥ 1, m ≥ 1).

We assume that q = 1, just for simplicity. We will use the fact that there is a constant L > 0 such that

∥M(x) − M(y)∥_{2} ≤ L|x − y|^{α} (∀x, y ∈ ℝ).

We use the notation Π_{i=0}^{n}M_{i} = M_{n}M_{n−1} ⋯ M_{1}M_{0} for the (noncommutative) product of the matrices M_{0}, …, M_{n}. Write

Z_{n+m} = ∑_{ϵ} ∫_{I(ϵ)} ∥ ∏_{k=0}^{m−1} M(β^{n+k}x) · ∏_{j=0}^{n−1} M(β^{j}x)∥ dx

≤ ∑_{ϵ} ∫_{I(ϵ)} ∥∏_{j=0}^{n−1} M(β^{j}x)∥ · ∥ ∏_{k=0}^{m−1} M(β^{n+k}x)∥ dx

where the sum is taken over all β-intervals I(ϵ) of level n (see Lemma 3.4). Let a_{ϵ} be the left endpoint of I(ϵ). The integral in the last sum , after the change of variables β^{n}(x − a_{ϵ}) = y, becomes

β^{−n} ∫_{0}^{β^{n}|I(ϵ)|} ∥∏_{j=0}^{n−1} M(β^{j}a_{ϵ} + β^{−n+j}y)∥ · ∥ ∏_{k=0}^{m−1} M(β^{k}y + β^{n+k}a_{ϵ})∥ dy.

Notice that if ϵ = (ϵ_{1}, ⋯, ϵ_{n}), then

β^{n+k}a_{ϵ} = β^{n+k} (\frac{ϵ_{1}}{β} + ⋯ + \frac{ϵ_{n}}{β^{n}} ) = β^{n+k−1}ϵ_{1} + ⋯ + β^{k}ϵ_{n}.

So, by Lemma 3.3, there is an integer n_{ϵ} such that

|β^{n+k}a_{ϵ} − n_{ϵ}| = O(ρ^{k} + ρ^{k+1} + ⋯ + ρ^{n+k−1}) = O(ρ^{k})  (4.5)

By the distortion lemma (Lemma 3.1, Lemma 3.2), we have

∥∏_{j=0}^{n−1} M(β^{j}a_{ϵ} + β^{−n+j}y)∥ ≤ C ∥∏_{j=0}^{n−1} M(β^{j}a_{ϵ})∥

∥ ∏_{k=0}^{m−1} M(β^{k}y + β^{n+k}a_{ϵ})∥ ≤ C ∥ ∏_{k=0}^{m−1} M(β^{k}y)∥

Therefore, we get

Z_{n+m} ≤ Cβ^{−n} ∑_{ϵ} P_{n}(a_{ϵ}) ≤ C′Z_{n}Z_{m}.

□

Corollary 4.8. Let F be the multiperiodic function defined by (1.2). Suppose that β > 1 is a PV–number and that f_{1}, ⋯, f_{d} are either identically zero or larger than a constant δ > 0. Suppose further that M_{x}^{ℓ} := M(β^{ℓ−1}x) ⋯ M(βx)M(x) has strictly positive entries for some integer ℓ > 0. Then for any q ∈ ℝ^{+}, the following limit exists

lim_{T →∞} \frac{1}{log T} ∫_{0}^{T} F (x)^{q}dx.

Proof. Without loss of generality, we may only consider the subsequence T_{n} = β^{n}. Since |G(x)| = ∑_{j=0}^{d−1} |F (x/β^{j})| where G is the solution of the associated vector equation (1.3), we have only to show the existence of the limit

lim_{n→∞} \frac{1}{n} ∫_{0}^{β^{n}} |G(x)|^{q}dx.

Making the change of variables x = β^{n}y, we are led to prove the existence of the limit

lim_{n→∞} \frac{1}{n} ∫_{0}^{1} |G(β^{n}x)|^{q}dx.

Notice that G(β^{n}x) = M_{x}^{n}G(x). Notice also that G(x) has strictly positive entries by the hypothesis on M_{x}^{ℓ}. So, for the non negative matrix M_{x}^{n} we have

C^{−1}∥M_{x}^{n}∥ ≤ |G(β^{n}x)| ≤ C∥M_{x}^{n}∥ (∀x ∈ [0, 1])

for some constant C > 0. By the proof of the last theorem, log ∫_{0}^{1} |G(β^{n}x)|^{q}dx is subadditive. □

Remark 4.9. Let M(x) be the matrix defined by (1.4). Let M̃ be the numerical matrix obtained by replacing f_{j}(x) in M(x) by 0 or 1 according to f_{j}(x) ≡ 0 or not. Then M̃^{ℓ} > 0 implies M_{x}^{ℓ} > 0. In particular, M̃^{d} > 0 if f_{j}(x) are all strictly positive.

Example 4.10. Let β > 1 be a PV–number. Let f_{1}(x) and f_{2}(x) be two strictly positive 1–periodic Hölder continuous functions such that f_{1}(0) + f_{2}(0) = 1. There is a unique multiperiodic function F defined by

F (x) = f_{1} (\frac{x}{β} ) F (\frac{x}{β} ) + f_{2} (\frac{x}{β^{2}} ) F (\frac{x}{β^{2}} )

For almost every x ∈ ℝ, n^{−1} log F (β^{n}x) has a limit as n → ∞; for any q ∈ ℝ^{+}, (log T)^{−1} ∫_{0}^{T} F (x)^{q}dx has a limit as T → ∞.

Example 4.11. Let β > 1 and a, b ∈ ℤ. Consider the contractive transformations on ℝ defined by

S_{1}x = \frac{x + a}{β}, S_{1}x = \frac{x + b}{β^{2}}.

For any 0 < p < 1, there exists a unique probability measure μ with compact support such that

μ = p μ ∘ S_{1}^{−1} + (1 − p) μ ∘ S_{2}^{−1}.

It is a self-similar measure. Its Fourier transform satisfies the equation

μ̂(x) = f_{1}(x/β)μ̂(x/β) + f_{2}(x/β^{2})μ̂(x/β^{2})

with f_{1}(x) = pe^{2πiax} and f_{2}(x) = qe^{2πibx} with q = 1 − p . This is a special case of the equation (1.2). The corresponding matrix defined by (1.4) and its inverse are respectively equal to

M(x) = ( \begin{matrix}pe^{2πiax} & qe^{2πibx/β} \\ 1 & 0 \end{matrix} ) , M(x)^{−1} = q^{−1}e^{−2πibx/β} ( \begin{matrix}0 & qe^{2πibx/β} \\ 1 & −pe^{2πiax} \end{matrix} )

If we take the norm |v| = max(|v_{1}|, |v_{2}|) on ℂ^{2}, the operator norms for M(x) and M(x)^{−1} are respectively ∥M(x)∥ = 1 and ∥M(x)^{−1}∥ = \frac{1+p}{1−p}. So, when β is a PV–number, under the condition \frac{1+p}{1−p} < \frac{1}{ρ}, for almost all x ∈ ℝ the following limit exists and does not depend on x

lim_{n→∞} \frac{1}{n} log (|μ̂(β^{n}x)| + |μ̂(β^{n−1}x)|) .

References

[Bl] F. Blanchard, β-expansions and symbolic dynamics, Theoretical Computer Sci. 8 (1989) 131–141.

[Bo] H. Bohr, Almost periodic functions, Chelsea Publishing Company, New York, 1947.

[D] I. Daubechies, Ten lectures on wavelets, CBMS Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, 1992.

[F] A.H. Fan, Asymptotic behaviour of multiperiodic functions scaled by Pisot numbers, J. Anal. Math. 86 (2002) 271–287.

[FL] A.H. Fan and K. S. Lau, Asymptotic behavior of multiperiodic functions G(x) = ∏_{n=1}^{∞} g(x/2^{n}), J. Fourier Anal. Appl. 4 (1998) 129–150.

[G] A. M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962) 409–432.

[JRS] P. Janardhan, D. Rosenblum and R. Strichartz, Numerical experiments in Fourier asymptotics of Cantor measures and wavelets, Experimental Math. 1, No 4 (1992) 249–273.

[Ki1] J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Royal Stat. Soc. B30 (1968) 499–510.

[Ki2] J. F. C. Kingman, Subadditive processes, in École d’Été de Probabilités de Saint-Flour, V–1975, 167–223. Lecture Notes in Math., 539, Springer, Berlin, 1976

[KL] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, New York, Wiley, 1974.

[KW] Y. Katznelson and B. Weiss, A simple proof of some ergodic theorems, Isr. J. of Math. 42 (1982) 291–296.

[P] W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung. 11 (1960) 401–416.

[Re] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957) 477–493.

[Ru] D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979) 27–58.

[S] R. Strichartz, Self-similarity in harmonic analysis, J. Fourier Anal. Appl. 1 (1994) 1–37.

[W] P. Walters, An introduction to ergodic theory, Springer-Verlag, 1982.

Ai-Hua FAN: CNRS UMR 6140 – LAMFA, Université de Picardie, 80039 Amiens, France

   E-mail address: ai-hua.fan@u-picardie.fr
   URL: http://www.mathinfo.u-picardie.fr/fan/

Benoît SAUSSOL: CNRS UMR 6140 – LAMFA, Université de Picardie, 80039 Amiens, France

   E-mail address: benoit.saussol@u-picardie.fr
   URL: http://www.mathinfo.u-picardie.fr/saussol/

Jörg SCHMELING: CNRS UMR 6140, LAMFA, Université de Picardie, 80039 Amiens, France & Department of Mathematics, University of Lund, P.O. Box 118, SE-221 00 LUND, Sweden

   E-mail address: joerg@maths.lth.se