1 Introduction↩︎

1.1 Objectives of the article and motivation↩︎

The aim of this article is to study the linear dynamics of random sequences of products of operators on a separable Fréchet space. That is, we would like to determine if the sequence \((T_n(.))_{n \geq 1}\) of the random operators \[\begin{align} \label{formeproduitopintro} T_n(\omega) = T(\tau^{n-1} \omega) \dotsm T(\tau \omega) T(\omega) \end{align}\tag{1}\] is universal, or topologically weakly mixing, or topologically mixing, or none of it, in the sense that the sequence \((T_n(\omega))_{n \geq 1}\) has this property for almost every \(\omega \in \mathbb{T}:= \mathbb{R}/ \mathbb{Z}\), or none of these properties. The probability space for our study will be the set \(\mathbb{T}\), equipped with its Borel subsets and with the normalized Lebesgue measure \(m\). The map \(\tau : \mathbb{T}\to \mathbb{T}\) will be a measure-preserving transformation on \(\mathbb{T}\), which means that \[m(\tau^{-1}(B)) = m(B)\] for every Borel subset \(B\) of \(\mathbb{T}\), and the map \(T : \mathbb{T}\to {\mathcal{B}}(X)\) will be given by \[\notag T(\omega) = \left\{ \begin{array}{ll} T_1 & if \quad \omega \in A_1 \\ T_2 & if \quad \omega \in A_2 \end{array}, \right.\] with \(T_1, T_2\) two operators on a separable Fréchet space \(X\), and with \(A_1, A_2\) two disjoint Borel subsets of \([0,1)\) such that \(A_1 \cup A_2 = [0,1)\) and \(m(A_k) > 0\) for \(k = 1,2\). Such types of products (1 ) appear when we consider linear cocycles \(F : \mathbb{T}\times X \to \mathbb{T}\times X, \, (\omega, x) \mapsto (\tau \omega, T(\omega) x)\). In this case, we have \(F^n(\omega, x) = (\tau^n x, T_n(\omega)x)\). Some work on the dynamical properties of linear cocycles, which is somewhat close to our study, are considered in [@BCH10], [@CH08] and [@MMPR].

The motivation of our work comes from the study of the asymptotic behavior of random products of matrices, which come into play, for example, in considering solutions to differential or difference equations with random coefficients. Bellman was the first to study random products of \(2 \times 2\) matrices with strictly positive entries ([@Bell]). Roughly speaking, he proved that the weak law of large numbers holds for these kinds of random products. A few years later, Furstenberg and Kesten also considered random products of matrices ([@FK]). In particular, they extended Bellman’s result by proving that the strong law of large numbers is applicable for random products of \(2\times 2\) matrices with strictly positive entries, under some assumptions on the coefficients of these matrices.

While random products of matrices and operators have been extensively studied from a probabilistic and statistical viewpoint - most notably via Lyapunov exponents (see, for instance, [@VIANA]) - the linear dynamics of such random products remains little explored. This work aims to explore the dynamical properties (e.g., orbit structure, universality) of random products of operators of the form (1 ).

In particular, our study includes the case of random products of independent variables on \(\mathbb{T}\) with a common distribution. Indeed, let \(A_1 = [0,1/2)\) and \(A_2 = [1/2, 1)\). Let \(\tau\) be the doubling map on \(\mathbb{T}\) defined by \(\tau(x) = 2x\). Let \(T : \mathbb{T}\to {\mathcal{B}}(X)\) be defined by \[\notag T(\omega) = \left\{ \begin{array}{ll} T_1 & if \quad \omega \in A_1 \\ T_2 & if \quad \omega \in A_2 \end{array}, \right.\] with \(T_1, T_2\) two bounded operators on a separable Fréchet space \(X\). For each \(n \geq 1\) and \(\omega\in \mathbb{T}\), let \(T_n(\omega) := T(\tau^{n-1} \omega) \dotsm T(\tau \omega) T(\omega)\). Let us denote by \(\varphi : \{ 0,1 \}^{\mathbb{N}_0} \to \mathbb{T}\) the map defined by \(\varphi((x_n)_{n \geq 0}) = \displaystyle \sum_{n \geq 0} \frac{x_n}{2^{n+1}}\) in \(\mathbb{T}\), and by \(\sigma : \{ 0 , 1 \}^{\mathbb{N}_0} \to \{ 0,1 \}^{\mathbb{N}_0}\) the backward shift defined by \(\sigma((x_n)_{n \geq 0}) = (x_{n+1})_{\geq 0}\). The map \(\varphi\) is a bijection from the set \(\Omega_0\) of sequences in \(\{ 0 , 1\}^{\mathbb{N}_0}\) not eventually equal to \(1\) onto \(\mathbb{T}\), and \(\tau \varphi = \varphi \sigma\). For every \(j \geq 0\), let \(\hat{X}_j : u = (u_k)_{k \geq 0} \in \{ 0 , 1 \}^{\mathbb{N}_0} \mapsto u_j\) be the \(j\)-th canonical projection. The random variables \(\hat{X}_j, j \geq 0,\) are independent on the probability space \((\{ 0,1 \}^{\mathbb{N}_0}, \mathcal{C}, \nu^{\otimes \mathbb{N}_0})\) and have the same distribution \(\nu := \frac{1}{2}(\delta_0 + \delta_1)\), where \(\mathcal{C}\) denotes the \(\sigma\)-algebra generated by the cylinders and \(\nu^{\otimes \mathbb{N}_0}\) denotes the product measure of \(\nu\). For every \(\omega= \varphi(u) \in \mathbb{T}\) with \(u \in \Omega_0\), and for every \(j \geq 0\), we have \(T(\tau^j \omega) = T(\varphi(\sigma^j u))\), which is equal to \(T_1\) if and only if \(\varphi(\sigma^ju) \in A_1\), that is, if and only if \(\hat{X_j}(u) = 0\), and is equal to \(T_2\) if and only if \(\hat{X_j}(u) = 1\). In particular, the random variables \(T(\tau^j \omega), j \geq 0,\) are independent on \(\mathbb{T}\) and have the same distribution, namely \(\frac{1}{2}(\delta_{T_1} + \delta_{T_2})\).

1.2 Linear dynamics↩︎

We now introduce the notions of linear dynamics that we will use for our study. For background on linear dynamics, we refer to Bayart’s and Matheron’s book [@BM], and to Grosse-Erdmann’s and Peris Manguillot’s book [@GEPM]. A good introduction to the notion of universality is given in [@GE].

A sequence of continuous maps \((T_n)_{n \geq 1}\) from a metric space \(X\) to itself is said to be universal if there is an element \(x \in X\) such that its orbit under \((T_n)_{n \geq 1}\), \[\textrm{orb}(x,(T_n)) := \{ T_n x : n \geq 1 \},\] is dense in \(X\). Fekete was the first to give an example of a universal family. He showed that there exists a real power series \(\displaystyle \sum_{n \geq 1} a_n \, x^n\) on \([-1,1]\) that diverges at every point \(x \ne 0\) in the worst possible way, in the sense that to every continuous function \(g\) on \([-1,1]\) with \(g(0) = 0\), there exists an increasing sequence of positive integers such that \(\displaystyle \sum_{n = 1}^{n_k} a_n \, x^n \underset{k\to \infty}{\longrightarrow} g\) uniformly on \([-1,1]\). In this case, \(T_n\) is the map that associates to \(f\) its partial Taylor series of order \(n\) at \(0\). This example of universality shows two aspects of universality: an aspect of maximal divergence, and the aspect of the existence of a single object which allows us to approximate a maximal class of objects.

The notion of topological transitivity also plays an important role in topological and linear dynamics. We say that a sequence of continuous maps \((T_n)_{n \geq 1}\) from a space \(X\) to itself is topologically transitive if, for any pair \(U,V \subset X\) of nonempty open sets, there exists an integer \(n \geq 1\) such that \(T_n(U) \cap V \ne \emptyset\). If this property holds for sufficiently large \(n \geq 1\), then we say that the sequence \((T_n)_{n \geq 1}\) is topologically mixing. Finally, we say that the sequence \((T_n)_{n \geq 1}\) is topologically weakly mixing if, for any nonempty open sets \(U_1, U_2, V_1, V_2\) of \(X\), there is some \(n \geq 1\) such that \(T_n(U_1) \cap V_1 \ne \emptyset\) and \(T_n(U_2) \cap V_2 \ne \emptyset\). Every topologically mixing sequence is obviously topologically weakly mixing, and every topologically weakly mixing sequence is topologically transitive. If the sequence \((T_n)_{n \geq 1}\) is topologically transitive and if the maps \(T_n\) are invertible, then the sequence \((T_n^{-1})_{n \geq 1}\) is also topologically transitive.

For a single continuous map \(T\) on \(X\), we say that \(T\) is topologically transitive if the sequence \((T^n)_{n \geq 1}\) is topologically transitive. When the space \(X\) is a separable complete metric space without isolated points, a continuous map \(T\) is topologically transitive if and only if there is a point \(x \in X\) with a dense orbit under \(T\). In fact, in this situation, the map \(T\) has a dense \(G_{\delta }\)-set of points with a dense orbit under \(T\). This is Birkhoff’s transitivity theorem ([@BM] and [@GEPM]). For a sequence of continuous and commuting maps \(T_n\) on \(X\), it is possible to generalize this result in the following way ([@GEPM]).

A topologically transitive continuous linear map on a separable Fréchet space \(T\) is also called a hypercyclic operator. In the linear setting, we have a more useful criterion than Birkhoff’s transitivity theorem to prove that a sequence of operators on a separable Fréchet space \(X\) is universal.

The criterion of Proposition 2 is called the Universality Criterion. This criterion will be a powerful tool for our study. Observe that the maps \(S_n\) are not required to be linear nor continuous.

An important example of operators in this paper will be the adjoint of the multiplication operator \(M_\phi\) on the Hardy space \(H^2(\mathbb{D})\). Its adjoint \((M_\phi)^*\) is hypercyclic on \(H^2(\mathbb{D})\) if and only if \(\phi(\mathbb{D}) \cap \mathbb{T}\ne \emptyset\), when \(\phi\) is a nonconstant bounded holomorphic function on \(\mathbb{D}\) ([@GodShapiro]). We will also consider entire functions of exponential type of the derivation operator \(D\) on the space of entire functions. These operators are exactly the operators on the space of entire functions that commute with \(D\), and they are always topologically mixing and hence hypercyclic, provided they are not a scalar multiple of the identity ([@GodShapiro]).

1.3 Main results↩︎

We study the linear dynamics of random sequences \((T_n(.))_{n \geq 1}\) when the operators \(T(\tau^i \omega), i \geq 0\), are adjoints of multiplication operators on the Hardy space \(H^2(\mathbb{D})\), or entire functions of exponential type of the derivation operator \(D\) on the space of entire functions. We also study a case of random products of operators on the space of entire functions, where the operators \(T(\tau^i \omega)\) do not commute. The ergodic properties of the transformation \(\tau : \mathbb{T}\to \mathbb{T}\) will play an important role in examining the universal behavior of the sequence \((T_n(\omega))_{n \geq 1}\), for almost every \(\omega \in \mathbb{T}\). We put together here some of our main results. The notations will be defined in Subsection 1.5.

Thanks to the fact that the adjoint of a multiplication operator has plenty of eigenvalues, we obtain a first sufficient criterion which gives the universality of the random sequence \((T_n(.))_{n \geq 1}\) in the case of adjoints of multiplication operators.

In the case \(m(A_1) = m(A_2)\), Conditions (2 ) and (3 ) are equivalent to \((\phi_1 \phi_2 )(\mathbb{D}) \cap \mathbb{T}\ne \emptyset\) and \(\phi_1 \phi_2\) nonconstant, by the open mapping theorem. This leads to two limit cases when \(m(A_1) = m(A_2)\): the case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{D}\), and the case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{C}\setminus \overline{\mathbb{D}}\). We will see that the study of these limit cases is not that easy.

Under certain conditions, we establish that the random sequence of operators \((T_n(.))_{n \geq 1}\) is universal when the operators \(T(\tau^i \omega)\) are entire functions of exponential type of the derivation operator \(D\). This will follow from the fact that the derivation operator on \(H(\mathbb{C})\) has suitable eigenvectors, namely the exponential functions \(e_\lambda : z \mapsto e^{\lambda z}\), with \(\lambda \in \mathbb{C}\).

In the case where \(m(A_1) = m(A_2) = 1/2\), Conditions (4 ) and (5 ) are equivalent to the fact that \(\varphi_1 \varphi_2\) is nonconstant, when \(\varphi_1\) and \(\varphi_2\) are two nonconstant entire functions of exponential type. This comes from the fact that every nonconstant entire function has a dense range in \(\mathbb{C}\). We will also study a situation where \(\varphi_1 \varphi_2\) is constant, showing in particular that the random sequence \((T_n(.))_{n \geq 1}\) can be non-universal even if \(T(\omega)\) is hypercyclic for every \(\omega\in \mathbb{T}\).

Until now, all the random products of operators \(T_n(.)\) we introduced before have the property that the operators \(T(\tau^i \omega), i \geq 0\), commute. We now consider a case of a random sequence of operators on \(H(\mathbb{C})\) for which these operators do not commute:

The operator \(T_{\lambda,b}\) is hypercyclic on \(H(\mathbb{C})\) precisely when \(\lambda = 1\) and \(b \ne 0\) (see, for instance, [@BeGoMR]). An interest of Theorem 5 is that the product \(T_n(\omega)\) can contain hypercyclic and non-hypercyclic operators \(T(\tau^i \omega)\) that do not commute.

We will also consider the case of an irrational rotation and of the doubling map on \(\mathbb{T}\). Given a real number \(\alpha\) in \((0,1)\), the rotation of parameter \(\alpha\) is defined by the transformation \(R_\alpha : x \mapsto x + \alpha \pmod 1\) on \([0,1)\), where \(x \pmod1\) is the fractional part of the real \(x\). This map can be seen as a map on \(\mathbb{T}\) by \(\tau(x) = x + \alpha\) in \(\mathbb{T}\), and can also be seen as a map on the unit circle by \(\tau(z) = e^{2 i \pi \alpha} z\). This map is ergodic precisely when \(\alpha\) is irrational. Moreover, this map is not weakly mixing. The doubling map is the map on \([0,1)\) defined by \(\tau(x) = 2 x \pmod1\). This map can also be seen as a map on \(\mathbb{T}\) by \(\tau(x) = 2 x \in \mathbb{T}\), and also as a map on the unit circle by \(\tau(z) = z^2\). Contrary to rotations, this map is mixing. These two transformations will appear many times in this article.

When the ergodic transformation involved in Theorems 4 and 5 is an irrational rotation and when \(A_1, A_2\) are intervals of \([0,1)\), the universality of the associated sequence \((T_n(\omega))_{n \geq 1}\) holds for every \(\omega \in \mathbb{T}\). This comes from the uniform distribution mod \(1\) of the sequence \((n \alpha)_{n \geq 1}\).

Regarding the limit case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{D}\) when \(m(A_1) = m(A_2) = 1/2\), we studied the situation where \(\phi_1 \phi_2\) is an inner function. We also generalized it to the situation where \(m(A_1) \ne m(A_2)\). In this context, for the doubling map on \(\mathbb{T}\), we obtained the following result.

The case of irrational rotations will be more harsh to study. This will strongly depend on the Diophantine properties of the irrational parameter \(\alpha\). For example, we will establish the following result.

However, in the case \(A_1 = [0,1/2)\) and \(A_2 = [1/2,1)\), we do not need any assumption on the irrational \(\alpha\).

For some other intervals of \([0,1)\), it will also be possible for the random sequence \((T_n(.))_{n \geq 1}\) to be non-universal. This is, for example, the case of the following result that we obtained.

These results show in particular that the random sequence \((T_n(.))_{n \geq 1}\) can be universal, even if both operators \(T_1\) and \(T_2\) are not hypercyclic (see Remark [remarkunivsequencebutnothcop]). We will also generalize Theorems 6, 7 and 9 for some other ergodic transformations on \((\mathbb{T},m)\) (see Section 3). A main tool in our study of the linear dynamics of random sequences of products of operators will be Birkhoff’s ergodic theorem.

We would like to point out the fact that the random variables \(f(\tau^n \omega)\) involved in Theorem 10 are not necessarily independent. Thus, Birkhoff’s ergodic theorem is a generalization of the law of large numbers. If the system is ergodic and if \(A \subseteq \Omega\) is a measurable set, then, applying Theorem 10 to the function \(f = \mathbb{1}_A\), we obtain \[\begin{align} \label{birkhoffpourindicatrices} \frac{1}{N} \, \textrm{Card}\{0 \leq n \leq N-1 : \tau^{n} \omega\in A \} \underset{N\to \infty}{\longrightarrow} \mu(A), \end{align}\tag{6}\] for almost every \(\omega \in \Omega\). If \(A_k\) is a subset of \([0,1)\), we denote by \(a_k(n,\omega)\) the quantity \[a_k(n,\omega) := \textrm{Card}\{0 \leq i \leq n-1 : \tau^i\omega \in A_k \}.\]

Let \(f : \mathbb{T}\to \mathbb{R}\) be a real function and let \(\tau\) be a measure-preserving transformation on \((\Omega, \mathcal{A},\mu)\). We denote by \[\mathbb{S}_n^\tau f(\omega) := \displaystyle \sum_{i=0}^{n-1} f(\tau^i \omega)\] the Birkhoff sum associated to \(f\) and \(\tau\). In particular, if \(A_1, A_2\) are two disjoint Borel subsets of \([0,1)\) with \(m(A_k) > 0\) for \(k = 1,2\), we have \[a_1(n,\omega) - \frac{m(A_1)}{m(A_2)} a_2(n,\omega) = \mathbb{S}_n^\tau f(\omega),\] where \(f\) is the real function on \(\mathbb{T}\) defined by \(f = \mathbb{1}_{A_1} - \tfrac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}\). This function \(f\) is centered, that is, \(\displaystyle \int_{\mathbb{T}} f \, dm = 0\).

When \(\Omega = \mathbb{T}\), when \(\tau = R_\alpha\) with \(\alpha\) an irrational and when \(A = [a,b)\) with \(0 \leq a < b \leq 1\), the convergence (6 ) holds for every \(\omega \in \mathbb{T}\). This follows from the uniform distribution mod \(1\) of the sequence \((n \alpha)_{n \geq 1}\) (see, for instance, [@Pet]).

1.4 Organization of the paper↩︎

The behavior of Birkhoff sums associated to the function \(f := \mathbb{1}_{A_1} - \frac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}\) on \(\mathbb{T}\), with \(A_1\) and \(A_2\) two disjoint Borel subsets of \([0,1)\), will play a crucial role in determining the universality of some random sequences of products of operators \((T_n(.))_{n \geq 1}\). The boundedness of these sums is linked to the so-called coboundary equation in \(L^\infty(\mathbb{T})\) for the function \(f\). We begin our study by presenting these results on the Birkhoff sums. We then focus on Birkhoff sums associated to this function \(f\) and to the doubling map and to irrational rotations. The question of whether these Birkhoff sums satisfy a central limit theorem will be examined. This will be done in Section 2. After this study of the behavior of Birkhoff sums, we will study the linear dynamics of random sequences of products of operators \((T_n(.))_{n \geq 1}\), when the random operator \(T(\omega)\) is equal to the adjoint of a multiplication operator \((M_{\phi_1})^*\) for every \(\omega\in A_1\) and equal to the adjoint of a multiplication operator \((M_{\phi_2})^*\) for every \(\omega\in A_2\), where \(A_1, A_2\) are two disjoint Borel subsets of \(\mathbb{T}\) such that \(A_1 \cup A_2 = [0,1)\) and \(m(A_k) > 0\) for \(k = 1,2\), and where \(\phi_1, \phi_2 \in H^\infty(\mathbb{D})\) are nonconstant. This will be done in Section 3. Finally, we study random products of operators on the space of entire functions \(H(\mathbb{C})\). First, we will study the situation where the operators \(T(\tau^i \omega), i \geq 0,\) are entire functions of exponential type of the derivation operator \(D\), and afterwards, we will consider a case of random products where the operators \(T(\tau^i \omega), i \geq 0,\) do not commute. This will be done in Section 4.

1.5 Notation and definition↩︎

We put together here some important definitions and notations for the comprehension of this article.

Basic ergodic facts

We denote by \(\mathbb{T}\) the set \(\mathbb{T}:= \mathbb{R}/ \mathbb{Z}\), that we will often identify as \([0,1)\), and by \(m\) the normalized Lebesgue measure on \(\mathbb{T}\). A transformation \(\tau : \mathbb{T}\to \mathbb{T}\) is measure-preserving if \(m(\tau^{-1}(B)) = m(B)\) for every Borel subset \(B\). A measure-preserving transformation \(\tau\) is said to be ergodic on \((\mathbb{T},m)\) if for every Borel subset \(A\) of \(\mathbb{T}\) such that \(\tau^{-1}(A) = A\), we have \(m(A) = 0\) or \(m(A) = 1\). Moreover, this transformation \(\tau\) is mixing whenever \[m(\tau^{-n}(A) \cap B) \underset{n\to \infty}{\longrightarrow} m(A) \, m(B)\] for every Borel sets \(A,B\) of \(\mathbb{T}\), and weakly mixing whenever \[\displaystyle \frac{1}{n} \sum_{j=0}^{n-1} \lvert m(\tau^{-j}(A) \cap B) - m(A) m(B) \rvert \underset{n\to \infty}{\longrightarrow} 0\] for every Borel subsets \(A,B\) of \(\mathbb{T}\). Mixing is a property of asymptotic decorrelation. Every mixing system is easily seen to be weakly mixing and ergodic. We refer to P. Walters’s book [@Wal] and to K. Petersen’s book [@Pet] for background on ergodic theory.

Functions with bounded variation, continued fractions

We denote by \(BV\) the space of real functions on \(\mathbb{T}\) with bounded variation, and by \(BV_0\) the space of real centered functions in \(BV\), that is, the functions \(f\) of \(BV\) such that \(\int_\mathbb{T}f \, dm = 0\). Any function defined on \(\mathbb{T}\) can be seen as a \(1\)-periodic function on \(\mathbb{R}\). If \(f \in BV_0\), we write \(\lVert f \rVert_2 := \left( \displaystyle \int_0^1 f(t)^2 \, dt \right)^{1/2}\) for the variance of \(f\). For \(f \in BV\), we denote by \(c_r(f)\) the complex Fourier coefficients of \(f\), with \(r \in \mathbb{Z}\). When \(f\) is in \(BV_0\), we always have \[c_r(f) = \frac{\gamma_r(f)}{r}, \quad \textrm{for r \ne 0},\] where \(\gamma_r(f)\) are complex numbers such that \(\displaystyle \sup_{r \ne 0} \vert \gamma_r(f) \rvert \leq \frac{V(f)}{2 \pi} < \infty\).

If \(\alpha \in (0,1)\) is an irrational number, we write \(\alpha = [0; a_1, a_2,...]\) its continued fraction expansion, where the \(a_j\)’s are positive integers. We write \((\frac{p_n}{q_n})_{n \geq 0}\) for the sequence of the convergents associated to \(\alpha\), where \(p_n\) and \(q_n\) are positive integers. The sequences \((p_n)_{n \geq 0}\) and \((q_n)_{n \geq 0}\) satisfy the following relations: \[\left\{ \begin{array}{ll} & p_0 = 0, \; p_1 = 1, \; q_0 = 1, \; q_1 = a_1\\ &q_{n+1} = a_{n+1} \, q_n + q_{n-1} \\ &p_{n+1} = a_{n+1} \, p_n + p_{n-1} \\ &p_{n-1} \, q_n - p_n \, q_{n-1} = (-1)^n \end{array}\;, \; n \geq 1. \right.\] We say that \(\alpha\) has bounded partial quotients if the sequence \((a_j)_{j \geq 1}\) is bounded. We refer to Khinchin’s book [@Khin] for more details on continued fractions.

Hardy spaces

In what follows, we define Hardy spaces \(H^p(\mathbb{D})\) for \(p = 2\) or \(p = \infty\), but we can also define them for \(1 \leq p \leq \infty\). We will just need these two cases for our study. We refer to Nikolski’s book [@N] and to Koosis’s book [@Koo] for background on Hardy spaces.

The Hardy space \(H^2(\mathbb{D})\) is the space of holomorphic functions \(f : \mathbb{D}\to \mathbb{C}\) such that \[\displaystyle \sup_{0 \leq r < 1} \frac{1}{2 \pi} \int_{0}^{2\pi} \lvert f(r e^{it})\rvert^2 dt = \lim_{r \nearrow 1} \frac{1}{2 \pi} \int_{0}^{2\pi} \lvert f(r e^{it})\rvert^2 dt < \infty.\] For every \(f,g \in H^2(\mathbb{D})\), we have \[\langle f,g \rangle= \lim_{r \nearrow 1} \frac{1}{2 \pi} \int_{0}^{2\pi} f(r e^{it}) \, \overline{g(r e^{it})} dt \quad \textrm{and} \quad \lVert f \rVert^2 = \displaystyle \sup_{0 \leq r < 1} \frac{1}{2 \pi} \int_{0}^{2\pi} \lvert f(r e^{it})\rvert^2 dt.\] Moreover, \(H^2(\mathbb{D})\) is isometrically isomorphic to the space \(\ell_2(\mathbb{N}_0)\). The space \(H^\infty(\mathbb{D})\) is the space of holomorphic functions \(f : \mathbb{D}\to \mathbb{C}\) such that \[\lVert f \rVert_\infty := \displaystyle \sup_{z \in \mathbb{D}} \lvert f(z) \rvert < \infty.\] We have \(H^\infty(\mathbb{D}) \subset H^2(\mathbb{D})\). For any \(\lambda \in \mathbb{D}\), the function \(k_\lambda\) defined by \[k_\lambda(z) = \frac{1}{1 - \overline{\lambda}z}\] satisfies \[\begin{align} f(\lambda) = \langle f, k_\lambda \rangle \end{align}\] for every \(f \in H^2(\mathbb{D})\), and is called the reproducing kernel of \(H^2(\mathbb{D})\). If \(f \in H^p(\mathbb{D})\) with \(p = 2\) or \(p = \infty\), the limits \[f^*(e^{it}) := \displaystyle \lim_{r \to 1} f (r e^{it})\] exist for almost every \(e^{it} \in \mathbb{T}\). We call the function \(f^*\) the boundary value of \(f\). An analytic function \(f : \mathbb{D}\to \mathbb{C}\) is said to be inner if \(f\) is bounded on \(\mathbb{D}\) and if \(\lvert f^* \rvert = 1\) almost everywhere on \(\mathbb{T}\). Any inner function \(f\) satisfies \(\lvert f(z) \rvert \leq 1\), for every \(z \in \mathbb{D}\). An analytic function \(F : \mathbb{D}\to \mathbb{C}\) is said to be outer if there is a constant \(c \in \mathbb{T}\) and a positive function \(h\) on \(\mathbb{T}\) satisfying \(\log h \in L^1(\mathbb{T})\), such that \[F(z) = c \, \exp \left(\frac{1}{2\pi} \int_0^{2\pi} \frac{e^{it}+z}{e^{it} - z} \, \log h(e^{it}) \, dt \right), \quad \textrm{for every} \; z \in \mathbb{D}.\]

If \(F,G\) are outer in \(H^2(\mathbb{D})\), then \(FG\) and \(1/F\) are still outer. The product of two inner functions of \(H^p(\mathbb{D})\), with \(p = 2\) or \(p = \infty\), remains an inner function. If \(f \in H^p(\mathbb{D})\) with \(p = 2\) or \(p = \infty\), the function \(Q_f\) defined on \(\mathbb{D}\) by \[Q_f(z) = \exp \left( \frac{1}{2\pi} \int_{0}^{2\pi} \frac{e^{it} + z}{e^{it} - z} \, \log \lvert f^*(e^{it}) \rvert \, dt \right)\] is outer and is called the outer factor of \(f\). An important result in Hardy spaces is the inner-outer factorization. This result states that if \(f \in H^p(\mathbb{D})\) with \(p = 2\) or \(p = \infty\), then \(f\) can be written as \(f = I F\), with \(I\) an inner function, and \(F\) an outer function in \(H^p(\mathbb{D})\). Moreover, we can take \(F = Q_f\), and the decomposition becomes unique. As a consequence of the inner-outer decomposition, if \(f \in H^\infty(\mathbb{D})\) and \(1/f \in H^\infty(\mathbb{D})\), then \(f\) is outer.

We define multiplication operators on \(H^2(\mathbb{D})\) as follows. If \(\phi\) is a function in \(H^\infty(\mathbb{D})\), the multiplication operator \(M_\phi\) on \(H^2(\mathbb{D})\) is given by \(M_\phi f = f \phi\). These operators satisfy \(\lVert M_\phi \rVert = \lVert \phi \rVert_{\infty} = \displaystyle \sup_{z \in \mathbb{D}} \lvert \phi(z) \rvert\). Moreover, we have the relation \[\begin{align} \label{eqvpadjointmultiplic} M_\phi^* \, k_\lambda = \overline{\phi(\lambda)} \, k_\lambda \quad \textrm{for every \lambda \in \mathbb{D}}. \end{align}\tag{7}\] It is well known that a multiplication operator \(M_\phi\) is never hypercyclic on \(H^2(\mathbb{D})\), but that its adjoint \(M_\phi^*\) is hypercyclic if and only if \(\phi(\mathbb{D}) \cap \mathbb{T}\ne \emptyset\), when \(\phi \in H^\infty(\mathbb{D})\) is nonconstant. This result was proved in [@GodShapiro]. See also [@GEPM] and [@BM].

2 Behavior of Birkhoff sums associated to a centered real function↩︎

Birkhoff sums of centered real functions will appear a lot in Section 3. These Birkhoff sums will influence the dynamical behavior of the random sequence \((T_n(.))_{n \geq 1}\). The aim of Section 2 is to study the behavior of such random variables. First, we state a criterion for Birkhoff sums to be unbounded on a set of measure \(1\), and next we investigate central limit theorems for such sums.

2.1 Boundedness of Birkhoff sums↩︎

Let \((\Omega, \mathcal{A},\mu, \tau)\) be an ergodic measure-preserving system and let \(f \in L_0^1(\Omega)\) be a real centered function. It is known that the sequence of Birkhoff sums \((\mathbb{S}_n^\tau f(\omega))_{n \geq 1}\) changes sign infinitely often for almost every \(\omega \in \Omega\), in the sense that it cannot be ultimately positive or negative ([@Ha]). Also, we have \(\displaystyle \liminf_{n \to \infty} \lvert \mathbb{S}_n^\tau f(\omega) \rvert = 0\) for almost every \(\omega \in \Omega\) ([@Atk], [@S]), and \(\displaystyle \limsup_{n \to \infty} \lvert \mathbb{S}_n^\tau f(\omega) \rvert > 0\) for almost every \(\omega \in \Omega\), if \(f\) is nonzero ([@Kachu]). More precisely, we have the following interesting result characterizing the boundedness of the sequence \((\mathbb{S}_n^\tau f(\omega))_{n \geq 1}\) for almost every \(\omega \in \Omega\).

Thus, Proposition 11 implies that \[\mu(\{ \omega \in \Omega : \displaystyle \limsup_{n \to \infty} \mathbb{S}_n^\tau f(\omega) = \infty \quad \textrm{or} \quad \liminf_{n \to \infty} \mathbb{S}_n^\tau f(\omega) = -\infty \}) = 1\] when the coboundary equation \(f = h - h \circ \tau\) has no solution \(h \in L^\infty(\Omega)\), thanks to the ergodicity of \(\tau\). However, it is not clear if this implies that \[\mu(\{ \omega \in \Omega : \displaystyle \limsup_{n \to \infty} \mathbb{S}_n^\tau f(\omega) = \infty \quad \textrm{and} \quad \liminf_{n \to \infty} \mathbb{S}_n^\tau f(\omega) = -\infty \}) = 1\] in this case. The aim of the next subsection will be to give a sufficient condition to replace the "or" by an "and" in this set. But first, we would like to mention the two following important results. The first one is due to Adams and Rosenblatt.

In the case of a real centered step function \(f\) on \(\mathbb{T}\), the authors of [@AdRos] even provide a description of this transformation \(\tau\) ([@AdRos]), of which they construct using Rokhlin towers. The second result studies when the sequence of Birkhoff sums associated to an irrational rotation and to a real step function is bounded at every point. If \(f : \mathbb{T}\to \mathbb{R}\) is a step function, we define the number \(\delta_f(x)\) for \(x \in \mathbb{T}\) by \[\delta_f(x) := f(x^+) - f(x^-) = \displaystyle \lim_{\varepsilon \searrow 0} f(x + \varepsilon) - f(x - \varepsilon),\] which corresponds to the jump of \(f\) at the point \(x\). We also write \(D_f := \{ x \in \mathbb{T}: \delta_f(x) \ne 0 \}\), which is a finite set, and we denote by \(\Delta_f(x)\) the following finite number: \[\Delta_f(x) := \displaystyle \sum_{k= -\infty}^{\infty} \delta_f(x + k \alpha).\] The second result is due to I. Oren and is the following.

We now apply Theorem 13 to the function \(f = \mathbb{1}_{A_1} - \tfrac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}\), with \(A_1 = [0,b)\) and \(A_2 = [b,1)\).

This shows in particular that when \(b \in \mathbb{Z}\alpha\), for the function \(f = \mathbb{1}_{A_1} - \tfrac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}\) with \(A_1 = [0,b)\) and \(A_2 = [b,1)\), the sequence of Birkhoff sums is bounded at every point. In the case \(b \notin \mathbb{Z}\alpha\), we know that at a given point \(\omega \in \mathbb{T}\), we have \[\displaystyle \limsup_{n \to \infty} \mathbb{S}_n^\tau f(\omega) = \infty \quad \textrm{or} \quad \limsup_{n \to \infty} \mathbb{S}_n^\tau f(\omega) = -\infty,\] but it does not seem clear whether both conditions can be satisfied simultaneously.

2.2 Central limit theorems for ergodic transformations↩︎

In this section, we discuss central limit theorems for the sequence \((\mathbb{S}_n^\tau f)_{n \geq 1}\) of Birkhoff sums associated to a real centered function \(f\) on \(\mathbb{T}\), where \(\tau\) is an ergodic transformation.

Given an ergodic transformation \(\tau\) on \((\mathbb{T}, m)\) and a real centered function \(f\) on \(\mathbb{T}\), we say that the sequence \((\mathbb{S}_n^\tau f)_{n \geq 1}\) satisfies a central limit theorem (CLT) if there exists a sequence \((a_n)_{n \geq 1}\) of positive real numbers with \(a_n \underset{n\to \infty}{\longrightarrow} \infty\) such that \[\frac{\mathbb{S}_n^\tau f}{a_n} \underset{n\to \infty}{\longrightarrow} \mathcal{N}(0,1) \quad \textrm{in distribution},\] where \(\mathcal{N}(0,1)\) is the random variable with density \(x \mapsto \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2}\) on \(\mathbb{R}\). Birkhoff sums satisfying a central limit theorem always satisfy \(\displaystyle \limsup_{n \to \infty} \mathbb{S}_n^\tau f(\omega) = \infty\) and \(\displaystyle \liminf_{n \to \infty} \mathbb{S}_n^\tau f(\omega) = -\infty\), for almost every \(\omega \in \mathbb{T}\).

We will now give examples of such transformations \(\tau\). First of all, Birkhoff sums associated to a real centered function \(f\) on \(\mathbb{T}\) and to an ergodic transformation \(\tau\) do not necessarily satisfy a CLT. Indeed, if \(\tau = R_\alpha\) is an irrational rotation with parameter \(\alpha\) and \(f\) is a real centered function on \(\mathbb{T}\) with bounded variation, the Denjoy-Koksma inequality ([@Her]) implies that \(\lVert \mathbb{S}_{q_k}^\tau f \rVert_{\infty} \leq V(f)\) along the subsequence of denominators of the convergents of \(\alpha\), where \(V(f)\) is the total variation of \(f\). Thus, the Birkhoff sums \((\mathbb{S}_n^\tau f)_{n \geq 1}\) cannot satisfy a CLT. In fact, for any aperiodic transformation \(\tau\) on \((\mathbb{T}, m)\) and for any distribution \(\nu\) on \(\mathbb{R}\), there is a measurable function \(f\) on \(\mathbb{T}\) such that \(\tfrac{\mathbb{S}_{n}^\tau f}{n}\) converges in distribution to \(\nu\). This result is due to J-P. Thouvenot and B. Weiss ([@TW]).

However, under certain assumptions on the function \(f\) and for an irrational rotation, there exist subsequences of \((\mathbb{S}_n^\tau f)_{n \geq 1}\) satisfying CLTs. An important result on this subject is the following, due to F. Huveneers ([@Huv]).

In particular:

It is important to note that Theorem 16 and Corollary 1 hold for any irrational \(\alpha\) in \((0,1)\).

In [@CIB], J-P. Conze, S. Isola, and S. Le Borgne also studied CLTs for rotations along subsequences of \((\mathbb{S}_n^\tau f)_{n \geq 1}\), when the parameter \(\alpha\) of the rotation has unbounded partial quotients, and when \(f\) is a certain real centered function with bounded variation. A few years later, J-P. Conze and S. Le Borgne continued the study of CLTs for irrational rotations and for a class of functions with bounded variation, which applies when the partial quotients of \(\alpha\) are bounded ([@CB]). Their method is close to the method used in [@Huv], and relies on an abstract CLT which is satisfied under some decorrelation conditions. It is also mentioned in [@CB] that when \(\tau = R_\alpha\) with \(\alpha\) an irrational, the behavior of the sequence \((\mathbb{S}_{n}^\tau f)_{n \geq 1}\) depends on the regularity of the function \(f\) on \(\mathbb{T}\). More precisely, under certain Diophantine conditions on \(\alpha \in \mathbb{R}\setminus \mathbb{Q}\), too much regularity on \(f\) will force \(f\) to be a coboundary in \(L^\infty(\mathbb{T})\). This is one of the reasons why they considered less regular functions, but with bounded variation, as for example step functions.

We explore in more depth the case of irrational rotations and the case of the doubling map on \(\mathbb{T}\). Recall that if \(\alpha\) is an irrational number, we denote by \((a_j)_{j \geq 1}\) the partial quotients of \(\alpha\), and by \((\frac{p_k}{q_k})_{k \geq 0}\) the sequence of convergents of \(\alpha\).

2.3 CLTs for an irrational rotation \(R_\alpha\) with \(\alpha\) having bounded partial quotients.↩︎

In this subsection, we discuss CLTs for an irrational rotation \(R_\alpha\), with \(\alpha\) satisfying \[\begin{align} \label{hypalphacasbpq} \exists \, A > 1,\, \exists \, 0 \leq p < 1/8 \,, \forall \, n \geq 1, \, a_n \leq A \, n^p. \end{align}\tag{8}\] In particular, the case of \(\alpha\) having bounded partial quotients is obtained with \(p = 0\).

The article [@CB] studies CLTs for irrational rotations, when the parameter \(\alpha\) satisfies the assumption (8 ). An important point in the approach [@CB] is the control of the variance \(\lVert \mathbb{S}_n^\tau f \rVert_2^2\). The authors of [@CB] proved that CLTs hold along certain subsets of integers with density one, on which the variance is large enough. In order to give a lower bound on the variance, they make the following assumption on the real centered functions \(f\) defined on \(\mathbb{T}\). \[\begin{align} \label{hypfonctionscasalphabpq} \exists \, M, \eta, \theta > 0 \quad \textrm{such that \; Card\{ 0 \leq j \leq N : a_{j+1} \lvert \gamma_{q_j}(f) \rvert \geq \eta \} \geq \theta \, N, \; \forall N \geq M.} \end{align}\tag{9}\]

Under (9 ), and when \(\alpha\) satisfies the assumption (8 ) (for example, when \(\alpha\) has bounded partial quotients), we have the following CLT.

Let us now apply Theorem 17 to the function \(f = \mathbb{1}_{A_1} - \tfrac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}\), when \(A_1 = [0,b)\) and \(A_2 = [b,1)\), and when \(b\) is a rational number in \((0,1)\). First, let us remark the following fact.

With this fact, we can prove the following lemma.

We thus deduce the following result.

For some intervals \(A_1\) and \(A_2\) with irrational endpoints, it is also possible to apply Theorem 17 to the function \(f\).

It also mentioned in [@CB] that if \(b = \displaystyle \sum_{n \geq 0} b_n q_n \alpha \; \textrm{mod 1}\) is the Ostrowski expansion of \(b\) associated to the denominators of \(\alpha\) and if \(\displaystyle \lim_{n \to \infty} \frac{\lvert b_n \rvert}{a_{n+1}} = 0\), then \(\displaystyle \lim_{k \to \infty} \lVert q_k b \rVert = 0\), where \(\lVert x \rVert\) is the distance of \(x\) to \(\mathbb{Z}\) (see [@GP]). In particular, under this assumption on \(b\) and when the partial quotients of \(\alpha\) are bounded, one can check that the sequence \((a_{j+1} \, \lvert \gamma_{q_j}(f) \rvert)_{j \geq 0}\) associated to our favorite function \(f\) converges to zero, and Condition (9 ) is not satisfied. We thus cannot apply Theorem 17. This shows the limits of the techniques used in the article [@CB].

2.4 CLTs for an irrational rotation \(R_\alpha\) with \(\alpha\) having unbounded partial quotients.↩︎

In the case where the irrational number \(\alpha\) does not satisfy (8 ), there are also CLTs for the rotation \(R_\alpha\) along subsequences, but the techniques used in [@CIB] are quite different. In this subsection, we focus on the case of an irrational number \(\alpha\) with unbounded partial quotients, which means that the sequence \((a_j)_{j \geq 1}\) of the convergents of \(\alpha\) is unbounded.

If \((t_k)_{k \geq 1}\) is an increasing sequence of positive integers, we set \(L_0 := 0\) and \(L_n = \displaystyle \sum_{k=1}^n q_{t_k}\) for \(n \geq 1\). We have the following version of the CLT for an irrational rotation \(R_\alpha\) when \(\alpha\) is such that the sequence \((a_{j+1})_{j \geq 0}\) is growing fast enough along a subsequence.

We now apply Theorem 20 to our favorite function \(f\).

We deduce the following consequence.

Exactly as in Proposition 19, when the partial quotients of \(\alpha\) are unbounded, we have the following result.

The same remark as in the case of an irrational number \(\alpha\) with bounded partial quotients holds: if \(b = \displaystyle \sum_{n \geq 0} b_n q_n \alpha \; \textrm{mod 1}\) is the Ostrowski expansion of \(b\) associated to the denominators of \(\alpha\) and if \(\displaystyle \lim_{n \to \infty} \frac{\lvert b_n \rvert}{a_{n+1}} = 0\), then Condition (11 ) is not satisfied for our favorite function \(f\).

2.5 CLTs for the doubling map↩︎

For weakly mixing transformations, the situation is considerably different. The first one to consider CLTs for weakly mixing transformations is Kac ([@K]), who studied CLTs for the doubling map and for integrable functions on \(\mathbb{T}\) whose Fourier coefficients verify a certain growth condition. More precisely, he proved the following result.

Many improvements regarding CLTs for weakly mixing transformations have been obtained later. For a class of piecewise monotonic weakly mixing transformations on \(\mathbb{T}\), S. Wong proved that CLTs hold for Hölder continuous functions defined on \([0,1)\) ([@Wong]). G. Keller and J. Rousseau-Egele also proved the existence of CLTs for such transformations, when the function \(f\) is in \(BV\), by using spectral properties of the Perron-Frobenius operator associated to the weakly mixing transformation \(\tau\) on \([0,1)\) ([@Ke], [@R-E]). J. Rousseau-Egele has even investigated the rate of convergence in the CLTs associated to such piecewise monotonic transformations and also proved a local limit theorem, when the function \(f\) is in \(BV\). For more information on CLTs for weakly mixing transformations, we refer to Denker’s article ([@Den]).

We consider now the case of the doubling map on \(\mathbb{T}\), that is, \(\tau(x) = 2 x \pmod1\). We consider the case of the two intervals \(A_1 = [0,b)\) and \(A_2 = [b,1)\) of \([0,1)\). We denote by \(f\) the following real function in \(BV_0\) given by \[f = \mathbb{1}_{A_1} - \tfrac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}.\] We will show, using Kac’s result (Theorem 23), that the sequence \((\mathbb{S}_n^\tau f)_{n \geq 1}\) satisfies a CLT with \(\sigma^2 > 0\), when \(b\in (0,1)\). Let us notice that this function \(f\) satisfies the assumptions of Theorem 23. The only thing to check is that \(\sigma^2 > 0\), that is, we have to prove that the coboundary equation (12 ) for \(f\) and \(\tau\) has no solution \(g \in L^2(\mathbb{T})\).

We thus deduce the following CLT for the doubling map.

3 Random products of adjoint multipliers on the Hardy space \(H^2(\mathbb{D})\)↩︎

Historically, the first examples of hypercyclic operators are Birkhoff’s operators on the space of entire functions, MacLane’s operator on the space of entire functions, and Rolewicz’s operators on \(X = \ell_p\) spaces, with \(1 \leq p < \infty\), or on \(X = c_0\). Precisely, Rolewicz’s operators are the operators \(T = \lambda B\), where \(\lambda \in \mathbb{C}\) is nonzero and \(B\) is the backward shift operator on \(X\) defined by \[B(x_1,x_2, \dotsc) = (x_2, x_3, \dotsc).\] These operators are hypercyclic precisely when \(\lvert \lambda \rvert > 1\). In the case \(\lvert \lambda \rvert \leq 1\), the operator \(T = \lambda B\) is just a contraction, so every orbit of \(T\) is bounded. Consequently, the operator \(T\) cannot be hypercyclic in this case. In the case \(\lvert \lambda \rvert > 1\), an application of the Hypercyclicity Criterion (Theorem 2) with the set \(\mathcal{D}_1 = c_{00}\) of finitely supported sequences as a dense subset of \(X\), and with the operators \(S_{k} = \lambda^{-k} S^k\), where \(S\) the forward shift on \(X\), shows that the operator \(T = \lambda B\) is topologically mixing on \(X\). The operator \(T = \lambda B\) on \(X = \ell_2(\mathbb{N}_0)\) is topologically conjugate to the adjoint multiplier operator \((M_{\lambda z})^*\) on \(H^2(\mathbb{D})\). Moreover, we have seen in the introduction that the adjoint of a multiplication operator \((M_\phi)^*\) is hypercyclic if and only if \(\phi(\mathbb{D}) \cap \mathbb{T}\ne \emptyset\), when \(\phi\) is nonconstant. These operators are classic in linear dynamics and since their linear dynamics is well known, it is natural to investigate the universality of a sequence of random products of operators defined by adjoints of multiplication operators on \(H^2(\mathbb{D})\). This section is devoted to this study. We will see that the study of the universality of random sequences \((T_n(.))_{n \geq 1}\) is not this simple compared to the study of the hypercyclicity of the adjoints of multiplication operators.

It is known that a multiplication operator \(M_\phi\) is never hypercyclic on \(H^2(\mathbb{D})\) (see, for instance, [@GEPM]). For random sequences of products of multiplication operators, we have the following remark. Let us recall that if \(A_k\) is a Borel subset of \([0,1)\), we denote by \(a_k(n,\omega)\) the number \[a_k(n,\omega) := \textrm{Card}\{0 \leq i \leq n-1 : \tau^i \omega\in A_k \}.\]

In this section, we investigate the linear dynamics of random sequences \((T_n(.))_{n \geq 1}\), with \[\label{eqcocycleadjointmultiplic} T(\omega) = \left\{ \begin{array}{ll} (M_{\phi_1})^* & if \quad \omega \in A_1 \\ (M_{\phi_2})^* & if \quad \omega \in A_2 \end{array}, \right.\tag{16}\] where \(\phi_1, \phi_2 \in H^\infty(\mathbb{D})\) are nonconstant functions, \(A_1, A_2\) are two disjoint Borel subsets of \([0,1)\) such that \(A_1 \cup A_2 = [0,1)\) and \(m(A_k) > 0\) for \(k = 1,2\), and where \(\tau\) an ergodic measure-preserving transformation on \((\mathbb{T},m)\). In this case, the operators \(T_n(\omega)\) are given by \[T_n(\omega) = (M_{\phi_1}^{a_1(n,\omega)})^* \, (M_{\phi_2}^{a_2(n,\omega)})^*,\] because the operators \(M_{\phi_1}\) and \(M_{\phi_2}\) commute. We will separate the two cases \(m(A_1) = m(A_2) = 1/2\) and \(m(A_1) \ne m(A_2)\), simply because the first case is simpler than the second case, and also because certain assumptions on the functions \(\phi_1\) and \(\phi_2\) will appear in the first case but will not appear in the second case. The behavior of the sequence \((T_n(\omega))_{n \geq 1}\) will depend very much on the ergodic transformation \(\tau\), as we will see.

3.1 Case \(m(A_1) = m(A_2) = 1/2\)↩︎

In this subsection, we consider \(A_1\) and \(A_2\) two disjoint Borel subsets of \([0,1)\) satisfying \(A_1 \cup A_2 = [0,1)\) and \(m(A_1) = m(A_2) = 1/2\).

The computation of \(\lVert T_n(\omega) \rVert\) gives a simple obstruction to the universality of the random sequence \((T_n(.))_{n \geq 1}\).

We will now see a sufficient condition for the random sequence \((T_n(.))_{n \geq 1}\) to be universal. The proof is based on the following useful criterion, which is mainly applied to operators with many eigenvalues. In what follows, if \(e \in X\) is an eigenvector of \(T\), we denote by \(\lambda(T,e)\) the corresponding eigenvalue. The linear span of a set \(\mathcal{A}\) of \(X\) will be written as \(\textrm{span}(\mathcal{A})\).

The sufficient condition for the universality of the random sequence is the following.

Proposition 27 leads to two limit cases: the case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{D}\) and the case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{C}\setminus \overline{\mathbb{D}}\), where \(\phi_1 \phi_2\) is nonconstant. We first treat some straightforward situations within these two cases.

As a consequence of Proposition 28, we treat the second easy situation.

In light of the forthcoming results, it is worth noting that if \(\phi_1, \phi_2 \in H^\infty(\mathbb{D})\) are nonconstant such that \(\lVert \phi_1 \rVert_\infty \leq 1\), \(\lVert \phi_2 \rVert_\infty \leq 1\) and \(\phi_1 \phi_2\) is inner, then \(\phi_1\) and \(\phi_2\) are both inner. Indeed, in this case, we have that \(1 \leq \lvert \phi_1^* \rvert = \frac{1}{ \lvert \phi_2^* \rvert} \leq 1\). Thus \(\lvert \phi_1^* \rvert =\lvert \phi_2^* \rvert =1\). The same results holds if \(\phi_1 \phi_2\) is inner, \(\phi_1(\mathbb{D}) \subseteq \mathbb{C}\setminus \mathbb{D}\) and \(\phi_2(\mathbb{D}) \subseteq \mathbb{C}\setminus \mathbb{D}\).

We now focus on the case where \(\phi_1 \phi_2\) is an inner function. Depending on the properties of the ergodic transformation \(\tau\) on \((\mathbb{T},m)\), we obtain sufficient conditions for the random sequence \((T_n(.))_{n \geq 1}\) to be universal. We first make the following observation.

We can now state the first sufficient condition.

In the case where \(f = \mathbb{1}_{A_1} - \mathbb{1}_{A_2}\) and the sequence \((\mathbb{S}_n^\tau f(\omega))_{n \geq 1}\) has a subsequence satisfying a CLT, we can omit the assumptions on the images of \(\phi_1\) and \(\phi_2\). The idea is to replace the dense subsets \(Z_1\) and \(Z_2\) of \(H^2(\mathbb{D})\) for the operators \(T_n(\omega)\), which are used to apply the Universality Criterion to the sequence \((T_n(\omega))_{n \geq 1}\), by another dense subset using model spaces. It can be seen as a particular case of Theorem 37, that we will present afterwards.

We can deduce from Theorem 31 the following results regarding the cases of the doubling map and the irrational rotations. The first result holds for the doubling map and follows from Corollary 4.

The second result holds for every irrational rotation and follows from Corollary 1.

We have seen that, in the case where \(\phi_1 \phi_2\) is nonconstant and such that \((\phi_1 \phi_2)(\mathbb{D}) \cap \mathbb{T}\ne \emptyset\), the sequence \((T_n(\omega))_{n \geq 1}\) is universal for almost every \(\omega \in \mathbb{T}\). Moreover, by Theorem 30, if \(\phi_1 \phi_2\) is constant on \(\mathbb{D}\) with \(\phi_1 \phi_2 \equiv \alpha\), \(\alpha \in \mathbb{T}\), and if \(\phi_1(\mathbb{D}) \cap \mathbb{T}\ne \emptyset\), then the random sequence \((T_n(.))_{n \geq 1}\) is universal. The following proposition shows that the converse is true in this case.

The proof of Proposition 32 relies on an important result of F. León-Saavedra and V. Müller, which we state here for the convenience of the reader.

In Proposition 33, a semigroup of operators is a subset \(\mathcal{M} \subset {\mathcal{B}}(X)\) containing the identity \(I\), and such that \(ST \in \mathcal{M}\) whenever \(S,T \in \mathcal{M}\). We can now prove Proposition 32.

In the case where \(m(A_1) \ne m(A_2)\), we will see that it is more difficult to identify limit cases, compared to the case where \(m(A_1) = m(A_2) = 1/2\). However, we can still obtain sufficient conditions, similar to those of Theorems 30 and 31, for the sequence \((T_n(\omega))_{ n \geq 1}\) to be universal for almost every \(\omega \in \mathbb{T}\).

3.2 Case where possibly \(m(A_1) \ne m(A_2)\)↩︎

In this subsection, we consider \(A_1\) and \(A_2\) two disjoint Borel subsets of \([0,1)\) satisfying \(A_1 \cup A_2 = [0,1)\) and \(\min\{m(A_1), m(A_2)\} > 0\). We study the dynamics of \((T_n(\omega))_{n \geq 1}\) for almost every \(\omega \in \mathbb{T}\), when \(T(\omega)\) is given by \((\ref{eqcocycleadjointmultiplic})\).

Exactly as in Proposition 25, the computation of \(\lVert T_n(\omega) \rVert\) provides a simple obstruction to the universality of random sequences \((T_n(.))_{ n \geq 1}\).

We also have an analog of Theorem 27 in the case where \(m(A_1)\) is not necessarily equal to \(m(A_2)\). This is the following result.

We would like to point out that Theorem 35 can also be deduced from [@CH08] in the case where \(\phi_1(z) = \lambda_1 z\) and \(\phi_2(z) = \lambda_2 z\) for every \(z \in \mathbb{D}\), in the setting of i.i.d products.

In the case where \(m(A_1) = m(A_2)\), if \(\phi_1 \phi_2\) is nonconstant, Conditions (\(\ref{eq1critvpscocyclemesnonegales}\)) and (\(\ref{eq2critvpscocyclemesnonegales}\)) are equivalent to \((\phi_1 \phi_2)(\mathbb{D}) \cap \mathbb{T}\ne \emptyset\), by the open mapping theorem. In the situation where \(m(A_1) \ne m(A_2)\), it is difficult to find a similar statement. The following two results are the analogs of Theorems 30 and 31 respectively. The first statement requires an assumption on the images of \(\phi_1\) and of \(\phi_2\), as in Theorem 30. The only difference with Theorem 30 is that here, we require both \(\displaystyle \limsup_{n \to \infty} \mathbb{S}_n^{\tau}f(\omega) = \infty\) and \(\displaystyle \liminf_{n \to \infty} \mathbb{S}_n^{\tau}f(\omega) = -\infty\) for almost every \(\omega\in \mathbb{T}\), and not just one of the two conditions for almost every \(\omega \in \mathbb{T}\).

The second theorem does not require any hypothesis on the images of \(\phi_1\) and of \(\phi_2\), as in Theorem 31, but requires \(\phi_1 \phi_2\) not to be outer. It is a more general situation than Theorem 31, since any function \(\phi \in H^\infty(\mathbb{D})\) such that \(1/\phi \in H^\infty(\mathbb{D})\) is outer.

We can apply Theorems 36 and 37 to the doubling map and to some irrational rotations. For the doubling map, this is the following result, which follows from Corollary 4.

For irrational rotations, we first have the following result, which follows from Corollary 2 and holds for example if the irrational \(\alpha\) has bounded partial quotients (in particular, for almost every irrational \(\alpha\)).

The case where \(\alpha\) as unbounded partial quotients is given by the following result, which follows from Corollary 3.

Finally, we have the following results for some irrational rotations and for some intervals of \([0,1)\), which follows from Propositions 19 and 22.

As we can see, the dynamics of the random sequence \((T_n(.))_{n \geq 1}\) is very dependent on the properties of the ergodic measure-preserving transformation \(\tau\) on \((\mathbb{T},m)\). We will now see that there exist ergodic transformations on \((\mathbb{T},m)\) for which the random sequence \((T_n(.))_{n \geq 1}\) is never universal.

In certain cases, for an irrational rotation, the sequence \((T_n(\omega))_{n \geq 1}\) can be non-universal, for every \(\omega\in \mathbb{T}\).

Thanks to Theorem 12, we also deduce the following result.

In the same line, we have the following result.

4 Other examples of random products on the space of entire functions↩︎

In this section, we study some natural examples of random products of operators defined on the space of entire functions. In particular, we will construct examples of universal random sequences \((T_n(.))_{n \geq 1}\) for which the operators \(T(\tau^i \omega)\), \(i \geq 0\), do not commute.

We denote by \(H(\mathbb{C})\) the space of entire functions, equipped with the topology of uniform convergence on compact subsets of \(\mathbb{C}\). This topology is generated by the seminorms \((p_n)_{n \geq 1}\), where \(p_n(f) = \displaystyle \sup_{ \lvert z \rvert \leq n } \lvert f(z) \rvert\). We denote by \(D\) the derivation operator on \(H(\mathbb{C})\) defined by \(D : f \mapsto f'\). Using Cauchy estimates, one can show that \(D\) is indeed an operator on \(H(\mathbb{C})\), that is, a continuous linear map on \(H(\mathbb{C})\).

We will first consider random products generated by entire functions of exponential type of \(D\). Let us recall some facts regarding entire functions of exponential type.

An entire function \(\varphi \in H(\mathbb{C})\) is said to be of exponential type if there exist constants \(M, A > 0\) such that \[\lvert \varphi(z) \rvert \leq M e^{A \lvert z \rvert}, \quad \textrm{for every z \in \mathbb{C}.}\] If \(\varphi : \mathbb{C}\to \mathbb{C}\) is an entire function of exponential type, with \(\varphi(z) = \displaystyle \sum_{n = 0}^{\infty} a_n z^n\), the expression \[\varphi(D) := \displaystyle \sum_{n \geq 0} a_n D^n\] defines an operator on \(H(\mathbb{C})\) ([@GEPM]). In particular, if \(a \ne 0\) is a complex number and if we denote by \(T_a : f \mapsto f(. + a)\) the translation by \(a\), we have \(T_a = \varphi(D)\) with \(\varphi(z) = e^{az}\). It is also known that if \(T\) is an operator on \(H(\mathbb{C})\) commuting with \(D\), then there exists an entire function \(\varphi\) of exponential type such that \(T = \varphi(D)\).

The linear dynamics of the operators \(D\) and \(T_a\), \(a \ne 0\), is well known (see, for instance, [@GEPM]). It is thus natural to have a look at the linear dynamics of random sequences \((T_n(.))_{n \geq 1}\) composed of random products of the operators \(D\) and \(T_a\) with \(a \ne 0\). This is the aim of the following result.

In the case where \(m(A_1) = m(A_2) = 1/2\) and where \(\varphi_1\) and \(\varphi_2\) are two nonconstant functions of exponential type, both Conditions 26 and 27 are simultaneously satisfied if and only if \(\varphi_1 \varphi_2\) is nonconstant. Indeed, this comes from the fact that the image of a nonconstant entire function is dense in \(\mathbb{C}\). In the situation where \(\varphi_1 \varphi_2\) is constant and nonzero, we must have \(\varphi_1(z) = e^{\alpha z + \beta}\) for some \(\alpha \ne 0\) and \(\beta \in \mathbb{C}\). Indeed, suppose that \(\lvert \varphi_1(z) \rvert \leq M e^{A \lvert z \rvert}\) for every \(z \in \mathbb{C}\), where \(A,M > 0\). Since \(\varphi_1\) does not vanish on \(\mathbb{C}\), we can write \(\varphi_1 = e^{\varphi}\) with \(\varphi\) an entire function. We thus obtain that \(\operatorname{Re}(\varphi(z)) \leq \log(M) + A \lvert z \rvert\) for every \(z \in \mathbb{C}\), and thus \(\varphi\) is a polynomial of degree at most one.

In this latter case, we have that \(\varphi_1(D) = e^\beta \, T_\alpha\) and \(\varphi_2(D) = c \, e^{-\beta} \, T_{-\alpha}\) for some \(c \ne 0\). We treat a case in the situation where \(\varphi_1 \varphi_2\) is constant on \(\mathbb{C}\). This is again linked to the behavior of Birkhoff sums. This shows in particular that a random sequence \((T_n(.))_{n \geq 1}\) can be non-universal even if \(T(\omega)\) is hypercyclic for every \(\omega\in \mathbb{T}\).

We now study an example of random sequences \((T_n(.))_{n \geq 1}\) for which the operators \(T(\tau^i \omega),\, i \geq 0\), do not commute.

If \(\lambda \in \mathbb{C}\) is nonzero and if \(b \in \mathbb{C}\), we denote by \(T_{\lambda, b}\) the operator on \(H(\mathbb{C})\) defined by \(T_{\lambda, b} \, f = f (\lambda\, . + b)\). Let us remark that \[\begin{align} T_{\lambda,b}^n \, f = f(\lambda^n \, . + b \displaystyle \sum_{k=0}^{n-1} \lambda^k) \end{align}\] for every \(n \geq 1\).

5 Comments and open problems↩︎

We end this article with some comments and open questions related to our previous results.

But before that, it is worth noting that many of the results stated in this paper remain valid for any measure-preserving ergodic dynamical system. This is the case, for example, of Propositions 15, 24, 25, 32, 34, 38, as well as Theorems 27, 30, 31, 35, 36, 37, 40 and 42.

5.1 A general remark on the universality of the random sequence \((T_n(.))_{n \geq 1}\)↩︎

Let us observe that in all of our results, the sequence of operators \((T_n(\omega))_{n \geq 1}\) is either universal for almost every \(\omega\in \mathbb{T}\), or not universal for almost every \(\omega\in \mathbb{T}\). Most of the time, this comes from Birkhoff’s Theorem 10, or this comes from the fact that the sequence of Birkhoff sums satisfies a CLT. We in fact have the following \(0-1\) law.

5.2 Comments on random products of adjoints of multiplication operators↩︎

Let us consider the sequence \((T_n(\omega))_{n \geq 1}\) given by \[\label{eqcocycleadjointmultipliccomments} T(\omega) = \left\{ \begin{array}{ll} (M_{\phi_1})^* & if \quad \omega \in A_1 \\ (M_{\phi_2})^* & if \quad \omega \in A_2 \end{array}, \right.\tag{31}\] where \(\phi_1, \phi_2 \in H^\infty(\mathbb{D})\) are nonconstant functions, \(A_1, A_2\) are two disjoint Borel subsets of \([0,1)\) such that \(A_1 \cup A_2 = [0,1)\) and \(m(A_1) = m(A_2) =1/2\), and with \(\tau\) an ergodic measure-preserving transformation on \((\mathbb{T},m)\). For example, let us suppose that \(A_1 = [0,1/2)\) and \(A_2 = [1/2,1)\).

We know that, thanks to Theorem 27, if \(\phi_1 \phi_2\) is nonconstant and such that \((\phi_1 \phi_2)(\mathbb{D}) \cap \mathbb{T}\ne \emptyset\), then the random sequence \((T_n(.))_{n \geq 1}\) is universal. This situation thus leads to two cases that we should call limit cases: the case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{D}\) and the case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{C}\setminus \overline{\mathbb{D}}\). We have also studied some cases that lead to one of these two limit cases (Proposition 28 and Corollary 5). These two sub-cases can be seen as trivial situations of the two limit cases. We have also studied the situation where \(\phi_1 \phi_2\) is inner, which is included in the case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{D}\) (Theorems 30 and 31). We can sum up our work on the two limit cases with the following table.

Certain situations are missing in our work. For example, we don’t know if the sequence \((T_n(.))_{n \geq 1}\) is universal when \(\phi_1, \phi_2 \in H^\infty(\mathbb{D})\) are nonconstant such that \(\phi_1(\mathbb{D}) \subset \mathbb{D}\), \(\phi_2(\mathbb{D}) \subset \mathbb{C}\setminus \overline{\mathbb{D}}\), \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{D}\), \(\phi_1 \phi_2\) is nonconstant and not inner, and \(\lVert \phi_1 \rVert_\infty \lVert \phi_2 \rVert_\infty \geq 1\). However, for certain types of functions \(\phi_1\) and \(\phi_2\) in this last case, we can observe that the sequence \((T_n(.))_{n \geq 1}\) is not universal. This is the case of the following example.

Example 1 gives a nontrivial situation where the random sequence \((T_n(.))_{n \geq 1}\) is non-universal, and where the limit cases hold. But this example is quite simple because we could use the norm of the inverse of \(\lVert T_n(\omega) \rVert\) to prove the non-universality. Unfortunately, it is not always possible to do it in general, since it depends heavily on the expression of the functions \(\phi_1\) and \(\phi_2\). Also, let us observe that the functions \(\phi_1\) and \(\phi_2\) in Example 1 are both outer. More generally, in the limit case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{C}\setminus \overline{\mathbb{D}}\) and \(\phi_1 \phi_2\) nonconstant, both functions \(\phi_1\) and \(\phi_2\) are outer since \(1/\phi_1\) and \(1/\phi_2\) belong to \(H^\infty(\mathbb{D})\). In this case, the inner parts \(I_1\) and \(I_2\) of \(\phi_1\) and \(\phi_2\) respectively are constant, and we thus cannot use the subspaces \(K := \displaystyle \bigcup_{n \geq 1} (I_1^n H^2(\mathbb{D}))^\perp\) and \(K' := \displaystyle \bigcup_{n \geq 1} (I_2^n H^2(\mathbb{D}))^\perp\) as in the proof of Theorem 31 to apply the Universality Criterion. This also is what makes the limit case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{C}\setminus \overline{\mathbb{D}}\) hard to solve.

Regarding the limit case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{D}\) and \(\phi_1 \phi_2\) nonconstant, we believe that a good understanding of the following example will help us to solve the limit case \((\phi_1 \phi_2)(\mathbb{D}) \subset \mathbb{D}\).

The following problem is thus still open.

When \(A_1, A_2\) are two disjoint Borel subsets of \([0,1)\) such that \(A_1 \cup A_2 = [0,1)\), Theorems 31, 36 and 37 require the Birkhoff sums \((\mathbb{S}_n^{\tau}f(\omega))_{n \geq 1}\) associated to the function \(f = \mathbb{1}_{A_1} - \frac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}\) to satisfy \[\displaystyle \limsup_{n \to \infty} \mathbb{S}_n^{\tau}f(\omega) = \infty \quad \textrm{and} \quad \displaystyle \liminf_{n \to \infty} \mathbb{S}_n^{\tau}f(\omega) = -\infty\] for almost every \(\omega \in \mathbb{T}\), in order to make the second part of the Universality Criterion on the right inverses work. This raises the following question.

Corollaries 8, 9 and 10 hold only for certain intervals of \([0,1)\), since their proofs make use of the computation of the Fourier coefficients of the function \(f = \mathbb{1}_{A_1} -\tfrac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}\) in order to apply CLTs for the doubling map and for the irrational rotations. Thus, the following question is natural.

In the same vein, we have seen in Corollary 12 that for some irrational rotations, the random sequence \((T_n(.))_{n \geq 1}\) won’t be universal. Thus, the following question is also natural.

In our work, we only use two disjoint Borel subsets \(A_1\) and \(A_2\) of \([0,1)\) such that \(A_1 \cup A_2 = [0,1)\) and \(m(A_k) > 0\) for \(k = 1,2\). It is therefore natural to be interested in the case of more than two Borel subsets of \([0,1)\).

Obviously, we can imagine similar results to Proposition 34 and Theorem 35 in the context of Problem 48. Finally, one can also look at the linear dynamics of random products \((T_n(.))_{n \geq 1}\) when the operator \(T(\omega)\) is another type of operator on a space of holomorphic functions.

For example, it would be natural to consider composition operators for \(T_1\) and \(T_2\) on a space of holomorphic functions and to study the linear dynamics of the corresponding sequence \((T_n(.))_{n \geq 1}\) in the situation of Problem 49, as well as weighted shifts on \(\ell_p\)-spaces, since the linear dynamics of these operators is rather well known and of interest for many problems in linear dynamics.

5.3 The case of a non-ergodic transformation↩︎

Almost all of our previous results in this work hold for an ergodic measure-preserving transformation on \((\mathbb{T},m)\). Without the assumption on the ergodicity of the transformation, Birkhoff’s theorem (Theorem 10) still holds, but the limit is not necessarily constant. Thus, our previous main results won’t hold in this context, except Proposition 28, Corollary 5 and Propositions 32 and 38. In order to investigate the problem of universality of random sequences of products of operators \((T_n(.))_{n \geq 1}\) for non-ergodic transformations on \((\mathbb{T},m)\), it is natural to start the investigation with the case of a rational rotation.

Let \(\alpha = \frac{p}{q}\) be a rational number in \((0,1)\), with \(p\) and \(q\) coprime. Let \(\tau\) be the rotation with parameter \(\alpha\), which is non-ergodic. In this case, for \(f\) a real centered bounded function on \(\mathbb{T}\), the coboundary equation \(f = h - h \circ \tau\) has a solution \(h \in L^\infty(\mathbb{T})\) if and only if \[\begin{align} \label{equationrotrationellefin} \displaystyle \sum_{j=0}^{q-1} f(x + j \alpha) = 0 \end{align}\tag{32}\] for almost every \(x \in [0,1)\). In this case, the function \[h(x) := -\frac{1}{q} \displaystyle \sum_{j=0}^{q-1} (j+1) f(x + j \alpha)\] provides a bounded solution of the coboundary equation. Thus, when Equation 32 has a solution for \(f = \mathbb{1}_{A_1} - \frac{m(A_1)}{m(A_2)} \mathbb{1}_{A_2}\), the Birkhoff sums associated to the function \(f\) are bounded. In particular, Proposition 38 applies in this case.