A TEMPORAL CENTRAL LIMIT THEOREM FOR REAL-VALUED COCYCLES OVER ROTATIONS MICHAEL BROMBERG AND CORINNA ULCIGRAI Abstract. We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by α where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point β. When α is badly approximable and β is badly approximable with respect to α, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D.Dolgopyat and O.Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when α is quadratic irrational, β is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig (Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use renormalization, but in order to treat irrational values of β, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains. 1. introduction and results The main result of this article is a temporal distributional limit theorem (see Section 1.1 below) for certain functions over an irrational rotation (Theorem 1.1 below). In order to introduce and motivate this result, in the first section, we first define two types of distributional limit theorems in the study of dynamical systems, namely spatial and temporal. Temporal limit theorems in dynamics are the focus of the recent paper [19] by D. Dolgopyat and O. Sarig; we refer the interested reader to [19] and the references therein for a comprehensive introduction to the subject, as well as for a list of examples of dynamical systems known up to date to satisfy temporal distributional limit theorems. In section 1.2 we then focus on irrational rotations, which are one of the most basic examples of low complexity dynamical systems, and recall previous results on temporal limit theorems for rotations, in particular Beck’s temporal CLT. Our main result in stated in section 1.3, followed by a description of the structure of the rest of the paper in section 1.4. 1.1. Temporal and Spatial Limits in dynamics. Distributional limit theorems appear often in the study of dynamical systems as follows. Let X be a complete separable metric space, m a Borel probability measure on X and denote by B is the Borel σ-algebra on X. Let T : X ! X be a Borel measurable map. We call the quadruple (X; B; m; T ) a probability preserving dynamical system and assume that T is ergodic with respect to m. Let f : X ! R be a Borel measurable function and set n−1 X arXiv:1705.06484v2 [math.DS] 22 May 2017 k Sn (T; f; x) := f ◦ T (x) k=0 We will also use the notation Sn (x), or Sn (f; x) instead of Sn (T; f; x), when it is clear from the context, what th is the underlying transformation or function. The function Sn(x) is called (the n ) Birkhoff sum (or also ergodic sum) of the function f over the transformation T . The study of Birkhoff sums, their growth and their behavior is one of the central themes in ergodic theory. When the transformation T is ergodic with respect to m; by the 1 R Birkhoff ergodic theorem, for any f 2 L (X; m), for m-almost every x 2 X, Sn(f; x)=n converges to fdm as n n grows; equivalently, one can say that the random variables Xn := f ◦ T where x is chosen randomly according to the measure m, satisfy the strong law of large numbers. We will now introduce some limit theorems which allow to study the error term in the Birkhoff ergodic theorem. The function f is said to satisfy a spatial distributional limit theorem (spatial DLT) if there exists a random variable with no atoms Y , and sequences of constants An;Bn 2 R, Bn ! 1, such that the random variables 1 2 MICHAEL BROMBERG AND CORINNA ULCIGRAI Sn(x)−An , where x is chosen randomly according to the measure m, converge in distribution to Y . In this case we Bn write S − A n n −!dist Y: Bn It is the case that many hyperbolic dynamical systems, under some regularity conditions on f, satisfy a spatial DLT with the limit being a Gaussian random variable. In the cases that we have in mind, the rate of mixing of n the sequence of random variables Xn := f ◦ T is sufficiently fast, in order for them to satisfy the Central Limit Theorem (CLT). On the other hand, in many classical examples of dynamical systems with zero entropy, for which n the random variables Xn := f ◦ T are highly correlated, the spatial DLT fails if f is sufficiently regular. For example, this is the case when T is an irrational rotation and f is of bounded variation. Perhaps surprisingly, many examples of dynamical systems with zero entropy satisfy a CLT when instead of averaging over the space X, one considers the Birkhoff sums Sn (x0) over a single orbit of some fixed initial condition x0 2 X. Fix an initial point x0 2 X and consider its orbit under T . One can define a sequence of occupation measures on R by 1 ν (F ) := # f1 ≤ k ≤ n : S (x ) 2 F g n n k 0 for every Borel measurable F ⊂ R. One can interpret the quantity νn (F ) as the fraction of time that the Birkhoff sums Sk (x0) spend in the set F , up to time n. Let Yn be a sequence of random variables distributed according to νn. We say that the pair (T; f) satisfies a temporal distributional limit theorem (temporal DLT) along the orbit of x0, if there exists a random variable with no atoms Y , and two sequences An 2 R and Bn ! 1 such that (Yn − An)=Bn converges in distribution to Y . In other words, the pair (T; f) satisfies a temporal DLT along the orbit of x0, if 1 S (x ) − A # 1 ≤ k ≤ n : k 0 n < a n−!!1 P rob (Y < a) n Bn for every a 2 R. If the limit Y is a Gaussian random variable, we call this type of behavior a temporal CLT along the orbit of x0. Note, that this type of result may be interpreted as convergence in distribution of a sequence of normalized random variables, obtained by considering the Birkhoff sums Sk (x0) for k = 1; ::; n and choosing k randomly uniformly. 1.2. Beck’s temporal CLT and its generalizations. One example of occurrence of a temporal CLT in dynamical systems with zero entropy is the following result by Beck, generalizations of which are the main topic of this paper. Let us denote by Rα the rotation on the interval T = R n Z by an irrational number α 2 R, given by Rα(x) = x + α mod 1: Let fβ : T ! R be the indicator of the interval [0; β) where 0 < β < 1, rescaled to have mean zero with respect to the Lebesgue measure on T, namely fβ(x) = 1[0,β) (x) − β: The sequence fSng of random variables given by the Birkhoff sums Sn(x) = Sn (Rα; fβ; x) , where x is taken uniformly with respect to the Lebesgue measure, is sometimes referred to in the literature as the deterministic random walk driven by an irrational rotation (see for example [6]). Beck proved [7, 8] that if α is a quadratic irrational, and β is rational, then the pair (Rα; fβ) satisfies a temporal DLT along the orbit of x0 = 0. More precisely, he shows that there exist constants C1 and C2 such that for all a; b 2 R, a < b b Z 2 1 Sk (Rα; fβ; 0) − C1 log n 1 − x # 1 ≤ k ≤ n : p 2 [a; b] ! p e 2 dx: n C2 log n 2π a Beck’s CLT relates to the theory of discrepancy in number theory as follows. If α 2 R is irrational, by unique ergodicity of the rotation Rα, the sequence of fjαg is equidistributed modulo one, i.e. in particular, for any β 2 [0; 1] if we set Nk(α; β) := # f0 ≤ j < k j 0 ≤ jα mod 1 < βg ; then Nk (α; β) =k converges to β, or, equivalently, Nk (α; β) = kβ + o(k). Discrepancy theory concerns the study of the error term in the expression Nk (α; β) = kβ + o(k). Beck’s result hence says that, when α is a quadratic A TEMPORAL CLT FOR COCYCLES OVER ROTATIONS 3 irrational and β is rational, the error term Nk (α; β) := Nk (α; β) − kβ , when k is chosen uniformly in f1; : : : ; ng, can be normalized so that it converges to the standard Gaussian distribution as n grows to infinity. Let us also remark that the Birkhoff sums in the statement of Beck’s theorem are related to the dynamics of the map Tfβ : T × R ! T × R, defined by Tfβ (x; y) = (Rα (x) ; y + fβ (x)) ; (x; y) 2 T × R; n n since one can see that the form of the iterates of Tf is Tf (x; y) = (Rα (x) ; y + Sn(fβ; x)).