@MISC{Griffin_limittheorems, author = {Jory Griffin and Jens Marklof}, title = {LIMIT THEOREMS FOR SKEW TRANSLATIONS}, year = {} }

Share

OpenURL

Abstract

Abstract. Bufetov, Bufetov-Forni and Bufetov-Solomyak have recently proved limit theorems for translation flows, horocycle flows and tiling flows, respectively. We present here analogous results for skew translations of a torus. §1. Bufetov [1] has recently established limit theorems for translation flows on flat surfaces and, in joint work with Forni [2], for horocycle flows on compact hyper-bolic surfaces. Bufetov and Solomyak [3] have proved analogous theorems for tiling flows. The striking feature of these results is that the central limit theorem (a com-mon feature of many “chaotic ” dynamical systems) fails. The limit laws are instead characterised in terms of the corresponding renormalisation dynamics. The pur-pose of the present note is to point out that an analogous result holds in the simpler case of skew translations of the torus. Our approach uses the modular invariance of theta sums as in [14, 15, 16]. The limit theorems proved in these papers may be in-terpreted as limit theorems for random, rather than fixed, skew translations, cf. also [9, 10, 11] and [6]. The approach by Flaminio and Forni [8] developed for nilflows may yield an alternative route to our main result, but we have not explored this further. Another class of systems which exhibit non-normal limit laws are random translations of the torus; we refer the reader to Kesten’s Cauchy limit theorem for circle rotations [12, 13] and the recent higher-dimensional generalisations by Dolgo-pyat and Fayad [4, 5]; see also Sinai and Ulcigrai [17] for limit theorems for circle rotations with non-integrable test functions. §2. Given α ∈ R, a skew translation of the torus T2 = R2/Z2 is defined by (1) Λα: T2 → T2, (p, q) 7 → (p + α, q + p). The nth iterate of this map is (2) Λnα(p, q) = p + nα, q + np + n(n − 1) 2