As Bleher, [B 92] observed the free flight vector of the planar, infinite horizon, periodic Lorentz process {Sn|n = 0, 1, 2, . . . } belongs to the nonstandard domain of attraction of the Gaussian law -actually with the √ n log n scaling. Our first aim is to establish his conjecture that, indeed, Sn √ n log n converges in distribution to the Gaussian law (a Global Limit Theorem). Here the recent method of Bálint and Gouëzel, [BG 06] helped us to essentially simplify the ideas of our earlier sketchy proof [SzV 04b]. Moreover, we can also derive a.) the local version of the Global Limit Theorem, b.) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally c.) the ergodicity of its infinite invariant measure.- ------------------------

control the probabilities that the Lorentz process S n in the moment of n th collision falls into a sequence of moderately increasing domain rather than into a domain of Prepared using etds.cls LLT for the Lorentz Process and Its Recurrence in the Plane 3 fixed size. These results, moreover, were restricted to the finite horizon case, i. e. to the case when there is no orbit without any collision.A novel -and surprising -approach appeared in 1998-1999, when independently Schmidt [Sch 98] and Conze [Con 99] were, indeed, able to deduce the recurrence from the global central limit theorem (CLT) of [BS 81] by adding (abstract) ergodic theoretic ideas. Their approach seems to be essentially restricted to the finite horizon case and to d = 2. Our main aim is to return to the probabilistic-dynamical approach and -still for the finite horizon case -we can first prove a true local central limit theorem (LCLT) for the planar Lorentz process S n .1.2. Statement of theorems As a matter of fact, beyond treating just the Lorentz process we are also able to obtain a LCLT in a much wider setup. Namely our LCLT is valid whenever Young obtains a CLT. Her systems, called in our paper as Young systems, are introduced in subsection 2.1. Roughly speaking, these are systems (X, T, ν)Remark Traditionally one formulates the LCLT for the absolutely continuous and for the arithmetic case separately. An advantage of our statement is that it is unified and beyond these two cases it also contains the mixed ones. Though for the absolutely continuous case it is slightly weaker than the LCLT for densities, nevertheless our variant, for instance, is still amply sufficient to treat recurrence properties.Turning to the Lorentz process, let us denote by (M, S R , µ) a two-dimensional dispersing billiard dynamical system with a finite horizon, the usual factor of the Lorentz process, where µ is the natural invariant probability measure (the Liouvilleone), and consider its Poincaré section (∂M, T, µ 1 ) (for formal definitions of billiards cf. section 5).In case one takes f as κ : ∂M → R 2 , the discrete free flight function of the planar Lorentz process, then this result combined with considerations of [KSz 85], and an asymptotic independence statement proved right after the main theorem immediately provide the recurrence of S n as well. It will be shown in section 5 that κ satisfies the conditions of the main theorem. Corollary 1.1. The planar Lorentz process with a finite horizon is almost surely recurrent. 1.3. Some history LCLT's for functions of a Markov chain were first obtained by Kolmogorov in 1955 using probabilistic ideas. Then, in 1957, Nagaev, [Nag 57]by using operator valued Fourier transforms and perturbation theory -could find a Prepared using etds.cls

The fundamental theorem (also called the local ergodic theorem) was introduced by Sinai and Chernov in 1987, see [S-Ch(1987)] and an improved version in [K-S-Sz(1990)]. It provides sufficient conditions on a phase point under which some neighborhood of that point belongs to one ergodic component. This theorem has been instrumental in many studies of ergodic properties of hyperbolic dynamical systems with singularities, both in 2-D and in higher dimensions. The existing proofs of this theorem implicitly use the assumption on the boundedness of the curvature of singularity manifolds. However, we found recently ([B-Ch-Sz-T(2000)]) that, in general, this assumption fails in multidimensional billiards. Here the fundamental theorem is established under a weaker assumption on singularities, which we call Lipschitz decomposability. Then we show that whenever the scatterers of the billiard are defined by algebraic equations, the singularities are Lipschitz decomposable. Therefore, the fundamental theorem still applies to physically important models -among others to hard ball systems, Lorentz gases with spherical scatterers, and Bunimovich-Reháček stadia. 452In several papers that appeared, singularities were assumed -either explicitly or implicitly -to consist of smooth 1-codim submanifolds of the phase space. Often, Vol. 3, 2002 Multi-Dimensional Semi-Dispersing Billiards 457

For billiards with a hyperbolic behavior, Fundamental Theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds. A "Transversal" Fundamental Theorem has recently been suggested by the present authors to prove global ergodicity (and then, as an easy consequence, the K-property) of semidispersing billiards, in particular, the global ergodicity of systems of N ^ 3 elastic hard balls conjectured by the celebrated Boltzmann-Sίnai ergodίc hypothesis. (In fact, the suggested "Transversal" Fundamental Theorem has been successfully applied by the authors in the cases N = 3 and 4.) The theorem generalizes the Fundamental Theorem of Chernov and Sinai that was really the fundamental tool to obtain local ergodicity of semi-dispersing billiards. Our theorem, however, is stronger even in their case, too, since its conditions are simpler and weaker. Moreover, a complete set of conditions is formulated under which the Fundamental Theorem and its consequences like the Zig-zag theorem are valid for general semi-dispersing billiards beyond the utmost interesting case of systems of elastic hard balls. As an application, we also give conditions for the ergodicity (and, consequently, the K-property) of dispersingbilliards. "Transversality" means the following: instead of the stable and unstable foliations occurring in the Chernov-Sinai formulation of the stable version of the Fundamental Theorem, we use the stable foliation and an arbitrary nice one transversal to the stable one.