Ergodic Theory and Dynamical Systems



Local limit theorem for the Lorentz process and its recurrence in the plane


DOMOKOS SZÁSZ a1 and TAMÁS VARJÚ a1
a1 Budapest University of Technology, Mathematical Institute and Center of Applied Mathematics, Budapest, Egry J. u. 1 Hungary H-1111 (e-mail: szasz@renyi.hu, kanya@math.bme.hu)

Article author query
szasz d   [Google Scholar] 
varju t   [Google Scholar] 
 

Abstract

For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Young's axioms (Lai-Sang Young, Ann. Math. 147 (1998), 585–650), which imply exponential decay of correlations and the central limit theorem (CLT), a local CLT is proven. In fact, a unified version of the local CLT is found, covering, among others, the absolutely continuous and arithmetic cases. For planar Lorentz process with a finite horizon, this result implies (a) a local CLT and (b) recurrence. For the latter case (d = 2, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt (C. R. Acad. Sci. Paris Ser. 1 Math. 372(9) (1998), 837–842) and J.-P. Conze (Ergod. Th. & Dynam. Sys. 19(5) (1999), 1233–1245) could already establish recurrence.

(Received April 30 2002)
(Revised August 3 2003)