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February  2008, 22(1&2): 247-325. doi: 10.3934/dcds.2008.22.247

Fluctuations of ergodic sums for horocycle flows on $\Z^d$--covers of finite volume surfaces

François Ledrappier 1, and  Omri Sarig 2,

1. 

Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame IN 46556-4618, United States
2. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received  September 2007 Published  June 2008

We study the almost sure asymptotic behavior of the ergodic sums of $L^1$--functions, for the infinite measure preserving system given by the horocycle flow on the unit tangent bundle of a $\Z^d$--cover of a hyperbolic surface of finite area, equipped with the volume measure. We prove rational ergodicity, identify the return sequence, and describe the fluctuations of the ergodic sums normalized by the return sequence. One application is a 'second order ergodic theorem': almost sure convergence of properly normalized ergodic sums, subject to a certain summability method (the ordinary pointwise ergodic theorem fails for infinite measure preserving systems).
Keywords: ergodic theorems., horocycle flow, geometrically infinite.
Mathematics Subject Classification: Primary: 37A17, 58F17; Secondary: 37A4.
Citation: François Ledrappier, Omri Sarig. Fluctuations of ergodic sums for horocycle flows on $\Z^d$--covers of finite volume surfaces. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 247-325. doi: 10.3934/dcds.2008.22.247
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