Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Quenched CLT for random toral automorphism

Pages: 331 - 348, Volume 24, Issue 2, June 2009      doi:10.3934/dcds.2009.24.331

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Arvind Ayyer - Department of Physics, Rutgers University, 136 Frelinghuysen Road, Piscataway, NJ 08854, United States (email)
Carlangelo Liverani - Dipartimento di Matematica, II Università di Roma (Tor Vergata), Via della Ricerca Scientifica, 00133 Roma, Italy (email)
Mikko Stenlund - Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, United States (email)

Abstract: We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps.

Keywords:  Central Limit Theorem, iterated maps, transfer operator.
Mathematics Subject Classification:  60F05, 37D20, 82C41, 82D30.

Received: February 2008;      Revised: September 2008;      Available Online: March 2009.