Ergodic Theory and Dynamical Systems

Research Article

Convergence of moments for Axiom A and non-uniformly hyperbolic flows

IAN MELBOURNEa1 and ANDREI TÖRÖKa2a3

a1 Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: ism@math.uh.edu)

a2 Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA (email: torok@math.uh.edu)

a3 Institute of Mathematics of the Romanian Academy, PO Box 1-764, RO-70700 Bucharest, Romania

Abstract

In the paper, we prove convergence of moments of all orders for Axiom A diffeomorphisms and flows. The same results hold for non-uniformly hyperbolic diffeomorphisms and flows modelled by Young towers with superpolynomial tails. For polynomial tails, we prove convergence of moments up to a certain order, and give examples where moments diverge when this order is exceeded. Non-uniformly hyperbolic systems covered by our result include Hénon-like attractors, Lorenz attractors, semidispersing billiards, finite horizon planar periodic Lorentz gases, and Pomeau–Manneville intermittency maps.

(Received October 12 2010)

(Revised March 01 2011)

(Online publication June 10 2011)