Large deviation bound for semiflows over a piecewise expanding base
by
Vitor Araújo
UFRJ, Brazil

We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a one-dimensional piecewise expanding base transformation with non-flat singularities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular set of the base map. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the semiflow, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast as time goes to infinity.

The arguments need the base transformation to exhibit exponential slow recurrence to the singular set which, in all known examples, implies exponential decay of correlations.

Suspension semiflows model the dynamics of flows admitting cross-sections, where the dynamics of the base is given by the Poincaré return map and the roof function is the return time to the cross-section. The results are applicable to semiflows modeling the geometric Lorenz attractors and the Lorenz flow.

Date received: July 3, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavj-91.