Mathematics – Dynamical Systems
Scientific paper
2006-07-29
Bull. Braz. Math. Soc. New Series 38(3), 335-376, 2007
Mathematics
Dynamical Systems
35 pages, 1 figure; revised the main theorem and corrected the proofs
Scientific paper
We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities or criticalities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/critical 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. Suspension semiflows model the dynamics of flows admitting cross-sections, where the dynamics of the base is given by the Poincar\'e return map and the roof function is the return time to the cross-section. The results are applicable in particular to semiflows modeling the geometric Lorenz attractors and the Lorenz flow, as well as other semiflows with multidimensional non-uniformly expanding base with non-flat singularities and/or criticalities under slow recurrence rate conditions to this singular/critical set.
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