Preprint D46/2007
Large deviations bound for semiﬂows over a non-uniformly expanding base

Vitor Araujo

**Keywords: **
non-uniform expansion | physical measures | hyperbolic times | large deviations | geometric lorenz flows | special flows

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.
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 Poincar? return map and the 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. We are also able to obtain exponentially fast escape rates from
subsets without full measure.