Mathematics – Dynamical Systems
Scientific paper
2011-06-23
Mathematics
Dynamical Systems
23 pages, no figure
Scientific paper
Since the pioneering works of Jakobson and Benedicks and Carleson, it has been known that positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating "Sejour probabilities" of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks and Carleson, to build various induced Markov maps and associated towers, to which the absolutely continuous measures can be lifted. Considering these various lifts altogether enables us to obtain a control of recurrence, sufficient to establish a level 2 large deviation principle, with the absolutely continuous measures as references. This full result encompasses dynamics far from equilibrium, and thus significantly extends presently known local large deviations results for quadratic maps.
Chung Yong Moo
Takahasi Hiroki
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