**abstract:**
After recalling what is an ASIP and how it can appear in dynamics, we will discuss and motivate the
following joint result with Magnus Aspenberg and Tomas Persson:
Consider the quadratic family T_{a}(x) = a x (1 - x), for x in 0, 1 and parameters a in (2,4). For any

transversal Misiurewicz parameter b, we find a positive measure subset Omega of mixing Collet- Eckmann parameters such that for any Holder function f with nonvanishing autocorrelation for b, the

functions f_{a}(T_{a}^{{k}}(1*2)) (where f _{a} is a suitable normalisation of f) for the normalised Lebesgue
measure on a positive measure subset of Omega (depending on f) satisfy an ASIP.*

Mon 5 Jun, 10:00 - 11:00, Aula Dini

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