For uniformly hyperbolic diffeomorphisms, the large deviation principle was established in the late 1980s by Takahashi, Orey and Pelikan, Kifer, and Young. We show that the (level-2) large deviation principle for empirical means holds for every logistic map, in spite of the fact that the critical point is a serious obstruction to uniform hyperbolicity. In particular, the large deviation principle holds for the logistic maps without a physical measure found by Hofbauer and Keller, and leads to a somewhat paradoxical conclusion: averaged statistics hold, even for some systems without average asymptotics.
This is a joint work with Yong Moo Chung and Hiroki Takahasi.