Existence of Large deviations rate function for any $S$-unimodal map

Abstract

For an arbitrary negative Schwarzian unimodal map with non-flat critical point, we establish the level-2 Large Deviation Principle (LDP) for empirical distributions. We also give an example of a multimodal map for which the level-2 LDP does not hold.

47 pages, 2 figures, to appear in Ergodic Theory and Dynamical Systems