Large deviation principle for S-unimodal maps with flat critical points

We study a topologically exact, negative Schwarzian unimodal map without neutral periodic points whose critical point is non-recurrent and flat. Assuming that the critical order is polynomial or logarithmic, we establish the large deviation principle and provide a partial description of the minimizers of the rate function. We apply our main results to a certain parametrized family of unimodal maps in the same topological conjugacy class, and determine the sets of minimizers.