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.
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