Examples of C^r interval map with large symbolic extension entropy

For any integer r ≥ 2 and any real ε > 0, we construct an explicit example of C^r interval map f with symbolic extension entropy h_sex(f) ≥ r/r-1 log ||f′||∞-ε and ||f′||∞ ≥ 2. T.Downarawicz and A.Maass [10] proved that for C^r interval maps with r > 1, the symbolic extension entropy was bounded above by r/r-1 log ||f′|infin; So our example proves this bound is sharp. Similar examples had been already built by T.Downarowicz and S.Newhouse for diffeomorphisms in higher dimension by using generic arguments on homoclinic tangencies.