Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy

For any integer $r\geq2$ and any real $\epsilon>0$, we construct an explicit example of $\mathcal{C}^r$ interval map $f$ with symbolic extension entropy $h_{sex}(f)\geq\frac{r}{r-1}\log\||f'\||_{\infty}-\epsilon$ and $\||f'\||_{\infty}\geq 2$. T.Downarawicz and A.Maass [10] proved that for $\mathcal{C}^r$ interval maps with $r>1$, the symbolic extension entropy was bounded above by $\frac{r}{r-1}\log\||f'\||_{\infty}$. 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.