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

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  • 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.
    Mathematics Subject Classification: Primary: 37E05, 37A35; Secondary: 37B10.


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