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

David Burguet (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

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 \cite{Dow} 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 prove 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.

Mots clés: entropie; extensions symboliques

Domaines: Systèmes dynamiques [math.DS]

David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete and Continuous Dynamical Systems - Series A, 2010, 26 (3), pp.873-899. ⟨10.3934/dcds.2010.26.873⟩. ⟨hal-00370669⟩