Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 873 - 899, Volume 26, Issue 3, March 2010      doi:10.3934/dcds.2010.26.873

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David Burguet - CMLS - CNRS UMR 7640, Ecole Polytechnique, 91128 Palaiseau Cedex, France (email)

Abstract: 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.

Keywords:  Entropy, Symbolic extension.
Mathematics Subject Classification:  Primary: 37E05, 37A35; Secondary: 37B10.

Received: April 2009;      Revised: October 2009;      Available Online: December 2009.