Open AccessExamples of $\mathcal{C}^r$ interval map with large symbolic extension entropy
Author(s) -
David Burguet
Publication year - 2010
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2010.26.873
Subject(s) - extension (predicate logic) , interval (graph theory) , bounded function , combinatorics , homoclinic orbit , mathematics , topological entropy , dimension (graph theory) , entropy (arrow of time) , symbolic dynamics , discrete mathematics , physics , pure mathematics , mathematical analysis , computer science , nonlinear system , programming language , quantum mechanics , bifurcation
20 pagesInternational audienceFor 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