The topological entropy versus level sets for interval maps (part II)

Jozef Bobok¹

¹ KM FSv ČVUT Thákurova 7 166 29 Praha 6, Czech Republic

Studia Mathematica, Tome 166 (2005) no. 1, pp. 11-27

Résumé

Let $f\colon\, [a,b]\to [a,b]$ be a continuous function of the compact real interval such that (i) $\mathop{\rm card} f^{-1}(y)\ge 2$ for every $y\in [a,b]$; (ii) for some $m\in\{\infty,2,3,\dots\}$ there is a countable set $L\subset [a,b]$ such that $\mathop{\rm card} f^{-1}(y)\ge m$ for every $y\in [a,b]\setminus L$. We show that the topological entropy of $f$ is greater than or equal to $\log m$. This generalizes our previous result for $m=2$.

Détail

DOI : 10.4064/sm166-1-2

Keywords: colon continuous function compact real interval mathop card every infty dots there countable set subset mathop card every setminus topological entropy greater equal log generalizes previous result