Interval maps of given topological entropy and Sharkovskii's type

Abstract

It is known that the topological entropy of a continuous interval map $f$ is positive if and only if the type of $f$ for Sharkovskii's order is $2^d p$ for some odd integer $p\ge 3$ and some $d\ge 0$; and in this case the topological entropy of $f$ is greater than or equal to $\frac{\log\lambda_p}{2^d}$, where $\lambda_p$ is the unique positive root of $X^p-2X^{p-2}-1$. For every odd $p\ge 3$, every $d\ge 0$ and every $\lambda\ge\lambda_p$, we build a piecewise monotone continuous interval map that is of type $2^dp$ for Sharkovskii's order and whose topological entropy is $\frac{\log\lambda}{2^d}$. This shows that, for a given type, every possible finite entropy above the minimum can be reached provided the type allows the map to have positive entropy. Moreover, if $d=0$ the map we build is topologically mixing.