On fair entropy of the tent family

Abstract

The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy $h(a)$ of the tent map $f_a$, as a function of the parameter $a=\exp(h_{top}(f_a))$, is continuous and strictly increasing on $[\sqrt{2},2]$. In this short note, we extend the last result and characterize regularity of the function $h$ precisely. We prove that $h$ is $\frac{1}{2}$-H\"{o}lder continuous on $[\sqrt{2},2]$ and identify its best H\"{o}lder exponent on each subinterval of $[\sqrt{2},2]$. On the other hand, parallel to a recent result on topological entropy of the quadratic family due to Dobbs and Mihalache, we give a formula of pointwise H\"{o}lder exponents of $h$ at parameters chosen in an explicitly constructed set of full measure. This formula particularly implies that the derivative of $h$ vanishes almost everywhere.