On fair entropy of the tent family

The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues [13] 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} $-Hölder continuous on $ [\sqrt{2},2] $ and identify its best Hölder 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 [7], we give a formula of pointwise Hölder 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.