On fair entropy of the tent family

TL;DR

This paper investigates the regularity of the fair entropy function for tent maps, proving it is 1/2-Hölder continuous and providing formulas for its pointwise exponents, revealing the derivative vanishes almost everywhere.

Contribution

It extends previous results by precisely characterizing the Hölder regularity of the fair entropy function for tent maps and deriving explicit formulas for pointwise exponents.

Findings

The fair entropy function is 1/2-Hölder continuous on [√2, 2].

The paper identifies the best Hölder exponent on subintervals of [√2, 2].

The derivative of the entropy function vanishes almost everywhere.

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.