On fair entropy of the tent family

Abstract

<p style='text-indent:20px;'>The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues [<xref ref-type="bibr" rid="b13">13</xref>] recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy <inline-formula><tex-math id="M1">\begin{document}$ h(a) $\end{document}</tex-math></inline-formula> of the tent map <inline-formula><tex-math id="M2">\begin{document}$ f_a $\end{document}</tex-math></inline-formula>, as a function of the parameter <inline-formula><tex-math id="M3">\begin{document}$ a = \exp(h_{top}(f_a)) $\end{document}</tex-math></inline-formula>, is continuous and strictly increasing on <inline-formula><tex-math id="M4">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula>. In this short note, we extend the last result and characterize regularity of the function <inline-formula><tex-math id="M5">\begin{document}$ h $\end{document}</tex-math></inline-formula> precisely. We prove that <inline-formula><tex-math id="M6">\begin{document}$ h $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula>-Hölder continuous on <inline-formula><tex-math id="M8">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula> and identify its best Hölder exponent on each subinterval of <inline-formula><tex-math id="M9">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula>. On the other hand, parallel to a recent result on topological entropy of the quadratic family due to Dobbs and Mihalache [<xref ref-type="bibr" rid="b7">7</xref>], we give a formula of pointwise Hölder exponents of <inline-formula><tex-math id="M10">\begin{document}$ h $\end{document}</tex-math></inline-formula> at parameters chosen in an explicitly constructed set of full measure. This formula particularly implies that the derivative of <inline-formula><tex-math id="M11">\begin{document}$ h $\end{document}</tex-math></inline-formula> vanishes almost everywhere.