Hausdorff and conformal measures for weakly expanding maps of the interval (Q1355056)

Weakly expanding maps of the interval \([0,1]\), except for some technical conditions caused by discontinuities, are defined as piecewise differentiable and monotone maps \(f\) which are expanding except in finitely many parabolic points. Let \(A\) be a closed invariant subset of \([0,1]\). The first result is that there is a unique \(z_A\) so that the pressure of \(-t\log| f'|\) is \(>0\) for \(t< z_a\) and \(\leq 0\) otherwise. Then there are \(z_A\)-conformal measures, which may be nonatomic or atomic (in the conformal case they are always nonatomic [\textit{J. Aaronson}, \textit{M. Denker} and \textit{M. Urbański}, Trans. Am. Math. Soc. 337, No. 2, 495-548 (1993; Zbl 0789.28010)]). Moreover, it is shown that \(z_A\) is the Hausdorff dimension of \(A\), provided \(z_A>0\). In the conformal case, such results are obtained in [\textit{M. Denker} and \textit{M. Urbański}, J. Lond. Math. Soc., II. Ser. 43, No. 1, 107-118 (1991; Zbl 0734.28007)], though the proofs are different for maps of the interval.