Piecewise smooth maps with absolutely continuous invariant measures and uniformly scaled Markov partitions (Q2778800)

The paper under review continues a long line of work beginning with a celebrated paper by the author [\textit{M. V. Jakobson}, Commun. Math. Phys. 81, 39-88 (1981; Zbl 0497.58017)] where, for the logistic family \(\{g_a\}_{a\in [0,4]}\), \(f_a(x)=ax(1-x)\), the existence of a positive Lebesgue measure set \(M\) of parameters \(a\) such that \(g_a\) has an absolutely continuous invariant measure for any \(a\in M\) was proved. The key idea behind this and subsequent works is that parameter values corresponding to Misiurewicz maps are density points of \(M\). NEWLINENEWLINENEWLINEIn the paper a different global approach is made. For some appropriate one-parameter families \(\{f_t\}_t\) of piecewise \(C^2\)-maps of the interval (the precise description involves a score of technical conditions, too complicated to be detailed here) the measure of the corresponding set \(M\) is estimated without assuming that \(t\) is close to some specific parameter value. Instead, the measure of a ``good'' subset of parameters of \(M\) is calculated using expansion, distortion and other properties of the family.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].