Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps (Q2821962)

Let \(f: [0,1]\to [0,1]\) be a piecewise expanding unimodal map of class \(C^{k+1}\) with \(k\geq 1\). A classical result of \textit{A. Lasota} and \textit{J. A. Yorke} [Trans. Am. Math. Soc. 186, 481--488 (1974; Zbl 0298.28015)]NEWLINE says that \(f\) admits a unique SRB measure \(\mu=\rho dx\) and the density \(\rho\) is of bounded variation. In this paper the authors study the regularity of \(\rho\). The following are the main results of this paper:NEWLINENEWLINE1) There is a sequence of functions \(\rho_0\), \(\rho_1\), \(\dotsc\), \(\rho_k\) of bounded variation such that \(\rho_0=\rho\) and for \(j<k\), \(\rho_j'=\rho_{j+1}\) almost everywhere.NEWLINENEWLINE2) (A) The set of points where \(\rho\) is non-differentiable has Hausdorff dimension zero.NEWLINENEWLINE(B) If a critical orbit is dense then the set of points where \(\rho\) is non-differential is uncountable.NEWLINENEWLINE(C) There is a set \(N\) with zero Hausdorff dimension such that \(\rho\) is \(k\) differentiable in the sense of Whitney on \([0,1]\setminus N\), that is, if \(\bar x \not\in N\), then NEWLINE\[NEWLINE \rho(x)-\rho(\bar x)=\sum_{m=1}^k\frac{\rho_m(\bar x)}{m!}(x-\bar x)^m+o((x-\bar x)^k). NEWLINE\]