Smoothness of invariant density for expanding transformations in higher dimensions (Q1566582)

For an expanding map \(T:\Omega\to\Omega\) with the property that the region \(\Omega\subset{\mathbb R}^d\) has a partition into finitely many closed regions, each with piecewise \(C^2\) boundaries of finite \((n-1)\)-dimensional volume, and on each partition \(T\) restricts to a \(C^k\) map, \(k\geq 2\), it is shown here that the absolutely continuous \(T\)-invariant measure on \(\Omega\) has density in \(C^{k-2}\).