Invariant measures for piesewise fractional linear maps

The first part deals with piecewise fractional linear maps with three branches. Given a map $T$ a map $S$ is called a related map if some branches of $T$ are replaced by a ‘flipped’ branch, namely a branch of $1-T$. The main question is if $T$ and $S$ have a common invariant measure. The short second part presents invariant measures of a new type.