For many fibred systems the existence of an invariant measure can be proved but considerably less is known about the shape of the density. In this note various examples of invariant densities are discussed: Piecewise fractional linear maps with four branches and maps which are associated to continued fractions with increasing digits. There are ergodic maps with a non-integrable density which do not have an indifferent fixed point and maps such that the set of points which miss the digit $k=1$ has positive Lebesgue measure.
"Some Results on Invariant Measures for $1$-dimensional Maps." Tokyo J. Math. Advance Publication 1 - 18, 2021. https://doi.org/10.3836/tjm/1502179353