Is there an invariant measure absolutely continuous wrt to the lebesgue measure for the map f

Question

Let $f:[0,1]\rightarrow[0,1]$ where $f(x)=x/2$ $(1-x)$, and let $\lambda$ be the lebesgue measure on [0,1]. Is there a probability measure $\mu$ that is invariant and absolutely continuous wrt to the lebesgue measure.

I had 2 ideas for possible measures $\mu() = \int_{} \lambda$ and $\mu()=1/ \sum_{=0}^{−1} ^{}_{*}()$, where $f_{}^{∗} \lambda ()=\lambda(^{-}())$ however I am not sure how to actually show these are invariant and absolutely continuous wrt to the lebesgue measure.

Answer 1

I think there is no such measure. Hint: what happens if you apply f over and over?