If $f$ is piecewise $C^2$, since your map is piecewise expanding it has an ACIM. See Lasota and York (Trans AMS, 1973). 

See also the *[Chaos, Fractals, and Noise][1]* by A. Lasota and M. Mackey. One of the theorems in Ch. 6 might work for your maps.


  [1]: http://books.google.it/books?id=2xsb6iveF9QC&lpg=PA167&ots=c7HDvka-L-&dq=chaos%2520fractals%2520and%2520noise%2520asymptotically%2520stable%25206.2.2&pg=PA148#v=onepage&q=theorem%25206.2.2&f=false