Are there any methods for finding an (infinite, absolutely continuous with respect to lebesgue measure) invariant measure $d\mu=f(u,v)dudv$ for something like the following?
$$ T:Q\to Q, \ Q=\{u^2+v^2\leq1, u\geq0,v\geq0\} $$ $$ T(u,v)= \left\{ \begin{array}{cc} \left(\frac{|u^2+(v-1/2)^2-1/4|}{u^2+v^2},\frac{|(u-1/2)^2+v^2-1/4|}{u^2+v^2}\right)&(u-1)^2+(v-1)^2\leq1,\\ \frac{1}{2}\left(\frac{|u^2+(v-1/2)^2-1/4|}{(u-1/2)^2+(v-1/2)^2},\frac{|(u-1/2)^2+v^2-1/4|}{(u-1/2)^2+(v-1/2)^2}\right)&(u-1)^2+(v-1)^2\geq1\\ \end{array} \right. $$
$T$ is continuous, piece-wise rational, and 4,5, or 6-to-1 (the domain is broken into seven regions, four of which get mapped onto all of $Q$ and three of which get mapped onto three of the regions).
Anyway, the details of my map aren't important. What I'm looking for is references or ideas for constructing an invariant measure. Thanks for any help.