  Let $f: [0,1]\times [0,1] \setminus {1/2} \to [0,1]\times [0,1]$ be the
$C^\infty$ endomorphism given by $$f(x,y)=(2x- [2x], y+ c/|x-1/2|- [y+
c/|x-1/2|]),$$ where $c$ is a positive real number. We prove that $f$ is
topologically mixing and if $c>1/4$ then $f$ is mixing with respect to Lebesgue
measure. Furthermore we prove that the speed of mixing is exponential.

Dynamical Systems