Is $h(y)=p(y)y+(1-p(y))f(y)$ a dipheomorphism?

Question

Given that $f$ is an $C^\infty$ dipheo, which is Anosov,conservative and $p$ is a bump function, can one show that \begin{equation}h(y)=p(y)y+(1-p(y))f(y)\end{equation} is a dipheomorphism?