Conservative Exact Rational Maps of the Sphere

In this paper we investigate conditions for conservativity and exactness of d-to-1 endomorphisms. We prove that all rational maps of degree greater than 1 are either conservative and exact with respect to normalized Lebesgue measure on the sphere, or have an attracting property inside the Julia set. We provide examples of rational maps which can be shown to be conservative and exact by using the previous result which was not formerly known to have this property.