In this chapter, we consider \(T: M \circlearrowleft \) where M is a compact Riemannian manifold without boundary (e.g. \(M=\mathbb T^d \cong \mathbb R^d/\mathbb Z^d\) , the d-dimensional torus), and T is a differentiable map. The normalized Riemannian or Lebesgue measure on M is denoted by \(\mu \) . We do not assume that \(\mu \) is invariant under T, but are interested in determining under what conditions there exists an invariant probability measure m in the Lebesgue measure class, i.e., admitting a density with respect to \(\mu \) .