Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space ${\mathbb{P}}^k$, k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number ${\kappa }_f>0$ such that if ϕ: J → ℝ is a Hölder continuous function with $sup\left(\varphi \right)-inf\left(\varphi \right)<{\kappa }_f$, then ϕ admits a unique equilibrium state ${\mu }_{\varphi }$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f,{\mu }_{\varphi })$ is K-mixing, whence ergodic. Proving...