I will discuss the problem of stability of pluricanonical maps. Even though it is a very natural question, very few results are known. I will show how some particular cases of this problem are related to an old conjecture of Fujita, usually called log spectrum conjecture. Roughly speaking, it says that the set of pseudo-effective thresholds of polarized pairs has only accumulation points from below. I will explain a proof of this conjecture using recent results in the minimal model program.