  We set out some general criteria to prove the K-property, refining the
assumptions used in arXiv:1906.09315 for the flow case, and introducing the
analogous discrete-time result. We also introduce one-sided
$\lambda$-decompositions, as well as multiple techniques for checking the
pressure gap required to show the K-property. We apply our results to the
family of Ma\~n\'e diffeomorphisms and the Katok map. Our argument builds on
the orbit decomposition theory of Climenhaga and Thompson.
