Comparison and pure infiniteness of crossed products

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Let α : G ↷ X \alpha : G\curvearrowright X be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action α \alpha is topologically free and has no G G -invariant regular Borel probability measure on X X , dynamical comparison implies that the reduced crossed product of α \alpha is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and the uniform tower property. Under the hypothesis that the action α \alpha is exact and essentially free, we show that paradoxical comparison together with the uniform tower property implies that the reduced crossed product of α \alpha is purely infinite. As applications, we provide new results on pure infiniteness of reduced crossed products in which the underlying spaces are not necessarily zero dimensional. Finally, we study the type semigroups of actions on the Cantor set in order to establish the equivalence of almost unperforation of the type semigroup and comparison. This sheds light on a question arising in a paper of Rørdam and Sierakowski.