KMS Spectra for Group Actions on Compact Spaces

Given a topologically free action of countable group G on compact metric space X, there is canonical correspondence between continuous 1-cocycles for this and diagonal 1-parameter groups automorphisms the reduced crossed product C $$^*$$ -algebra. The KMS spectrum defined as set inverse temperatures which exists state. We prove that possible spectra depend heavily nature acting G. For subexponential growth, we only are $$\{0\}$$ , $$[0,+\infty )$$ $$(-\infty ,0]$$ $${\mathbb {R}}$$ . certain wreath groups, amenable exponential any closed subset containing zero arises spectrum. Finally, nonamenable including with infinitely many generators, may arise. Besides uncovering surprising relation geometric theoretic properties spectra, our results provide two simple -algebras following universality property: (containing, resp. not zero)