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Abstract A probability measure-preserving action of a discrete amenable groupGis said to bedominantif it is isomorphic to a generic extension of itself. Recently, it was shown that for$$G = \mathbb {Z}$$, an action is dominant if and only if it has positive entropy and that for anyG, positive entropy implies dominance. In this paper, we show that the converse also holds for anyG, that is, that zero entropy implies non-dominance.
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We extend results of Richard Holley beyond the integer lattice to a large class of countable groups which includes free groups and all amenable groups: for nearest-neighbor interactions on the Cayley graphs of such groups, we show that a shift-invariant measure is Gibbs if and only if it is Glauber-invariant. Moreover, any shift-invariant measure converges weakly to the set of Gibbs measures when evolved under the corresponding Glauber dynamics. These results are proven using a notion of free energy density relative to a sofic approximation by homomorphisms, which avoids the boundary problems which appear when applying a standard free energy method in a nonamenable setting. We also show that any measure which minimizes this free energy density is Gibbs.more » « less
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An ergodic dynamical system $$\mathbf{X}$$ is called dominant if it is isomorphic to a generic extension of itself. It was shown by Glasner et al [On some generic classes of ergodic measure preserving transformations. Trans. Moscow Math. Soc. 82 (1) (2021), 15–36] that Bernoulli systems with finite entropy are dominant. In this work, we show first that every ergodic system with positive entropy is dominant, and then that if $$\mathbf{X}$$ has zero entropy, then it is not dominant.more » « less
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Abstract The f -invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen, who first used it to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only if their base measures have the same Shannon entropy. Bowen also showed that the f -invariant is a variant of sofic entropy; in particular, it is the exponential growth rate of the expected number of good models over a uniform random homomorphism. In this paper we present an analogous formula for the relative f -invariant and use it to prove a formula for the exponential growth rate of the expected number of good models over a random sofic approximation which is a type of stochastic block model.more » « less
