The Entropy Function of an Invariant Measure

摘要

Given a countable relational language L. we consider probability measures on the space of L-structures with underlying set N that are invariant under the logic action. We study the growth rate of the entropy function of such a measure, defined to be the function sending n ε N to the entropy of the measure induced by restrictions to L-structures on (0,...,n - 1). When L has finitely many relation symbols, all of arity k ≥ 1. and the measure has a property called non-redundance, we show that the entropy function is of the form Cn~k + o(n~k). generalizing a result of Aldous and Janson. When k ≥ 2, we show that there are invariant measures whose entropy functions grow arbitrarily fast in o(n~k), extending a result of Hatami-Norine. For possibly infinite languages L. we give an explicit upper bound on the entropy functions of non-redundant invariant measures in terms of the number of relation symbols in L of each arity: this implies that finite-valued entropy functions can grow arbitrarily fast.