Shearer’s inequality and the Infimum Rule

 Abstract:

We review subbadditivity properties of Shannon entropy, in particular, from the Shearer’€™s inequality we derive the €œinfimum rule†for actions of amenable groups. We briefly discuss applicability of the €œinfimum formula to actions of other groups. Then we pass to topological entropy of a cover. We prove Shearer’€™s inequality for disjoint covers and give counterexamples otherwise. We also prove that, for actions of amenable groups, the supremum over all open covers of the €œinfimum fomula gives correct value of topological entropy. Joint work with Tomasz Downarowicz and Bartosz Frej.