Entropy of Cohen-Lenstra measures: the u-aspect

2025/01/16 by Artane Siad, Siad, Artane
Mathematics · #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Approximation Theory and Sequence Spaces

paper · pdf · doi:10.48550/arxiv.2501.09222

Abstract

Let \rm H(ν^u_{\rm CL}) be the entropy of the Cohen-Lenstra measure on finite abelian p-groups associated to an integral unit-rank 0 ≤ u ∈ ℕ. In this note, we show that 0 < \rm H(ν^u_{\rm CL}) < ∞ for all u, \rm H(ν^u_{\rm CL}) is a strictly decreasing function of u ≥ 0, and \rm H(ν^u_{\rm CL}) \xrightarrowu → ∞ 0. In particular, this shows that the groupoid measure is an entropy maximizer in the class of Cohen-Lenstra measures of varying integral unit-rank on finite abelian p-groups.

Related

• Ergodic measures in minimal group actions with finite topological sequence entropy
• Entropy of Schur-Weyl Measures
• Entropy Density of Uniquely Ergodic Measures for Full Shifts over Amenable Residually Finite Groups
• On groups, slow heat kernel decay yields Liouville property and sharp entropy bounds
• The Fried average entropy and slow entropy for actions of higher rank abelian groups