{"paper":{"title":"Quantitative Logarithmic Equidistribution of the Crucial Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Kenneth Jacobs","submitted_at":"2015-07-13T13:50:06Z","abstract_excerpt":"Let $K$ be a algebraically closed field of characteristic 0 that is complete with respect to a non-Archimedean absolute value. Let $\\phi\\in K(z)$ with $\\textrm{deg}(\\phi)\\geq 2$. In this paper we establish uniform logarithmic equidistribution of the crucial measures $\\nu_{\\phi^n}$ attached to the iterates of $\\phi$. These measures were introduced by Rumely in his study of the Minimal Resultant Locus of $\\phi$. Our equidistribution result comes from a bound on the diameter of points in $\\textrm{supp}(\\nu_{\\phi^n})$ that depends only on $n$ and $\\phi$. We also show that the sets $\\textrm{MinResL"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03460","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}