{"paper":{"title":"Random Iteration of Rational Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"David Simmons","submitted_at":"2013-03-11T22:20:40Z","abstract_excerpt":"It is a theorem of Denker and Urba\\'nski ('91) that if $T:\\mathbb C\\to\\mathbb C$ is a rational map of degree at least two and if $\\phi:\\mathbb C\\to\\mathbb R$ is H\\\"older continuous and satisfies the \"thermodynamic expanding\" condition $P(T,\\phi) > \\sup(\\phi)$, then there exists exactly one equilibrium state $\\mu$ for $T$ and $\\phi$, and furthermore $(\\mathbb C,T,\\mu)$ is metrically exact. We extend these results to the case of a holomorphic random dynamical system on $\\mathbb C$, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bog"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2705","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"}