{"paper":{"title":"Graph-directed systems and self-similar measures on limit spaces of self-similar groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.GR","authors_text":"Ievgen Bondarenko, Rostyslav Kravchenko","submitted_at":"2010-01-13T20:12:37Z","abstract_excerpt":"Let $G$ be a group and $\\phi:H\\to G$ be a contracting homomorphism from a subgroup $H<G$ of finite index. V.Nekrashevych [25] associated with the pair $(G,\\phi)$ the limit dynamical system $(\\lims,\\si)$ and the limit $G$-space $\\limGs$ together with the covering $\\cup_{g\\in G}\\tile\\cdot g$ by the tile $\\tile$. We develop the theory of self-similar measures $\\mu$ on these limit spaces. It is shown that $(\\lims,\\si,\\mu)$ is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile $\\tile$ has integer measure and we give an algorithmic way to compute it. In additio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.2291","kind":"arxiv","version":2},"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"}