{"paper":{"title":"Critical exponents of induced Dirichlet forms on self-similar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Ka-Sing Lau, Shi-Lei Kong","submitted_at":"2016-12-06T08:52:16Z","abstract_excerpt":"In a previous paper [arXiv:1604.05440], we studied certain random walks on the hyperbolic graphs $X$ associated with the self-similar sets $K$, and showed that the discrete energy ${\\mathcal E}_X$ on $X$ has an induced energy form ${\\mathcal E}_K$ on $K$ that is a Gagliardo-type integral. The domain of ${\\mathcal E}_K$ is a Besov space $\\Lambda^{\\alpha, \\beta/2}_{2,2}$ where $\\alpha$ is the Hausdorff dimension of $K$ and $\\beta$ is a parameter determined by the \"return ratio\" of the random walk. In this paper, we study the functional relationship of ${\\mathcal E}_X$ and ${\\mathcal E}_K$. In pa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01708","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"}