{"paper":{"title":"On the dimension of Furstenberg measure for $SL_2(R)$ random matrix products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DS","authors_text":"Boris Solomyak, Michael Hochman","submitted_at":"2016-10-09T07:59:43Z","abstract_excerpt":"Let $\\mu$ be a measure on $SL_{2}(\\mathbb{R})$ generating a non-compact and totally irreducible subgroup, let $\\chi>0$ denote its Lyapunov exponent, and let $\\nu$ be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if $\\mu$ is supported on finitely many matrices with algebraic entries, then \\[ \\dim\\nu=\\min\\{1,\\frac{h_{\\textrm{RW}}(\\mu)}{2\\chi}\\} \\] where $h_{\\textrm{RW}}(\\mu)$ is the random walk entropy of $\\mu$, and $\\dim$ denotes pointwise dimension. In particular, for every $\\delta>0$, there is a neighborhood $U$ of the identity in $SL_{2}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02641","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"}