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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}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1610.02641","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-10-09T07:59:43Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"94475802c30efb2b2559c72eb1db7d3469efac2981fb9d69814b916d7f506df8","abstract_canon_sha256":"b9967c185da0175ff35cb612cc78a30521da5295e8c8e3ee79b9719db082ad76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:47.841700Z","signature_b64":"TcenDtP/vlwF8KpkcgsNtV7Di4HytbJvXcgA+vB8vTW+LsvkIZ/AayRQXSx9sVZOxrRM2EGwuTYSGVnH0dX8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c640104295425214c9a4d6fa7b3d7a82f7a0c701040aa6536bc6b1ca7c40200d","last_reissued_at":"2026-05-18T00:36:47.841074Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:47.841074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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