{"paper":{"title":"Singular measures of circle homeomorphisms with two break points","license":"","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DS","authors_text":"Akhtam Dzhalilov, Dieter Mayer, Isabelle Liousse","submitted_at":"2007-07-24T11:35:06Z","abstract_excerpt":"Let $T_{f}$ be a circle homeomorphism with two break points $a_{b},c_{b}$ and irrational rotation number $\\varrho_{f}$. Suppose that the derivative $Df$ of its lift $f$ is absolutely continuous on every connected interval of the set $S^{1}\\backslash\\{a_{b},c_{b}\\}$, that $DlogDf \\in L^{1}$ and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\\neq1$. We prove that the unique $T_{f}$- invariant probability measure $\\mu_{f}$ is then singular with respect to Lebesgue measure $l$ on $S^{1}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.3528","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"}