{"paper":{"title":"A formula for the Entropy of the Convolution of Gibbs probabilities on the circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.DS","authors_text":"Artur O. Lopes","submitted_at":"2017-02-10T11:20:28Z","abstract_excerpt":"Consider the transformation $T:S^1 \\to S^1$, such that $T(x)=2\\, x$ (mod 1), and where $S^1$ is the unitary circle. Suppose $J:S^1 \\to \\mathbb{R}$ is Holder continuous and positive, and moreover that, for any $y\\in S^1$, we have that $\\sum_{x\\,\\,\\text{such that}\\,\\,\\, T(x)= y} \\, J(x)=1.$\n  We say that $\\rho$ is a Gibbs probability for the Holder continuous potential $\\log J$, if $\\mathcal{L}_{\\log J}^* \\,(\\rho)=\\rho ,$ where $\\mathcal{L}_{\\log J}$ is the Ruelle operator for $\\log J$. We call $J$ the Jacobian of $\\rho$.\n  Suppose $\\nu=\\mu_1*\\mu_2$ is the convolution of two Gibbs probabilities "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03134","kind":"arxiv","version":3},"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"}