{"paper":{"title":"Cycle Doubling, Merging And Renormalization in the Tangent Family","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Linda Keen, Tao Chen, Yunping Jiang","submitted_at":"2017-08-05T19:05:20Z","abstract_excerpt":"In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\\{ T_t(z)=i t\\tan z\\}_{0< t\\leq \\pi}$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as $t$ increases we find single instances of \"period quadrupling\", \"period splitting\" and standard \"period doubling\"; there follows a general pattern of \"period merging\" where two attracting cycles of period $2^n$ \"merge\" into one attracting cycle of period $2^{n+1}$, and \"cycle doublin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.01808","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"}