{"paper":{"title":"An example of unbounded chaos","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bau-Sen Du","submitted_at":"2010-06-03T10:08:16Z","abstract_excerpt":"Let $\\phi(x) = |1 - \\frac 1x|$ for all $x > 0$. Then we extend $\\phi(x)$ in the usual way to become a continuous map from the compact topological (but not metric) space $[0, \\infty]$ onto itself which also maps the set of irrational points in $(0, \\infty)$ onto itself. In this note, we show that (1) on $[0, \\infty]$, $\\phi(x)$ is topologically mixing, has dense irrational periodic points, and has topological entropy $\\log \\lambda$, where $\\lambda$ is the unique positive zero of the polynomial $x^3 - 2x -1$; (2) $\\phi(x)$ has bounded uncountable {\\it invariant} 2-scrambled sets of irrational po"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.0604","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"}