{"paper":{"title":"Typical points for one-parameter families of piecewise expanding maps of the interval","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniel Schnellmann","submitted_at":"2009-11-28T19:08:41Z","abstract_excerpt":"Let $I\\subset\\mathbb{R}$ be an interval and $T_a:[0,1]\\to[0,1]$, $a\\in I$, a one-parameter family of piecewise expanding maps such that for each $a\\in I$ the map $T_a$ admits a unique absolutely continuous invariant probability measure $\\mu_a$. We establish sufficient conditions on such a one-parameter family such that a given point $x\\in[0,1]$ is typical for $\\mu_a$ for a full Lebesgue measure set of parameters $a$, i.e. $$ \\frac{1}{n}\\sum_{i=0}^{n-1}\\delta_{T_a^i(x)} \\overset{\\text{weak-}*}{\\longrightarrow}\\mu_a,\\qquad\\text{as} n\\to\\infty, $$ for Lebesgue almost every $a\\in I$. In particular"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.5411","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"}