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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0911.5411","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-11-28T19:08:41Z","cross_cats_sorted":[],"title_canon_sha256":"d71e730a08ed311fce72380012431fb180a9e0c3653bae587b09b5a7d85b1978","abstract_canon_sha256":"8a708366688118444a82bdecf90978a5b6dab96bfdfc1003245619144252ebf3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:04.853914Z","signature_b64":"U/7gEIU8MQtuFgKvJb5nDWPPgNb9VZLucdmkwg4S/ubMAn+wqdFwkN9PUQ2XfiGrsAuJDglIwkBNR/gXPmczBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89800633b09e38514a5b0595190f0c3c9fe9a0dcbdbb0feb24f7c554cb4a36af","last_reissued_at":"2026-05-18T04:18:04.853281Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:04.853281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $a\\in I$. 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