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In particular, if g is a piecewise-analytic map preserving mu then there is an open g-invariant set U supporting mu such that g is piecewise-linear on U and has slopes which are rational powers of a. In a similar vein, for mu as above, if c is another integer and b,c are not powe"},"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":"1008.3548","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-08-20T18:14:20Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"9eeda87f9101eaf60c53767de27441a078cdac785a6dcaf2186a89e2c7034ddf","abstract_canon_sha256":"cf5e4b9dd36088a08d98773ffa2ac4f4e4a241a248bc6760c2c09a679afb044a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:15.487121Z","signature_b64":"je4VXsmmVU1gddJU5nfanvNa4PIdpa4bIJyFbSYfJkZiNm0yDvoktXCBn9O2lapmnnQDOOZ5W9wUZ75WNo95Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b06a0c1d19c5b836daa9b0477bb0a0b30fe7f9a8051fdb7744421b00d21b92f","last_reissued_at":"2026-05-18T02:42:15.486494Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:15.486494Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric rigidity of times-m invariant measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DS","authors_text":"Michael Hochman","submitted_at":"2010-08-20T18:14:20Z","abstract_excerpt":"Let b be an integer and mu a probability measure on [0,1] which is invariant and ergodic multiplication by b mod 1, and 0<dim(mu)<1. 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