{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:22OSRO5TM5RLE3B7WBEF2RW6GH","short_pith_number":"pith:22OSRO5T","canonical_record":{"source":{"id":"1712.08882","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-12-24T06:45:34Z","cross_cats_sorted":[],"title_canon_sha256":"257f630627f3fd64bf1e2fc80d174156797eeee227e31f939625021874661fe1","abstract_canon_sha256":"31bfb62bad9f8d9fa134d24772fa4bb9686dee3567aa0464963f4170e09233db"},"schema_version":"1.0"},"canonical_sha256":"d69d28bbb36762b26c3fb0485d46de31c6cda50fbcb6c5c6cf55954968fe955b","source":{"kind":"arxiv","id":"1712.08882","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.08882","created_at":"2026-05-18T00:01:19Z"},{"alias_kind":"arxiv_version","alias_value":"1712.08882v2","created_at":"2026-05-18T00:01:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08882","created_at":"2026-05-18T00:01:19Z"},{"alias_kind":"pith_short_12","alias_value":"22OSRO5TM5RL","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"22OSRO5TM5RLE3B7","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"22OSRO5T","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:22OSRO5TM5RLE3B7WBEF2RW6GH","target":"record","payload":{"canonical_record":{"source":{"id":"1712.08882","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-12-24T06:45:34Z","cross_cats_sorted":[],"title_canon_sha256":"257f630627f3fd64bf1e2fc80d174156797eeee227e31f939625021874661fe1","abstract_canon_sha256":"31bfb62bad9f8d9fa134d24772fa4bb9686dee3567aa0464963f4170e09233db"},"schema_version":"1.0"},"canonical_sha256":"d69d28bbb36762b26c3fb0485d46de31c6cda50fbcb6c5c6cf55954968fe955b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:19.691533Z","signature_b64":"aC5eRIiONGkuDhy5YvVeNZBv9r3fhrZ5aSpHjS4iAih744HE2HmoqRzS4VZX2U/Qq3xzqOAbdrIlTpbp4WyoCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d69d28bbb36762b26c3fb0485d46de31c6cda50fbcb6c5c6cf55954968fe955b","last_reissued_at":"2026-05-18T00:01:19.691049Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:19.691049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.08882","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YJrlB/xt6IIlWmR/fhaswGqXSvhJteDXD9knY7yZPk4/N7aShX6W4jQlJ8KuppvtoimAt8OrJzz2lT6jpW84BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T02:03:51.223773Z"},"content_sha256":"6dc2c00d360c49a10e672530eb9b634b7826eed5a8bd7e1d1dbe9344c2c6f06b","schema_version":"1.0","event_id":"sha256:6dc2c00d360c49a10e672530eb9b634b7826eed5a8bd7e1d1dbe9344c2c6f06b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:22OSRO5TM5RLE3B7WBEF2RW6GH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Smooth symmetries of $\\times a$-invariant sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Michael Hochman","submitted_at":"2017-12-24T06:45:34Z","abstract_excerpt":"We study the smooth self-maps $f$ of $\\times a$-invariant sets $X\\subseteq[0,1]$. Under various assumptions we show that this forces $\\log f'(x)/\\log a\\in\\mathbb{Q}$ at many points in $X$. Our method combines scenery flow methods and equidistribution results in the positive entropy case, where we improve previous work of the author and Shmerkin, with a new topological variant of the scenery flow which applies in the zero-entropy case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08882","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Dv/CCL0TiyCSjKkaeECdC26rQpRvf6S4KqGfwk/FjxNA6h+2+0R92aF5uqwMEKYfy/KzgmH5PZdt65RtEJ+BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T02:03:51.224544Z"},"content_sha256":"bda16026a6314ffdb14fe646774fbfce76be8232fc359b545d8fe3d974d28ebe","schema_version":"1.0","event_id":"sha256:bda16026a6314ffdb14fe646774fbfce76be8232fc359b545d8fe3d974d28ebe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/22OSRO5TM5RLE3B7WBEF2RW6GH/bundle.json","state_url":"https://pith.science/pith/22OSRO5TM5RLE3B7WBEF2RW6GH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/22OSRO5TM5RLE3B7WBEF2RW6GH/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T02:03:51Z","links":{"resolver":"https://pith.science/pith/22OSRO5TM5RLE3B7WBEF2RW6GH","bundle":"https://pith.science/pith/22OSRO5TM5RLE3B7WBEF2RW6GH/bundle.json","state":"https://pith.science/pith/22OSRO5TM5RLE3B7WBEF2RW6GH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/22OSRO5TM5RLE3B7WBEF2RW6GH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:22OSRO5TM5RLE3B7WBEF2RW6GH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"31bfb62bad9f8d9fa134d24772fa4bb9686dee3567aa0464963f4170e09233db","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-12-24T06:45:34Z","title_canon_sha256":"257f630627f3fd64bf1e2fc80d174156797eeee227e31f939625021874661fe1"},"schema_version":"1.0","source":{"id":"1712.08882","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.08882","created_at":"2026-05-18T00:01:19Z"},{"alias_kind":"arxiv_version","alias_value":"1712.08882v2","created_at":"2026-05-18T00:01:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08882","created_at":"2026-05-18T00:01:19Z"},{"alias_kind":"pith_short_12","alias_value":"22OSRO5TM5RL","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"22OSRO5TM5RLE3B7","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"22OSRO5T","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:bda16026a6314ffdb14fe646774fbfce76be8232fc359b545d8fe3d974d28ebe","target":"graph","created_at":"2026-05-18T00:01:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the smooth self-maps $f$ of $\\times a$-invariant sets $X\\subseteq[0,1]$. Under various assumptions we show that this forces $\\log f'(x)/\\log a\\in\\mathbb{Q}$ at many points in $X$. Our method combines scenery flow methods and equidistribution results in the positive entropy case, where we improve previous work of the author and Shmerkin, with a new topological variant of the scenery flow which applies in the zero-entropy case.","authors_text":"Michael Hochman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-12-24T06:45:34Z","title":"Smooth symmetries of $\\times a$-invariant sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08882","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6dc2c00d360c49a10e672530eb9b634b7826eed5a8bd7e1d1dbe9344c2c6f06b","target":"record","created_at":"2026-05-18T00:01:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"31bfb62bad9f8d9fa134d24772fa4bb9686dee3567aa0464963f4170e09233db","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-12-24T06:45:34Z","title_canon_sha256":"257f630627f3fd64bf1e2fc80d174156797eeee227e31f939625021874661fe1"},"schema_version":"1.0","source":{"id":"1712.08882","kind":"arxiv","version":2}},"canonical_sha256":"d69d28bbb36762b26c3fb0485d46de31c6cda50fbcb6c5c6cf55954968fe955b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d69d28bbb36762b26c3fb0485d46de31c6cda50fbcb6c5c6cf55954968fe955b","first_computed_at":"2026-05-18T00:01:19.691049Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:19.691049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aC5eRIiONGkuDhy5YvVeNZBv9r3fhrZ5aSpHjS4iAih744HE2HmoqRzS4VZX2U/Qq3xzqOAbdrIlTpbp4WyoCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:19.691533Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.08882","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6dc2c00d360c49a10e672530eb9b634b7826eed5a8bd7e1d1dbe9344c2c6f06b","sha256:bda16026a6314ffdb14fe646774fbfce76be8232fc359b545d8fe3d974d28ebe"],"state_sha256":"6eff672f548e76eeabbb164e5c1765431327158984bbe142c96a6d9b6b3a26ca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m0t5mysCIOYdp9FIKrloBUGSrn7vZFmmp7cgu14kKY3eIwX3EWIoU/KIp5xJNW74EUuBho6edSP/9cjgdHuEBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T02:03:51.228482Z","bundle_sha256":"127c50b2729641737c7df104fafff0b64fdd363664cba314eca36b511b0bcbb3"}}