{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:DZDVSJXKNWGKA3MMKTMXNP6MYO","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":"5e7a294cac9cc3f1e3bafe36acf201ba76f2865a69f83303975aeb45e16f73be","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-18T02:49:38Z","title_canon_sha256":"70f6a236e8e153899968da1cb15b1671fef91d35505bfd2f83f2373a260ffc45"},"schema_version":"1.0","source":{"id":"1507.05140","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.05140","created_at":"2026-05-18T01:30:34Z"},{"alias_kind":"arxiv_version","alias_value":"1507.05140v3","created_at":"2026-05-18T01:30:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.05140","created_at":"2026-05-18T01:30:34Z"},{"alias_kind":"pith_short_12","alias_value":"DZDVSJXKNWGK","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"DZDVSJXKNWGKA3MM","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"DZDVSJXK","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:0c16f3e2ea106762ae67bac1903bb36bca701e27f5c99c076a62b2957fecd9d6","target":"graph","created_at":"2026-05-18T01:30:34Z","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":"In 1987, J. H. Elton, has proved the first fundamental result in convergence of IFS, the Elton's Ergodic Theorem. In this work we prove the natural extension of this theorem to the projected Hutchinson measure $\\mu_{\\alpha}$ associated to a GIFSpdp $\\mathcal{S}=\\left(X, (\\phi_j:X^{m} \\to X)_{j=0,1, ..., n-1}, (p_j)_{j=0,1, ..., n-1}\\right),$ in a compact metric space $(X,d)$. More precisely, the average along of the trajectories $x_{n}(a)$ of the GIFS, starting in any initial points $x_0, ..., x_{m-1} \\in X$ satisfies, for any $f \\in C(X , \\mathbb{R})$, $$\\lim_{N\\to +\\infty} \\frac{1}{N}\\sum_{n","authors_text":"Elismar R. Oliveira","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-18T02:49:38Z","title":"The Ergodic Theorem for a new kind of attractor of a GIFS"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05140","kind":"arxiv","version":3},"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":"sha256:076404d11b784a1248db2bee794ed55655eb12214f0521b27a28c9cd6a290cb4","target":"record","created_at":"2026-05-18T01:30:34Z","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":"5e7a294cac9cc3f1e3bafe36acf201ba76f2865a69f83303975aeb45e16f73be","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-18T02:49:38Z","title_canon_sha256":"70f6a236e8e153899968da1cb15b1671fef91d35505bfd2f83f2373a260ffc45"},"schema_version":"1.0","source":{"id":"1507.05140","kind":"arxiv","version":3}},"canonical_sha256":"1e475926ea6d8ca06d8c54d976bfccc384e026edd9b01eb66507c908ced699cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1e475926ea6d8ca06d8c54d976bfccc384e026edd9b01eb66507c908ced699cb","first_computed_at":"2026-05-18T01:30:34.212349Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:34.212349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cDwRqwb3u6KRbrCXZ1q/IE5INOT6+urUmsIJ2K+Yt4zKSDPMrXrBf3Pb/CbJ3b/ONWhzx42X5b+9O9n6SFt6Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:34.213005Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.05140","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:076404d11b784a1248db2bee794ed55655eb12214f0521b27a28c9cd6a290cb4","sha256:0c16f3e2ea106762ae67bac1903bb36bca701e27f5c99c076a62b2957fecd9d6"],"state_sha256":"2c08b8114e4b25930fcd03dd81571c8b712c65a49bc0e90f35caa3caa6049965"}