{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:OGRDUBNQHJI2RQCMB7DTF5EO2C","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":"a85438ffc3cea5f448b736c2017cbe7d4cb965367b471161fadaa76c322527f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-28T17:55:20Z","title_canon_sha256":"f99bcb2f16bf904ae265f6f34e8b0d0017fab7174d4801f571f18ff4d76ffda3"},"schema_version":"1.0","source":{"id":"1509.08413","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.08413","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"arxiv_version","alias_value":"1509.08413v1","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.08413","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"pith_short_12","alias_value":"OGRDUBNQHJI2","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OGRDUBNQHJI2RQCM","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OGRDUBNQ","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:8c8a8e730ac53d9b6a4b6389704ad0117af6189d62109218d0b939024d4fe3e1","target":"graph","created_at":"2026-05-18T01:31:53Z","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":"Topological entropy is a widely studied indicator of chaos in topological dynamics. Here we give a generalized definition of topological entropy which may be applied to set-valued functions. We demonstrate that some of the well-known results concerning topological entropy of continuous (single-valued) functions extend naturally to set-valued functions while others must be altered. We also present sufficient conditions for a set-valued function to have positive or infinite topological entropy.","authors_text":"James Kelly, Tim Tennant","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-28T17:55:20Z","title":"Topological entropy on set-valued functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.08413","kind":"arxiv","version":1},"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:e789b7ad6dd767ced067ba10bcc38c339f8ba92e698b01d0e73f179973083ac3","target":"record","created_at":"2026-05-18T01:31:53Z","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":"a85438ffc3cea5f448b736c2017cbe7d4cb965367b471161fadaa76c322527f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-09-28T17:55:20Z","title_canon_sha256":"f99bcb2f16bf904ae265f6f34e8b0d0017fab7174d4801f571f18ff4d76ffda3"},"schema_version":"1.0","source":{"id":"1509.08413","kind":"arxiv","version":1}},"canonical_sha256":"71a23a05b03a51a8c04c0fc732f48ed0ab3dcf0e79fd8599a25d7b1bfa4ee866","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"71a23a05b03a51a8c04c0fc732f48ed0ab3dcf0e79fd8599a25d7b1bfa4ee866","first_computed_at":"2026-05-18T01:31:53.925157Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:53.925157Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DiG6ekfT3KpMoNIAWQLDyhckcYfa3hOmwCM5RuPTGoUZov96xPDMnKALyv4NY0SmClfja32vFnY+MnuqO9hSBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:53.925701Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.08413","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e789b7ad6dd767ced067ba10bcc38c339f8ba92e698b01d0e73f179973083ac3","sha256:8c8a8e730ac53d9b6a4b6389704ad0117af6189d62109218d0b939024d4fe3e1"],"state_sha256":"f362e631cefef5312a4a7131235d36f9413bc777578fac00aa73b0b9f20a42db"}