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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"},"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":"1507.05140","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-07-18T02:49:38Z","cross_cats_sorted":[],"title_canon_sha256":"70f6a236e8e153899968da1cb15b1671fef91d35505bfd2f83f2373a260ffc45","abstract_canon_sha256":"5e7a294cac9cc3f1e3bafe36acf201ba76f2865a69f83303975aeb45e16f73be"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:34.213005Z","signature_b64":"cDwRqwb3u6KRbrCXZ1q/IE5INOT6+urUmsIJ2K+Yt4zKSDPMrXrBf3Pb/CbJ3b/ONWhzx42X5b+9O9n6SFt6Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e475926ea6d8ca06d8c54d976bfccc384e026edd9b01eb66507c908ced699cb","last_reissued_at":"2026-05-18T01:30:34.212349Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:34.212349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Ergodic Theorem for a new kind of attractor of a GIFS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Elismar R. Oliveira","submitted_at":"2015-07-18T02:49:38Z","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 $(X,d)$. 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