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On the other hand, we show that there is no universal effective $Z^d$-system for $d>1$, and in particular SFTs cannot be universal for subactions of rank $d>1$. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs.\n  We also discu"},"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":"0901.3600","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-01-23T04:40:01Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"ba19a1b739f55d8e62570c74f737b43c125dc3029f740dd9f90eff56907548bb","abstract_canon_sha256":"0ffdf374c278f342ab0281b402b2e9db88fb3865813a64332dbd02a8ea66f4b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:15.584993Z","signature_b64":"KFyHQu9U1rBK7WeFa7Wo2jT86+/MKZQ+blu58wnFwj+1/llnGIVurFG4uHcDyGkgNO6b3bZoVpiFKeUZp3XuDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a763badb0c4745392b3423752a4f3e60682d383a3298cb0e77b51ba4d6a94f10","last_reissued_at":"2026-05-18T02:42:15.584298Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:15.584298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universality in Multidimensional Symbolic Dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.DS","authors_text":"Michael Hochman","submitted_at":"2009-01-23T04:40:01Z","abstract_excerpt":"We show that in the category of effective $Z$ dynamical systems there is a universal system, i.e. one that factors onto every other effective system. 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