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The ${\\bf a}$-weighted topological pressure of $f$, denoted by $P^{\\bf a}(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$ P^{\\bf a}(X, f)=\\sup\\left\\{a_1h_\\mu(T)+a_2h_{\\mu\\circ\\pi^{-1}}(S)+\\int f \\;d\\mu\\right\\}, $$ where the supremum is taken over the $T$-invariant measures on $X$. It not only generalizes the variational princ"},"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":"1412.0078","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-29T07:15:26Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"605e03e8b37fcabc8831ca004d3d3da0382c7a2b4ae21d2d82b447e3ea5ea248","abstract_canon_sha256":"1ba6a9b35d2b0d04aae8143842220d5d918bfe131653f2fd0dc78b21f63b8e00"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:31.416591Z","signature_b64":"MHL/b0VBdg9Fkjsdu8wEg6Z+7LzdFQe1wtb5wKgiQGKFsfH596Lo8kOD1Di5C5yQgseC/Q+K3zJI4Hzml8EGDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ea4289e279a13b158bf62309007d8ceb2268712a89f47ddeeba4aaf2e7a5091","last_reissued_at":"2026-05-18T02:32:31.416240Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:31.416240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Variational principle for weighted topological pressure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DS","authors_text":"De-Jun Feng, Wen Huang","submitted_at":"2014-11-29T07:15:26Z","abstract_excerpt":"Let $\\pi:X\\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let ${\\bf a}=(a_1,a_2)\\in {\\Bbb R}^2$ with $a_1>0$ and $a_2\\geq 0$, and $f\\in C(X)$. The ${\\bf a}$-weighted topological pressure of $f$, denoted by $P^{\\bf a}(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$ P^{\\bf a}(X, f)=\\sup\\left\\{a_1h_\\mu(T)+a_2h_{\\mu\\circ\\pi^{-1}}(S)+\\int f \\;d\\mu\\right\\}, $$ where the supremum is taken over the $T$-invariant measures on $X$. 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