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But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(\\phi)$ associated to a left invariant distance $d$ just provides an upper bound to $h(\\phi)$, which is characterized by the so called variational principle. We prove that \\[ h\\left(\\phi\\right) = h\\left(\\phi|_{T(G_\\phi)}\\right) \\] where $G_\\phi$ is the maximal connected subgroup of $G$ such that $\\phi(G_"},"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":"1711.02562","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-11-06T18:44:37Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"2a243c7546e6b5a244cddd9c64712b3b1bc0436f04a142c45112abc7ca39b220","abstract_canon_sha256":"81277f2cf0713fdb393910670c0a6daeff332ec4b22a4d857a82f4f7a2a0232e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:18.958848Z","signature_b64":"4lgBq9tTU7+BWdTjmEkYGXzr8Im0f0AlQTY7Kueza++XUigw2Vywk2HeAiRUr8CSO8M+eTGB+iJZAcLrbafZBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76f2170d7a80b7327ae5795b432e7b291780cff5fa7c7b3ca561c7471690aff7","last_reissued_at":"2026-05-18T00:17:18.958406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:18.958406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The topological entropy of endomorphisms of Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.DS","authors_text":"Mauro Patr\\~ao","submitted_at":"2017-11-06T18:44:37Z","abstract_excerpt":"In this paper, we determine the topological entropy $h(\\phi)$ of a continuous endomorphism $\\phi$ of a Lie group $G$. 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