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According to \\cite{GL}, after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\\Lambda$ of the action, included in $Fix(f)$.\n  Here, we suppose that $f$ and $h$ are $C^1$ in neighbourhood of $\\Lambda$ and any point $x\\in\\Lambda$ admits an $h$-unstable manifold $W^u(x)$. Using Bonatti's techniques, we prove that either there exists an integer $N$ such that $W^u(x)$ is included in $Fix(f^N)$ or there is "},"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":"1703.09037","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-03-27T12:41:45Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"e500d20fd13cd3302dc161d432b70dc1e36af54008c3c6223a81db86352fb20f","abstract_canon_sha256":"6c91d9932097cca4c436063db97127aa53e7747c2bf9d07294b446d62f70d673"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:54.504499Z","signature_b64":"6HvGVOxIu7NJWtkvHhHvt9ekQmcfAkPJK6EzOduueT0JdXYKTuhq+pfQzg+fKSosnT8ZxTCOFWh9lyywIWaOAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d0e791c35acff7d0689241a09f8f57907d8b213ad8f2df14b97242025784d21","last_reissued_at":"2026-05-18T00:47:54.503799Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:54.503799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Any Baumslag-Solitar action on surfaces with a pseudo-Anosov element has a finite orbit","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.DS","authors_text":"Isabelle Liousse, Nancy Guelman","submitted_at":"2017-03-27T12:41:45Z","abstract_excerpt":"We consider $f, h$ homeomorphims generating a faithful $BS(1,n)$-action on a closed surface $S$, that is, $h f h^{-1} = f^n$, for some $ n\\geq 2$. According to \\cite{GL}, after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\\Lambda$ of the action, included in $Fix(f)$.\n  Here, we suppose that $f$ and $h$ are $C^1$ in neighbourhood of $\\Lambda$ and any point $x\\in\\Lambda$ admits an $h$-unstable manifold $W^u(x)$. 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