{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T05:20:26Z","timestamp":1769059226914,"version":"3.49.0"},"reference-count":37,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T00:00:00Z","timestamp":1760659200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00297\/2020 and UIDP\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00297\/2020 and UIDP\/00297\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Aust. Math. Soc."],"published-print":{"date-parts":[[2026,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We prove that Brinkmann\u2019s problems are decidable for endomorphisms of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788725101195_inline1.png\"\/>\n                        <jats:tex-math>$F_n\\times F_m$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    : given\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788725101195_inline2.png\"\/>\n                        <jats:tex-math>$(x,y),(z,w)\\in F_n\\times F_m$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788725101195_inline3.png\"\/>\n                        <jats:tex-math>$\\Phi \\in \\mathrm {End}(F_n\\times F_m)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , it is decidable whether there is some\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788725101195_inline4.png\"\/>\n                        <jats:tex-math>$k\\in \\mathbb {N}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788725101195_inline5.png\"\/>\n                        <jats:tex-math>$(x,y)\\Phi ^k=(z,w)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    (or\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788725101195_inline6.png\"\/>\n                        <jats:tex-math>$(x,y)\\Phi ^k\\sim (z,w)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ). We also prove decidability of a two-sided version of Brinkmann\u2019s conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788725101195_inline7.png\"\/>\n                        <jats:tex-math>$F_n\\times F_m$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Finally, we study the dynamics of automorphisms of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1446788725101195_inline8.png\"\/>\n                        <jats:tex-math>$F_n\\times F_m$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.\n                  <\/jats:p>","DOI":"10.1017\/s1446788725101195","type":"journal-article","created":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T07:11:25Z","timestamp":1760685085000},"page":"34-56","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["ON THE DYNAMICS OF ENDOMORPHISMS OF THE DIRECT PRODUCT OF TWO FREE GROUPS"],"prefix":"10.1017","volume":"120","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1585-9443","authenticated-orcid":false,"given":"ANDR\u00c9","family":"CARVALHO","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02xankh89","id-type":"ROR","asserted-by":"publisher"}],"name":"Faculdade de Ci\u00eancias da Universidade do Porto"}]}],"member":"56","published-online":{"date-parts":[[2025,10,17]]},"reference":[{"key":"S1446788725101195_r16","doi-asserted-by":"publisher","DOI":"10.5802\/aif.2447"},{"key":"S1446788725101195_r13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2024.08.023"},{"key":"S1446788725101195_r23","doi-asserted-by":"publisher","DOI":"10.1145\/6490.6496"},{"key":"S1446788725101195_r17","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196709005111"},{"key":"S1446788725101195_r26","unstructured":"[26] Logan, A. 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