{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/24264"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/24264","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Dynamics of irreducible endomorphisms of F_n","abstract":"We consider the class non-surjective irreducible endomorphisms of the free group F_n. We show that such an endomorphism \\phi is topologically represented by a simplicial immersion f:G \\rightarrow G of a marked graph G; along the way we classify the dynamics of \\partial \\phi acting on \\partial F_n: there are at most 2n fixed points, all of which are attracting. After imposing a necessary additional hypothesis on \\phi, we consider the action of \\phi on the closure \\overline{CV}_n of the Culler-Vogtmann Outer space. We show that \\phi acts on \\overline{CV}_n with ``sink'' dynamics: there is a unique fixed point [T_{\\phi}], which is attracting; for any compact neighborhood N of [T_{\\phi}], there is K=K(N), such that \\overline{CV}_n\\phi^{K(N)} \\subseteq N. The proof uses certian projections of trees coming from invariant length measures. These ideas are extended to show how to decompose a tree T in the boundary of Outer space by considering the space of invariant length measures on T; this gives a decomposition that generalizes the decomposition of geometric trees coming from Imanishi's theorem. The proof of our main dynamics result uses a result of independent interest regarding certain actions in the boundary of Outer space. Let T be an \\mathbb{R}-tree, equipped with a very small action of the rank n free group F_n, and let H \\leq F_n be finitely generated. We consider the case where the action F_n \\curvearrowright T is indecomposable--this is a strong mixing property introduced by Guirardel. In this case, we show that the action of H on its minimal invariant subtree T_H has dense orbits if and only if H is finite index in F_n. There is an interesting application to dual algebraic laminations; we show that for T free and indecomposable and for H \\leq F_n finitely generated, H carries a leaf of the dual lamination of T if and only if H is finite index in F_n. This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.","abstract_html":"We consider the class non-surjective irreducible endomorphisms of the free group F_n. We show that such an endomorphism \\phi is topologically represented by a simplicial immersion f:G \\rightarrow G of a marked graph G; along the way we classify the dynamics of \\partial \\phi acting on \\partial F_n: there are at most 2n fixed points, all of which are attracting. After imposing a necessary additional hypothesis on \\phi, we consider the action of \\phi on the closure \\overline{CV}_n of the Culler-Vogtmann Outer space. We show that \\phi acts on \\overline{CV}_n with ``sink'' dynamics: there is a unique fixed point [T_{\\phi}], which is attracting; for any compact neighborhood N of [T_{\\phi}], there is K=K(N), such that \\overline{CV}_n\\phi^{K(N)} \\subseteq N. The proof uses certian projections of trees coming from invariant length measures. These ideas are extended to show how to decompose a tree T in the boundary of Outer space by considering the space of invariant length measures on T; this gives a decomposition that generalizes the decomposition of geometric trees coming from Imanishi's theorem. The proof of our main dynamics result uses a result of independent interest regarding certain actions in the boundary of Outer space. Let T be an \\mathbb{R}-tree, equipped with a very small action of the rank n free group F_n, and let H \\leq F_n be finitely generated. We consider the case where the action F_n \\curvearrowright T is indecomposable--this is a strong mixing property introduced by Guirardel. In this case, we show that the action of H on its minimal invariant subtree T_H has dense orbits if and only if H is finite index in F_n. There is an interesting application to dual algebraic laminations; we show that for T free and indecomposable and for H \\leq F_n finitely generated, H carries a leaf of the dual lamination of T if and only if H is finite index in F_n. This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.","abstract_has_math":false,"creators":["Reynolds, Patrick R."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Kapovitch, Ilia","Leininger, Christopher J.","Dunfield, Nathan M.","Mineyev, Igor"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-25T15:02:44Z","date_published":"2011-05-25T15:02:44Z","updated_at":"2026-07-22T22:25:23Z","subjects":["irreducible endomorphism","outer space","R-tree","length measure","free group"],"languages":["en"],"rights":["Copyright 2011 Patrick Reese Reynolds"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/24264","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Kapovitch, Ilia","Leininger, Christopher J.","Dunfield, Nathan M.","Mineyev, Igor"]},{"key":"dc:creator","label":"Author","values":["Reynolds, Patrick R."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-25T15:02:44Z","2011-05"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["irreducible endomorphism","outer space","R-tree","length measure","free group"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2011 Patrick Reese Reynolds"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/24264"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We consider the class non-surjective irreducible endomorphisms of the free group F_n. We show that such an endomorphism \\phi is topologically represented by a simplicial immersion f:G \\rightarrow G of a marked graph G; along the way we classify the dynamics of \\partial \\phi acting on \\partial F_n: there are at most 2n fixed points, all of which are attracting. After imposing a necessary additional hypothesis on \\phi, we consider the action of \\phi on the closure \\overline{CV}_n of the Culler-Vogtmann Outer space. We show that \\phi acts on \\overline{CV}_n with ``sink'' dynamics: there is a unique fixed point [T_{\\phi}], which is attracting; for any compact neighborhood N of [T_{\\phi}], there is K=K(N), such that \\overline{CV}_n\\phi^{K(N)} \\subseteq N. The proof uses certian projections of trees coming from invariant length measures. These ideas are extended to show how to decompose a tree T in the boundary of Outer space by considering the space of invariant length measures on T; this gives a decomposition that generalizes the decomposition of geometric trees coming from Imanishi's theorem. The proof of our main dynamics result uses a result of independent interest regarding certain actions in the boundary of Outer space. Let T be an \\mathbb{R}-tree, equipped with a very small action of the rank n free group F_n, and let H \\leq F_n be finitely generated. We consider the case where the action F_n \\curvearrowright T is indecomposable--this is a strong mixing property introduced by Guirardel. In this case, we show that the action of H on its minimal invariant subtree T_H has dense orbits if and only if H is finite index in F_n. There is an interesting application to dual algebraic laminations; we show that for T free and indecomposable and for H \\leq F_n finitely generated, H carries a leaf of the dual lamination of T if and only if H is finite index in F_n. This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-04-18T19:26:05Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Reynolds_Patrick.pdf: 538671 bytes, checksum: f98f07c54aecb9bef959f39019af6bd5 (MD5)","Made available in DSpace on 2011-05-25T15:02:44Z (GMT). No. of bitstreams: 2 Reynolds_Patrick.pdf: 538589 bytes, checksum: 693699173b9e0b5937fab2045f0d5dd8 (MD5) license.txt: 4066 bytes, checksum: 848d902a98c330a57d685a9de6efa60a (MD5)"]},{"key":"dc:title","label":"Title","values":["Dynamics of irreducible endomorphisms of F_n"]}]}],"canonical_facts":{"dc:contributor":["Kapovitch, Ilia","Leininger, Christopher J.","Dunfield, Nathan M.","Mineyev, Igor"],"dc:creator":["Reynolds, Patrick R."],"dc:date":["2011-05-25T15:02:44Z","2011-05"],"dc:description":["We consider the class non-surjective irreducible endomorphisms of the free group F_n. We show that such an endomorphism \\phi is topologically represented by a simplicial immersion f:G \\rightarrow G of a marked graph G; along the way we classify the dynamics of \\partial \\phi acting on \\partial F_n: there are at most 2n fixed points, all of which are attracting. After imposing a necessary additional hypothesis on \\phi, we consider the action of \\phi on the closure \\overline{CV}_n of the Culler-Vogtmann Outer space. We show that \\phi acts on \\overline{CV}_n with ``sink'' dynamics: there is a unique fixed point [T_{\\phi}], which is attracting; for any compact neighborhood N of [T_{\\phi}], there is K=K(N), such that \\overline{CV}_n\\phi^{K(N)} \\subseteq N. The proof uses certian projections of trees coming from invariant length measures. These ideas are extended to show how to decompose a tree T in the boundary of Outer space by considering the space of invariant length measures on T; this gives a decomposition that generalizes the decomposition of geometric trees coming from Imanishi's theorem. The proof of our main dynamics result uses a result of independent interest regarding certain actions in the boundary of Outer space. Let T be an \\mathbb{R}-tree, equipped with a very small action of the rank n free group F_n, and let H \\leq F_n be finitely generated. We consider the case where the action F_n \\curvearrowright T is indecomposable--this is a strong mixing property introduced by Guirardel. In this case, we show that the action of H on its minimal invariant subtree T_H has dense orbits if and only if H is finite index in F_n. There is an interesting application to dual algebraic laminations; we show that for T free and indecomposable and for H \\leq F_n finitely generated, H carries a leaf of the dual lamination of T if and only if H is finite index in F_n. This generalizes a result of Bestvina-Feighn-Handel regarding stable trees of fully irreducible automorphisms.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-04-18T19:26:05Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Reynolds_Patrick.pdf: 538671 bytes, checksum: f98f07c54aecb9bef959f39019af6bd5 (MD5)","Made available in DSpace on 2011-05-25T15:02:44Z (GMT). No. of bitstreams: 2 Reynolds_Patrick.pdf: 538589 bytes, checksum: 693699173b9e0b5937fab2045f0d5dd8 (MD5) license.txt: 4066 bytes, checksum: 848d902a98c330a57d685a9de6efa60a (MD5)"],"dc:identifier":["http://hdl.handle.net/2142/24264"],"dc:language":["en"],"dc:rights":["Copyright 2011 Patrick Reese Reynolds"],"dc:subject":["irreducible endomorphism","outer space","R-tree","length measure","free group"],"dc:title":["Dynamics of irreducible endomorphisms of F_n"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:23Z"}