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oai:www.ideals.illinois.edu:2142/24264 2023-07-10 col_2142_16340 col_2142_5131 com_2142_16339 com_2142_8903 com_2142_5130

irreducible endomorphism outer space R-tree length measure Dynamics of irreducible endomorphisms of F_n Kapovitch, Ilia Leininger, Christopher J. Kapovitch, Ilia Dunfield, Nathan M. Mineyev, Igor Reynolds, Patrick R. 2011-05-25T15:02:44Z 2011-05-25T15:02:44Z 2011-05-25T15:02:44Z 2011-05 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 Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-04-18T19:26:05Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: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) http://hdl.handle.net/2142/24264 en Copyright 2011 Patrick Reese Reynolds free group Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD