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. We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the cor- responding stratum of the bordification, and in that deformation space it may be irreducible (sometimes this is referred as relative irreducibility).

[1]  A. Martino,et al.  Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks , 2018, 1807.02781.

[2]  Ilya Kapovich,et al.  Detecting Fully Irreducible Automorphisms: A Polynomial Time Algorithm , 2016, Exp. Math..

[3]  Camille Horbez.  Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings , 2014, 1408.0544.

[4]  M. Handel,et al.  The free splitting complex of a free group, II: Loxodromic outer automorphisms , 2014, Transactions of the American Mathematical Society.

[5]  A. Martino,et al.  Stretching factors, metrics and train tracks for free products , 2013, 1312.4172.

[6]  S. Meinert.  The Lipschitz metric on deformation spaces of G–trees , 2013, 1312.1829.

[7]  M. Bestvina,et al.  The boundary of the complex of free factors , 2012, 1211.3608.

[8]  Camille Horbez,et al.  The hyperbolicity of the sphere complex via surgery paths , 2012, 1210.6183.

[9]  Ilya Kapovich.  Algorithmic detectability of iwip automorphisms , 2012, 1209.3732.

[10]  Ilya Kapovich,et al.  On hyperbolicity of free splitting and free factor complexes , 2012, 1206.3626.

[11]  M. Handel,et al.  The free splitting complex of a free group I: Hyperbolicity , 2011, 1111.1994.

[12]  M. Kassabov,et al.  Hairy graphs and the unstable homology of Mod(g, s), Out(Fn) and Aut(Fn) , 2011, 1107.4839.

[13]  M. Bestvina,et al.  Hyperbolicity of the complex of free factors , 2011, 1107.3308.

[14]  M. Bestvina.  A Bers-like proof of the existence of train tracks for free group automorphisms , 2010, 1001.0325.

[15]  A. Martino,et al.  The isometry group of Outer Space , 2009, 0912.0299.

[16]  A. Martino,et al.  METRIC PROPERTIES OF OUTER SPACE , 2008, 0803.0640.

[17]  G. Levitt,et al.  The outer space of a free product , 2005, math/0501288.

[18]  J. Los.  On the conjugacy problem for automorphisms of free groups , 1996 .

[19]  A. Hatcher,et al.  Homological stability for automorphism groups of free groups , 1995 .

[20]  H. Bass.  Covering theory for graphs of groups , 1993 .

[21]  S. M. Gersten,et al.  Fixed points of automorphisms of free groups , 1987 .

[22]  K. Vogtmann,et al.  Moduli of graphs and automorphisms of free groups , 1986 .

[23]  John R. Stallings,et al.  Topology of finite graphs , 1983 .

[24]  M. Lustig.  Conjugacy and Centralizers for Iwip Automorphisms of Free Groups , 2007 .

[25]  Mladen Bestvina,et al.  Train tracks and automorphisms of free groups , 1992 .

[26]  S. Gersten.  On Fixed Points of Certain Automorphisms of Free Groups , 1984 .