Asymmetric dynamics of outer automorphisms

2016/08/04 by Bell, Mark C.
#20E05 #37E25 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1608.01550

Abstract

We consider the action of an irreducible outer automorphism ϕ on the closure of Culler--Vogtmann Outer space. This action has north-south dynamics and so, under iteration, points converge exponentially to [T^ϕ₊]. For each N ≥ 3, we give a family of outer automorphisms ϕ_k ∈ \textrmOut(\mathbbF_N) such that as, k goes to infinity, the rate of convergence of ϕ_k goes to infinity while the rate of convergence of ϕ_k^-1 goes to one. Even if we only require the rate of convergence of ϕ_k to remain bounded away from one, no such family can be constructed when N < 3. This family also provides an explicit example of a property described by Handel and Mosher: that there is no uniform upper bound on the distance between the axes of an automorphism and its inverse.

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