Expansive automorphisms of totally disconnected, locally compact groups

2013/12/20 by Glockner, Helge, Raja, C. R. E. · 2 citations
#22D05 (primary) #22D45 #22E20 #37A25 #37P20 (secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1312.5875

Abstract

We study automorphisms α of a totally disconnected, locally compact group G which are expansive in the sense that, for some identity neighbourhood U, the sets αⁿ(U) (for integers n) intersect in the trivial group. Notably, we prove that the automorphism induced by α on G/N for an α-stable closed normal subgroup N of G is always expansive. Further results involve the associated contraction groups U_α consisting of all x in G such that αⁿ(x) → e as n tends to infinity. If α is expansive, then W := U_αU_α^-1 is an open identity neighbourhood in G. We give examples where W fails to be a subgroup. However, W is a nilpotent open subgroup whenever G is a closed subgroup of a general linear group over the p-adic numbers. Further results are devoted to the divisible and torsion parts of U_α, and to the so-called "nub" U₀ of an expansive automorphism α (the intersection of the closures of U_α and U_α^-1).

Cited by

• Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications
• Decidability and Characterization of Expansivity for Group Cellular Automata

Related

• Totally disconnected locally compact groups with just infinite locally normal subgroups
• Homomorphisms into totally disconnected, locally compact groups with dense image
• Flat groups of automorphisms of totally disconnected, locally compact groups
• Contraction Groups and Passage to Subgroups and Quotients for Endomorphisms of Totally Disconnected Locally Compact Groups
• Contraction groups for tidy automorphisms of totally disconnected groups