Dynamical Systems of Algebraic Origin pp 1-33 | Cite as

# Group actions by automorphisms of compact groups

Chapter

## Abstract

Throughout this book the term for all

*compact group*will denote a compact, metrizable group, and a*compact Lie group*will be a compact (possibly finite) subgroup of some finite-dimensional matrix group over the complex numbers. Any metric*δ*on a compact group X is assumed to be invariant, i.e.*δ(x, x’) = δ(yx, yx’) = δ(xy, x’y)*for all*x, x’, y ∈ X*, and to induce the topology of*X*. The identity element of a group*X*will usually be written as**1, 0**,**1**_{ X }, or**0**_{ X }, depending on whether*X*is multiplicative or additive, and whether there is danger of confusion. If*X*is a topological group we write*X°*for its connected component of the identity,*C(X)*for its centre, Aut(*X*) for the group of continuous automorphisms of*X*, and Inn(*X*) ⊂ Aut(*X*) for the normal subgroup of inner automorphisms of*X*. The trivial automorphism of*X*is denoted by id_{ X }=**1**_{Aut(X)}, and we set Out(*X*) = Aut(*X*)/Inn(*X*). If*X*is compact and*δ*is a metric on*X*we define a metric*on Aut(***δ***X*) by setting$$ \partial \left( {\alpha, \beta } \right)\mathop{{\max }}\limits_{{x \in X}} \left( {\partial \left( {\alpha (x),\beta (x)} \right) + } \right)\partial \left( {{\alpha^{{ - 1}}}(x),{\beta^{{ - 1}}}\left( {x)} \right)} \right) $$

*α, β*∈ Aut(*X*); the topology on Aut(*X*) induced by*is called the uniform topology. If***δ***X*is a compact Lie group then Aut(*X*) is again a Lie group in the uniform topology, Inn(*X*) is an open subgroup of*X*, and Out(*X*) is therefore discrete. For an arbitrary compact group*X*, the group Out(*X*) is zero-dimensional in the uniform topology ([32]).## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Birkhäuser Verlag 1995