By Gelbart S.

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**Example text**

It follows that p(U ), X/G\p(U) ¯ neighbourhood U of x such that U are disjoint open sets separating p(x) and p(y). 2 If G is not compact, X/G need not be Hausdorﬀ. Highly pathological examples may be constructed using smooth R-actions. 1 Regard the 2-torus T2 as R2 /Z2 . Given α ∈ R , deﬁne the R-action Φt : T2 → T2 by Φt (θ, ψ) = (θ + t, ψ + αt), mod Z2 . If α is irrational, then every orbit of Φt is dense in T2 . In this case the orbit space T2 /R has only one nonempty open set: T2 /R. For this, observe that if q : T2 → T2 /R denotes the orbit map then U ⊂ T2 /R is open if and only if q −1 (U ) is an open and Φt -invariant subset of T2 .

Since exp |L is a local diﬀeomorphism at the origin, we may choose L so that exp maps L diﬀeomorphically onto an open neighbourhood D of e ∈ H. 6 Haar measure We conclude the chapter by recalling the simple proof of the existence of Haar measure for a compact Lie group. 1 Let G be a compact Lie group. There exists a unique Borel probability measure on G which is invariant under both left and right translations. Proof. We prove existence and leave uniqueness to the reader. Suppose that dim(G) = m.

These results hold for real or June 21, 2007 9:41 40 WSPC/Book Trim Size for 9in x 6in DynamicsSymmetry Dynamics and Symmetry complex representations. 2 Although we make little use of characters in this book, we emphasize that character theory is a fundamental tool for the analysis of group representations, especially over the complex ﬁeld. 1 The F-representation (V, G) is irreducible if the only G-invariant F-linear subspaces of V are {0} and V . If there exist nontrivial invariant subspaces, (V, G) is reducible.