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I am looking for references on the automorphism group $\mathrm{Aut}(X)$ of a compact abelian group $X$. By automorphisms I mean topological group automorphisms. Some particular questions are as follows. How can $\mathrm{Aut}(X)$ be given topological structure? With this structure, can one approximate an automorphism by periodic automorphisms? When is a conjugacy class dense in $\mathrm{Aut}(X)$? Noting that automorphisms of $X$ preserve Haar measure, what ergodic-theoretic properties are generic in $\mathrm{Aut}(X)$?

I am most concerned with the case of a general abstract compact abelian group $X$, but I would also be very interested in references for specific $X$, even the circle.

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    $\begingroup$ By Pontryagin duality, $X$ has the same automorphisms as its character group, which is discrete. In particular, for instance, the automorphisms of a torus are just $GL_n(\mathbb{Z})$. $\endgroup$
    – Eric Wofsey
    Mar 26, 2014 at 14:09

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See Lian B. Smythe's paper on arXiv.org (2011).

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