Given an automorphism τ of a compact group G, we study the factorization of C(τ,K), the contraction group of τ modulo a closed τ -invariant subgroup K, into the product C(τ)K, of the contraction group C(τ) of τ, and K. We prove that the factorization C(τ,K) = C(τ)K holds for every closed τ - invariant subgroup K if and only if G contains arbitrarily small closed normal τ -invariant subgroups N with finite-dimensional quotients G/N. For metrizable groups, we obtain that C(τ)K is a dense subgroup of C(τ,K), for every closed τ -invariant subgroup K. These results are used to link the contraction group to the properties of the dynamical system (G, τ). It follows that τ is distal if and only if C(τ) is trivial, while ergodicity of τ implies that C(τ) is nontrivial. When G is metrizable, the closure of C(τ) is the largest closed τ -invariant subgroup on which τ acts ergodically and, at the same time, it is the smallest among closed normal τ -invariant subgroups N such that τ acts distally on G/N. If τ is ergodic, then its restriction to any closed connected normal τ -invariant subgroup N with finite-dimensional quotient G/N is also ergodic. Moreover, when G is connected, the largest closed τ -invariant subgroup on which τ acts ergodically is necessarily connected.

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Journal Illinois Journal of Mathematics
Jaworski, W. (2012). Contraction groups, ergodicity, and distal properties of automorphisms of compact groups. Illinois Journal of Mathematics, 56(4), 1023–1084.