$-trivial for $ \bQ( \eps \in dg ) $-almost every $ g \in G $.

The subgroup $ H_\mu^s $ {#sec: H mu s}
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To begin with, we recall the well-known [*Peter–Weyl theorem*]{} for compact groups (see, e.g., [@Sugiura Chapter 1] and [@JS Corollary 13]).

Let $ G $ be a compact group and let $ \nu $ denote the normalized Haar measure on $ G $. Let $ \sG $ denote the totality of irreducible unitary representations $ \rho $ of $ G $ on a finite-dimensional linear space $ V^{\rho} $. Then the following holds: [ *The family $$( \rho_{i,j} : \, 1 \le i,j \le \dim \rho , \ \rho \in \sG )$$ forms a total family in the space of continuous functions on $ G $*]{}. Here $ ( \rho_{i,j} ) $ denotes the matrix element of a representation $ \rho \in \sG $.

In what follows we consider the STYE on a compact group $ G $ with a fixed noise law $ \mu $.

To characterize the trichotomy, we introduce a subset $ H^s_\mu $ of $ G $ as follows. For an extremal point $ \bP^o \in \ext $ we define $$\sH^s_\mu(\bP^o) 
