the equation is equivalent to the simple Tsirelson–Yor equation on $ \Z/2\Z $: $$\sgn {\left( \eta_k \right)} = \sgn {\left( \eta_{k-1} \right)} \cdot \sgn {\left( \xi_k \right)} , 
\quad k \in -\N . 
\label{STYE Z/2Z}$$ This equation will be dealt with in Example \[ex: Z/2Z\].

 

The present paper is organized as follows. In section \[sec: pre\], we give the precise definition of the simple Tsirelson–Yor equation on a compact group and introduce the notions of a solution, a strong solution and uniqueness in law. We also prepare some preliminary facts about the set of solutions. Section \[sec: exist\] is devoted to the proof of Theorem \[Intro exist\]. In section \[sec: hom sp\], we give a precise statement and the proof of Theorem \[Intro action\]. In section \[sec: H mu s\], we introduce a subgroup $ H_\mu^s $ which appears in Theorem \[Intro trichotomy\] and restate Theorem \[Intro isot group\] as Theorem \[thm: isot group\]. In section \[sec: example\], we investigate the equation in detail and consider another typical example of the simple Tsirelson–Yor equation. Section \[sec: trichotomy\] is devoted to the proofs of Theorem \[Intro trichotomy\] and the propositions which are given in section \[sec: H mu s\]. The Tsirelson–Yor 