\label{T-Yor eq 2}$$ If we write $ {\widehat}{\eta}_k=\theta(\eta_k) $ and $ {\widehat}{\xi}_k=\theta(\xi_k) $, then the equation is exactly the simple Tsirelson–Yor equation on $ G $.

We will prove in Proposition \[reduction\] that in order to study the Tsirelson–Yor equation with given laws it is sufficient to investigate the simple Tsirelson–Yor equation on $ G $.

We will show in Example \[ex: Yor eq\] that Yor’s equation is an example of our Tsirelson–Yor equation. Proposition \[reduction\] says that the equation is essentially ‘equivalent’ to the equation $$\alpha (\eta_k) = \alpha (\eta_{k-1}) + \alpha (\xi_k) \quad \mbox{modulo 1}, 
\quad k \in -\N , 
\label{STYE torus}$$ which is nothing but the equation .

In addition, we will show in Example \[ex: Tanaka eq\] that the following equation is also an example: $$\eta_k = \sgn {\left( \eta_{k-1} \right)} \cdot \xi_k , 
\quad k \in -\N 
\label{Tanaka}$$ provided that the noise law has no point mass at $ x=0 $. Here $ \sgn (x) = 1 $ if $ x \ge 0 $ and $ \sgn (x) = -1 $ if $ x<0 $. This equation has been dealt with in [@R-Y Chapter IX, Exercise 3.18] and [@LGY pp. 87–88]. Proposition \[reduction\] says that 