the fixed point.

For $1< q <2$, there are only two dominant operators as the time $t \to \infty$, ${\cal D}$,${\cal C}$. In this case one obtains a conventional logarithmic conformal algebra of two-point functions near the fixed point: $$\begin{aligned}
\label{logalg}
&~& \langle {\cal D}^{(q)}(z)~{\cal D}(0)^{(q)} 
\rangle_c = \langle {\cal D}(z)~{\cal D}(0) \rangle_c  
\sim {\tilde N}^2_q~\frac{\sqrt{\pi}}{\sqrt{\xi_0}}~f'_q(2\xi_0)2
{\rm ln}|z/L|^2~, \nonumber \\
&~& \epsilon~q~\langle {\cal D}^{(q-1)}(z)~D^{(q)}(0) \rangle_c = 
\langle {\cal C}(z)~{\cal D}(0) \rangle_c \sim {\tilde N}^2_q~
\left(h^2_q(\xi_0) - h_{q-1}~h_q (\xi_0)\right)~, \end{aligned}$$ and all the other correlators are subleading as $t \to \infty$.

Therefore, the [*on shell algebra*]{} is of the conventional [*logarithmic form*]{} [@lcft], between a pair of operators, and hence, ${\cal D}^{(q-2)}$ and subsequent operators, which owe their existence to the non-trivial RW metric, do not modify the two-point correlators of the standard logarithmic algebra of ‘recoil’ (impulse) [@kmw] [^2].

Next, we consider the case where $2< q < 3$. In this case, from (\[offshellalg\]) we observe that there are now three operators which dominate in the limit $t \to \infty$, ${\cal D}$, ${\cal C}$ and ${\cal B} = \epsilon^2~q~(q-1)
{\cal D}^{(q-2)}$, whose form is implied from (\[tdope\]), in analogy with ${\cal C}$. The corresponding algebra of correlators consists of parts 