For $t\le1$, we have $U_tv=T_tv$ and so $$\begin{aligned}
 \int_{\tilde Y} (U_tv)\,w\,d\tilde\mu  & = \int_{\tilde Y} v\,w\circ f_t\,d\tilde\mu = \int_{\tilde Y} v(y,u)\,w(y,u+t)\,d\tilde\mu \\ & 
=\int_Y \int_0^{1-t} v(y,u) w(y,u+t)\,du\,d\mu 
=\int_Y \int_t^1 v(y,u-t) w(y,u)\,du\,d\mu \\ &
=\int_{\tilde Y} v(y,u-t)1_{[t,1]}(u)w(y,u)\,d\tilde\mu.\end{aligned}$$ For $t>1$, $$\begin{aligned}
&  \int_{\tilde Y} (U_tv)\,w\,d\tilde\mu   = 
\int_{\tilde Y} T_t(1_{\{\tilde\varphi>t\}}v)\,w\,d\tilde\mu  = 
\int_{\tilde Y} 1_{\{\tilde\varphi>t\}}v\,w\circ f_t\,d\tilde\mu    
\\ & \qquad  = \int_{\tilde Y} (1_{\{\tilde\varphi>t\}}v)(y,u)w(Fy,u+t-\varphi(y))\,d\tilde\mu  i
 \\ &  \qquad
= \int_Y\int_{\varphi(y)-t}^1 1_{\{\varphi(y)>t\}}v(y,u)w(Fy,u+t-\varphi(y)))\,du\,d\mu   \\ &  \qquad
= \int_Y\int_0^{1+t-\varphi(y)} 1_{\{\varphi(y)>t\}}v(y,u-t+\varphi(y))w(Fy,u))\,du\,d\mu   \\ &  \qquad
= \int_{\tilde Y} 1_{\{t<\varphi(y)<t+1-u\}}v(y,u-t+\varphi(y))w\circ \tilde F(y,u)\,d\tilde\mu   
= \int_{\tilde Y} (\tilde R v_t)(y,u)   w(y,u)\,d\tilde\mu,\end{aligned}$$ as required.

In Section \[sec-GM\], we defined the symbolic metric $d_\theta$ on $Y$ and the Banach space $F_\theta(Y)$ of $d_\theta$-Lipschitz functions $v:Y\to{{\mathbb R}}$ The next result makes use of the Gibbs-Markov structure and a weakened version of condition (A2).

\[prop-H2\] Let $\theta\in(0,1)$. Viewing the family of twisted transfer operators $\hat R_0(s)$ as operators on $F_\theta(Y)$,

-   The spectral radius of $\hat R_0(s)$ is less than $1$ for $s\in\overline{{\mathbb H}}-\{0\}$ and is equal to $1$ for $s=0$.

-   $1$ is a simple eigenvalue for $\hat R_0(0)$ and is isolated in the spectrum of $\hat R_0(0)$.

This is standard. By for example the proof of [@MT12 Proposition 11.4], $\hat R_0(s)$ has spectral radius at 