most $1$ and essential spectral radius at most $\theta$ for all $s\in\overline{{\mathbb H}}$. Also the spectral radius is less than $1$ for all $s\in{{\mathbb H}}$. Hence it suffices to consider eigenvalues at $1$ for $s=ib$. It follows from ergodicity of $F$ that $1$ is a simple eigenvalue for its transfer operator $\hat R_0(0)$, so it remains to rule out $1$ as an eigenvalue for $\hat R_0(ib)$, $b\neq0$.

Consider the family of operators $M_b:F_\theta(X)\to F_\theta(X)$ given by $M_bv=e^{ib\varphi}v\circ F$. The operators $\hat R_0(ib)$ and $M_b$ are $L^2$ adjoints so it is equivalent to show that $1$ is not an eigenvalue for $M_b$. We claim that if $1$ is an eigenvalue, then every period $\tau$ corresponding to a periodic orbit for the semiflow lies in $(2\pi/b){{\mathbb Z}}$ which violates condition (A2). (For this proposition it suffices to have two irrationally related periods.)

To prove the claim, suppose that $M_bv=v$ for some $v\in F_\theta(X)$, $v\not\equiv0$. In other words, $e^{ib\varphi}v\circ F=v$. In particular, $|v|\circ F=|v|$ and it follows by ergodicity that $|v|$ is constant. Hence $v$ is nonvanishing. Iterating, we have $e^{ib\sum_{j=0}^{k-1}\varphi\circ F^j}v\circ F^k=v$. Now suppose that $y$ is a periodic point for $F$ of period $k$. The period $\tau$ of the corresponding periodic 