Generic spectral-radius conjecture for complex transfer operators

Let $I$ be the underlying expanding dynamical system, let $\tau$ be a cocycle, and let $\boldsymbol{G}=?$. In the one-dimensional torus case $\boldsymbol{G}=?$, for $q e 0$ define the complex transfer operator

$${?}$$

where $\rho_{\operatorname{sp}}$ denotes its spectral radius on a sufficiently smooth function space. Generic spectral-radius conjecture. For a generic choice of $\tau$, for every $?$, there exists $?$ such that, for every $?$,

$$e^{\frac{1}{2}P(2\varphi)-\epsilon}\leq\rho_{\operatorname{sp}}({\mathcal M}_q)\leq e^{\frac{1}{2}P(2\varphi)+\epsilon}.$$

This conjecture proposes a precise asymptotic scale for the spectral radii of the high-frequency complex transfer operators. The supplied passage does not define the meaning of “generic,” the precise smooth function spaces, or the pressure normalization beyond the surrounding setup, so these details require verification.