Asymptotic spectral gap for open partially expanding maps

We consider a simple model of an open partially expanding map. Its trapped set is a fractal set. We are interested by the quantity $\gamma_{asympt.}:=\limsup_{\hbar\rightarrow0}\log\left(r_{s}\left(\mathcal{L}_{\hbar}\right)\right)$, namely the log of the spectral radius of the transfer operator $\mathcal{L}_{\hbar}$ in the limit of high frequencies $1/\hbar$ in the neutral direction. Under some hypothesis it is known from D. Dolgopyat 2002 that $\exists\epsilon>0, \gamma_{asympt.}\leq\gamma_{Gibbs}-\epsilon$ with $\gamma_{Gibbs}=\mathrm{Pr}\left(V-J\right)$ and using semiclassical analysis that $\gamma_{asympt.}\leq\gamma_{sc}=\mathrm{sup}\left(V-\frac{1}{2}J\right)$, where $\mathrm{Pr}\left(.\right)$ is the topological pressure, $J>0$ is the expansion rate function and $V$ is the potential function which enter in the definition of the transfer operator. In this paper we show $\gamma_{asympt}\leq\gamma_{up}:=\frac{1}{2}\mathrm{Pr}\left(2\left(V-J\right)\right)+\frac{1}{4}\left\langle J\right\rangle $ where $\left\langle J\right\rangle$ is an averaged expansion rate given in the text. To get these results, we introduce some new techniques such as a global normal form for the dynamical system, a semiclassical expression beyond the Ehrenfest time that expresses the transfer operator at large time as a sum over rank one operators (each is associated to one orbit) and establish the validity of the so-called diagonal approximation up to twice the local Ehrenfest time. Finally, with an heuristic random phases approximation we get the conjecture that generically $\gamma_{asympt}=\gamma_{conj}:=\frac{1}{2}\mathrm{Pr}\left(2\left(V-J\right)\right)$.