Flat trace statistics of the transfer operator of a random partially expanding map

Abstract

We consider the skew-product of an expanding map $E$ on the circle $\mathbb{T}$ with an almost surely ${\mathcal{C}}^k$ random perturbation $\tau ={\tau }_0+\delta \tau$ of a deterministic function ${\tau }_0$: \[F :\left\{\begin{array}{rcl} \mathbb T \times \mathbb R & \longrightarrow & \mathbb T \times \mathbb R\\ (x,y)& \longmapsto & (E(x), y+τ(x))\\ \end{array} \right.\] The associated transfer operator $\mathcal{L}:u\in {\mathcal{C}}^k(\mathbb{T}\times \mathbb{R})\mapsto u\circ F$ can be decomposed with respect to frequency in the $y$ variable into a family of operators acting on functions on the circle: \[\mathcal L_ξ:\left\{\begin{array}{rcl} \mathcal C^k(\mathbb T) & \longrightarrow & \mathcal C^k(\mathbb T)\\ u & \longmapsto & e^{iξτ}u\circ E \\ \end{array} \right.\] We show that the flat traces of ${\mathcal{L}}_{\xi }^n$ behave as normal distributions in the semiclassical limit $n,\xi \to \infty$ up to the Ehrenfest time $n\le c_k\log \xi$.

References in corpus (2)

• On the spectra of quenched random perturbations of partially expanding maps on the torus
• On the rate of mixing of circle extensions of Anosov maps

Cited by in corpus (1)

• A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics