Anderson Localization for Schr\"{o}dinger Operators with Monotone Potentials Generated by the Doubling Map

In this paper, we consider the Schr\"{o}dinger operators on $\ell^{2}(\N)$, defined for all $x\in\mathbb{T}$ by \begin{equation} (H(x)u)_n = u_{n+1} + u_{n-1} + \lambda f(2^{n} x) u_n, \quad \text{for } n \geq 0,\notag \end{equation} with the Dirichlet boundary condition $u_{-1}=0$. Building on Zhang's recent breakthrough work [Comm.Math.Phys.405:231(2024)] that resolved Damanik's open problem [Proc.Sympos. Pure Math.76,Amer.Math.Soc.(2007)] on the uniform positivity of the Lyapunov exponent, for the potential $f \in C^{1}(0,1)$ with $\|f\|_{C^{1}(0,1)} < C$ and $\inf_{x \in (0,1)} |f^{\prime}(x)| > c>0$, we obtain the large deviation estimate and prove that for a.e. $x \in \mathbb{T}$ and sufficiently large $\lambda > \lambda_{0}$, the operators $H(x)$ display Anderson localization. Furthermore, if the potentials also have zero mean, our analysis reveals that the doubling map models can exhibit localization behavior for both small and large coupling constants $\lambda$.