Anderson Localization for Schrödinger Operators with Monotone Potentials Generated by the Doubling Map

In this paper, we consider the Schrödinger operators on ${\mathrm{\ell }}^2(\mathbb{N})$, defined for all $x\in \mathbb{T}$ by \begin{equation} (H(x)u)n = u{n+1} + u_{n-1} + λf(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 $\mathrm{\parallel }f{\mathrm{\parallel }}_{C^1(0,1)}<C$ and ${\inf }_{x\in (0,1)}\mathrm{\mid }{f}^{\mathrm{\prime }}(x)\mathrm{\mid }>c>0$, 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 $\lambda$.