Arithmetic Landscape Functions of a Discrete Cat Map

Abstract

We study the diagonal Green function $\widetilde{u}(x)=[L_N^{-1}]_{x,x}$ of the operator $L_N=I-\alpha P$ on the finite torus $(\mathbb{Z}/N\mathbb{Z})^2$, where $P$ is the transfer operator of the discrete cat map $T_N(x)=Ax \bmod N$. We prove the exact formula $\widetilde{u}(x)=(1-\alpha^{k_x})^{-1}$, where $k_x$ is the minimal period of $x$ under $T_N$. This formula appears to be new. It shows that the diagonal landscape is a complete spectral invariant of the orbit structure, depending on each point only through its orbit length. Since $\det(A-I)=-1$ is a unit in $\mathbb{Z}/N\mathbb{Z}$ for every $N\ge2$, the origin is the unique fixed point of $T_N$ and the unique global maximum of $\widetilde{u}$. The resulting localization is driven by arithmetic alone, with no disorder and no broken symmetry, a mechanism distinct from classical Anderson theory and from Filoche--Mayboroda landscape theory. We further establish the Chandra Green--Zeta Identity, showing that the Green trace satisfies $\operatorname{tr}(G_N)=N^2-\alpha\frac{d}{d\alpha}\log Z_N(\alpha)$, where $Z_N$ is the dynamical zeta function of $T_N$, and that a Laplacian perturbation degrades the localization gap at first order in $\varepsilon$. All results are verified computationally.