Uniform positivity of the Lyapunov exponent for $C^1$ monotone potentials generated by the cat map

TL;DR

This work proves that for a $C^1$ potential $v$ on the torus with the Arnold cat map base, if the directional derivative in the unstable direction satisfies $D_{\vec{u}}v(\omega)>c$, then the Lyapunov exponent of the associated one-dimensional Schrödinger operator is uniformly positive across all energies when the coupling $\lambda$ is sufficiently large, and in fact satisfies $L(E;\lambda) > \log\lambda - C_0(v)$ for all $E$ and $\lambda>0$. The approach reduces the two-dimensional cocycle problem to one-dimensional dynamics along local unstable leaves using Fubini and a leafwise analysis, centered on a careful study of the phase $\theta_n(\omega)$ via a polar decomposition $A(\omega,t,\lambda)=\Lambda(T\omega)O(\omega)$. A key technical achievement is controlling derivatives $D_{\vec{u}}\theta_n$ and the distribution of discontinuities of $\theta_n$ along leaves, yielding a leafwise lower bound and, through integration over leaves, the global uniform bound. The results extend to general hyperbolic $\mathrm{SL}(2,\mathbb{Z})$ base maps with eigenvalue $\beta>1$, highlighting the role of hyperbolicity and directional derivatives in guaranteeing Anderson localization indicators and robust positivity of Lyapunov exponents in deterministic, non-random settings.

Abstract

We consider an Arnold's Cat Map generated $C^1$ bounded potential with the directional derivative in the unstable direction bounded away from zero. We show that the Lyapunov exponent for the associated Shrödinger Operator is uniformly positive for all energies provided the coupling is sufficiently large.

Theorems & Definitions (18)

• Example 1
• Example 2
• Example 3
• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• proof : Proof of Theorem \ref{['t.plepd']}.
• Lemma 1
• proof