Spectral Gaps of Almost Mathieu Operator in Exponential Regime

For almost Mathieu operator $(H_{\lambda,\alpha,\theta}u)n=u{n+1}+u_{n-1}+2\lambda \cos2\pi(\theta+n\alpha)u_n$, the dry version of Ten Martini problem predicts that the spectrum ${\mathrm{\Sigma }}_{\lambda ,\alpha }$ of $H_{\lambda ,\alpha ,\theta }$ has all gaps open for all $\lambda \mathrm{\ne }0$ and $ \alpha \in $\mathbb{R}$\backslash $\mathbb{Q}$$. Avila and Jitomirskaya prove that ${\mathrm{\Sigma }}_{\lambda ,\alpha }$ has all gaps open for Diophantine $\alpha$ and $0<\mathrm{\mid }\lambda \mathrm{\mid }<1$. In the present paper, we show that ${\mathrm{\Sigma }}_{\lambda ,\alpha }$ has all gaps open for all $\alpha \in \mathbb{R}\mathrm{\backslash }\mathbb{Q}$ with small $\lambda$.