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Dirac spectral flow on contact three manifolds II: Thurston--Winkelnkemper contact forms

Given an open book decomposition $(Σ,τ)$ of a three manifold $Y$, Thurston and Winkelnkemper [TW] construct a specific contact form $a$ on $Y$. Given a spin-c Dirac operator $D$ on $Y$, the contact form naturally associates a one parameter family of Dirac operators $D_r = D - \frac{ir}{2}\cl(a)$ for $r\geq0$. When $r>>1$, we prove that the spectrum of $D_r = D_0 - \frac{ir}{2}\cl(a)$ within $[-(r^{1/2})/2, (r^{1/2})/2]$ are almost uniformly distributed. With the result in Part I, it implies that the subleading order term of the spectral flow from $D_0$ to $D_r$ is of order $r (\log r)^{\frac{9}{2}}$. Besides the interests of the spectral flow, the method of this paper provide a tool to analyze the Dirac operator on an open book decomposition.