A FINITE APPROXIMATE LIVŠIC THEOREM FOR ANOSOV DIFFEOMORPHISMS

In this paper, we prove a finite approximate version of the Livšic theorem for Anosov diffeomorphisms. Let $f$ be a transitive Anosov diffeomorphism and $\varphi\in C^{\alpha}(M)$. We show that there exist $0<\beta\le \alpha, C>0$ and $\tau>0$ such that for any $\varepsilon>0$, if $\left|\sum_{i=0}^{n}\varphi\left(f^i(p)\right)\right|\le \varepsilon$ for each periodic point $ p=f^n(p)$ with $n\le \varepsilon^{-\frac{1}{2}}$, then there exist $u\in C^{\beta}(M)$ and $h\in C^{\beta}(M)$ such that $\varphi=u\circ f-u+h. $ Moreover, $\|u\|_{C^\beta}\le C$ and $\|h\|_{C^\beta}\le C\varepsilon^\tau.$