Abstract

We study the ergodic integrals of the horocycle flows $h_{\rho }$ of $C^r$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^3$ contact Anosov flow with orientable strong-stable distribution $E_{-}$, we show that $\left|\frac{1}{T}{\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->}_0^T\phi \circ h_{\rho }(x)d<!--FUNCTION\; APPLICATION-->\rho -\mu (\phi )\right|\le \frac{C}{T^{𝜖}}\parallel \phi {\parallel }_{C^3}$ for some $𝜖>0$, with $\mu$ the invariant measure of $h_{\rho }$. We thereby implement the toy model program of Giulietti–Liverani (2017) in the natural setting of geodesic flows in variable negative curvature, where nontrivial resonances exist.

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