Pointwise convergence of some continuous-time polynomial ergodic averages

In this paper, we study the pointwise convergence of centain continuous-time polynomial ergodic averages. Our approach is based on the topological models of measurable flows. One of the main results of this paper is as follows: Let $a\in \mathbb{R}$, $Q\in \mathbb{R}[t]$ with $\text{deg}\ Q\ge 2$. Let $(X,\mathcal{X},\mu, (T^{t})_{t\in \mathbb{R}})$ and $(X,\mathcal{X},\mu, (S^{t})_{t\in \mathbb{R}})$ be two measurable flows. Then for any $f_1, f_2, g\in L^{\infty}(\mu)$, the limit \begin{equation*} \lim\limits_{M\to\infty}\frac{1}{M}\int_{0}^{M}f_1(T^{t}x)f_2(T^{at}x)g(S^{Q(t)}x)dt \end{equation*} exists for $\mu$-a.e. $x\in X$. In particular, we are able to build a pointwise ergodic theorem involving geodesic flow and horocycle flow.