On the convergence of the time average for skew-product structure and multiple ergodic system

In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation and observation function are neglected in some measure-theoretical sense. The theorems extend the classical results which have been established for continuous dynamical systems or continuous observation functions. Meanwhile, on the torus $\mathbb{T}^{d}$ with special rotation, we prove the pointwise convergence of multiple ergodic average $\disp \f 1 N \sum_{n=0}^{N-1} f_{1}(R_{\alpha}^{n}x)f_{2}(R_{\alpha}^{2n}x)$ in $\mathbb{T}^{d}$.