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

TL;DR

This paper extends ergodic theorems to certain discontinuous skew-product systems and proves pointwise convergence of multiple ergodic averages on tori with special rotations, broadening classical results.

Contribution

It introduces uniform and semi-uniform ergodic theorems for discontinuous skew-product transformations under measure-theoretic conditions, and establishes convergence of multiple ergodic averages on tori.

Findings

Extended classical ergodic theorems to discontinuous systems.

Proved pointwise convergence of multiple ergodic averages on tori.

Demonstrated the applicability of ergodic results beyond continuous systems.

Abstract

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 $\mathbb{T}^{d}$.