We prove that for a minimal rotation T on a 2-step nilmanifold and any measure mu, the push-forward T^n(mu) of mu under T^n tends toward Haar measure if and only if mu projects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density 1. These results strengthen Parry's result that such systems are uniquely ergodic. Extending the work of Furstenberg, we prove an analogous theorem for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally, we characterize limits of T^n(mu) for some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties which strengthen unique ergodicity in a way analogous to how mixing and weak mixing strengthen ergodicity for measure preserving systems.
Integer part independent polynomial averages and applications along primes
利用多项式序列的等分布特性，遵循Leibman开发的方法（“在nilmanifold上平移的多项式序列的遍历平均的逐点收敛。遍历论动力学Systems，25（2005）no. 1，201-213”）和Frantzikinakis（“Multiple recurrence and convergence for Hardy field sequences of polynomial growth. Journal d 'Analyse Mathematique，112（2010），79-135”和“在nilmanifold上稀疏序列的等分布。Journal d 'Analyse Mathematique，109（2009），353-395”），我们证明了遍历平均值与由实值强独立多项式的整数部分给出的迭代，在平均值中收敛到“右”期望极限。这些结果，通过Furstenberg的对应原理，立即组合应用，同时结合这些结果与方法从“多项式多维Szemerédi定理沿移位素数。Israel J. Math.，194（2013），no. 1，331-348”和“Closest integer polynomial multiple recurrence along shifted primes.遍历论动力学系统，1-20。doi：10.1017/etds.2016.40”我们得到了单个序列以及沿着…
