**Title:** Equidistribution of affine random walks on tori and nilmanifolds

**Speaker:** Tsviqa Lakrec

**Speaker Info:** Universitat Zurich

**Brief Description:**

**Special Note**:

**Abstract:**

I will discuss a quantitative equidistribution result for the random walk on a torus arising from the action of the group of affine transformations, and a generalization of these results to some nilmanifolds. This is a joint work with Weikun He and Elon Lindenstrauss.Under the assumption that the Zariski closure of the group generated by the linear part acts strongly irreducibly on $\mathbb{R}^d$, we give quantitative estimates (depending only on the linear part of the random walk) for how fast this random walk equidistributes, unless the initial point and the translation part of the affine transformations can be perturbed so that the random walk is trapped in a finite orbit of small cardinality. In particular, this shows that the random walk equidistributes in law to the Haar measure if and only if the random walk is not trapped in a finite orbit. Under a certain condition, we can generalize this theorem to affine and linear random walks on a nilmanifold. This essentially extends the results of Bourgain-Furman-Lindenstrauss-Mozes and He-SaxcĂ© for linear random walks on the torus.

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