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2024, Volume 44, Issue 4: 983-996. Doi: 10.3934/dcds.2023135

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Ergodicity for higher dimension cylindrical cascades

Patricia Cirilo

Universidade Federal de São Paulo, Brazil

Received: May 2022

Revised: October 2023

Early access: November 2023

Published: April 2024

Partially supported by FAPESP-Brazil grant #2013/16553-9, #2019/09045-3, #2019/10269-3

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Abstract

In this paper we study non-compact extensions of toral translations. We prove that the generic higher dimensional cylindrical cascade (i.e, a skew product in $ \mathbb{T}^d \times \mathbb{R}^{r} $) above a Liouvillian irrational translation is ergodic. Since Liouvillian vectors form a residual set, as a corollary we show that ergodicity is a generic property in the set of skew products over rigid rotations.

Keywords:

Cylindrical cascades,
skew product,
toral translations,
ergodicity,
Liouvillian rotation,
infinite-measure preserving transformations.

Mathematics Subject Classification: Primary: 37A05, 37B20; Secondary: 37A20, 37A40, 28D05.

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Figure 1. Position of the rectangle for a given $ m_n $

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References

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