Strictly Ergodic Models Under Face and Parallelepiped Group Actions
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Strictly Ergodic Models Under Face and Parallelepiped Group Actions Wen Huang1,2 · Song Shao1,2 · Xiangdong Ye1,2
Received: 12 September 2016 / Revised: 24 January 2017 / Accepted: 16 February 2017 / Published online: 13 March 2017 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2017
Abstract The Jewett–Krieger theorem states that each ergodic system has a strictly ergodic topological model. In this article, we show that for an ergodic system one may require more properties on its strictly ergodic model. For example, the orbit closure of points in diagonal under face transforms may be also strictly ergodic. As an application, we show the pointwise convergence of ergodic averages along cubes, which was firstly proved by Assani (J Anal Math 110:241–269, 2010). Keywords Ergodic averages · Model · Cubes · Face transformations Mathematics Subject Classification 37B05 · 37A05
Huang is partially supported by NNSF for Distinguished Young Schooler (11225105), and all authors are supported by NNSF of China (11371339, 11431012, 11571335) and by the Fundamental Research Funds for the Central Universities.
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Song Shao [email protected] Wen Huang [email protected] Xiangdong Ye [email protected]
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Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei 230026, Anhui, People’s Republic of China
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Department of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China
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1 Introduction In this section, we will state the main results of the paper and give main ideas of proofs. 1.1 Main Results Throughout this paper, by a topological dynamical system (t.d.s. for short) we mean a pair (X, T ), where X is a compact metric space and T is a homeomorphism from X to itself. A measurable system (m.p.t. for short) is a quadruple (X, X , μ, T ), where (X, X , μ) is a Lebesgue probability space and T : X → X is an invertible measurepreserving transformation. Let (X, X , μ, T ) be an ergodic m.p.t. We say that ( Xˆ , Tˆ ) is a topological model (or just a model) for (X, X , μ, T ) if ( Xˆ , Tˆ ) is a t.d.s. and there exists an invariant probability measure μˆ on the Borel σ -algebra B( Xˆ ) such that the systems (X, X , μ, T ) and ( Xˆ , B( Xˆ ), μ, ˆ Tˆ ) are measure theoretically isomorphic. The well-known Jewett–Krieger’s theorem [14,15] states that every ergodic system has a strictly ergodic model. We note that one can add some additional properties to the topological model. For example, in [16] Lehrer showed that the strictly ergodic model can be required to be a topological (strongly) mixing system in addition. Let ( Xˆ , Tˆ ) be a t.d.s. Write (x, . . . , x) (2d times) as x [d] . Let F [d] , G [d] and Q[d] ( Xˆ ) be the face group of dimension d, the parallelepiped group of dimension d and the dynamical parallelepiped of dimension d, respectively (see Sect. 2 for definitions). The orbit closure of x [d] under the face group action will be denoted by F [d
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