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Minimal Dynamical Systems on Connected Odd Dimensional Spaces

Published online by Cambridge University Press:  20 November 2018

Huaxin Lin
Huaxin Lin*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai, China 20062 and , Department of Mathematics, University of Oregon, Eugene, Oregon 97402, USA e-mail: hlin@uoregon.edu
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Abstract

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Let $\beta:\,{{S}^{2n+1}}\,\to \,{{S}^{2n+1}}$ be a minimal homeomorphism $\left( n\,\ge \,1 \right)$. We show that the crossed product $C\left( {{S}^{2n+1}} \right)\,{{\rtimes }_{\beta}}\mathbb{Z}$ has rational tracial rank at most one. Let $\Omega $ be a connected, compact, metric space with finite covering dimension and with ${{H}^{1}}\left( \Omega ,\,\mathbb{Z} \right)\,=\,\left\{ 0 \right\}$. Suppose that ${{K}_{i}}\left( C\left( \Omega \right) \right)\,=\,\mathbb{Z}\,\oplus \,{{G}_{i}}$, where ${{G}_{i}}$ is a finite abelian group, $i\,=\,0,\,1$. Let $\beta :\,\Omega \,\to \,\Omega $ be a minimal homeomorphism. We also show that $A\,=\,C\left( \Omega \right)\,{{\rtimes }_{\beta}}\,\mathbb{Z}$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on $X\,\times \,\Omega $, where $X$ is the Cantor set.

Keywords

46L3546L05Minimal dynamical systems
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 4 , 01 August 2015 , pp. 870 - 892
Copyright
Copyright © Canadian Mathematical Society 2015

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