Minimal Dynamical Systems on Connected Odd Dimensional Spaces

Lin, Huaxin

Canadian journal of mathematics, Tome 67 (2015) no. 4, pp. 870-892

Résumé

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.

Détail

DOI : 10.4153/CJM-2014-035-7

Mots-clés : 46L35, 46L05, Minimal dynamical systems