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 Post subject: Ergodicity
PostPosted: Mon, 30 Nov 2009 16:42:56 UTC 
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For every n\geq 1, does there exist a probability space (\Omega,\mathcal{F},\mathbb{P}) and a map T\colon\Omega\to\Omega such that T^{\times n}\colon\prod^n\Omega\to\prod^n\Omega is ergodic but T^{\times(n+1)} is not?

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


Last edited by outermeasure on Tue, 1 Dec 2009 01:40:21 UTC, edited 1 time in total.

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 Post subject: Re: Ergodicity
PostPosted: Mon, 30 Nov 2009 20:27:36 UTC 
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outermeasure wrote:
For every n\geq 1, does there exist a probability space (\Omega,\mathcal{F},\mathbb{P}) and a map T\colon X\to X such that T^{\times n}\colon\prod^n\Omega\to\prod^n\Omega is ergodic but T^{\times(n+1)} is not?


May I assume X=(\Omega, \mathcal{F},\mathbb{P})?

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 Post subject:
PostPosted: Tue, 1 Dec 2009 01:39:40 UTC 
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Oops... corrected

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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