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Stochastic graphs, completely simple semigroups and one-sided Markov shifts

00:00 22-12-2010

&lt;p&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;We study stochastic graphs and one-sided Markov shifts corresponding to positively recurrent Markov chains with countable (finite or infinite) state spaces. &lt;/span&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: navy; FONT-SIZE: 14px"&gt;&lt;/span&gt;&lt;/font&gt;&lt;/p&gt; &lt;p&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;The following classification problem is considered: &lt;/span&gt;&lt;span style="COLOR: black"&gt;&lt;/span&gt;&lt;/font&gt;&lt;/p&gt; &lt;p&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;when two one-sided Markov shifts are isomorphic up to a measure preserving isomorphism?&lt;/span&gt;&lt;span style="COLOR: black"&gt;&lt;/span&gt;&lt;/font&gt;&lt;/p&gt; &lt;p&gt;&lt;span style="COLOR: black"&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;font size="2"&gt;&amp;nbsp;&lt;/font&gt;&lt;/font&gt;&lt;/span&gt;&lt;/p&gt; &lt;p&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;We show that every ergodic one-sided Markov shift&amp;nbsp;$T=T_G$&lt;i&gt; &lt;/i&gt;corresponding to stochastic&lt;/span&gt;&lt;span style="COLOR: black"&gt;&lt;/span&gt;&lt;/font&gt;&lt;/p&gt; &lt;p&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;graph&amp;nbsp;$G$&lt;i&gt; &lt;/i&gt;can be represented in a canonical form&amp;nbsp;$T=T_{H(G)}$ by means of a canonical &lt;/span&gt;&lt;span style="COLOR: black"&gt;&lt;/span&gt;&lt;/font&gt;&lt;/p&gt; &lt;p&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;(uniquely determined by $T$) stochastic graph $H(G)$. In the canonical form, two such shifts&amp;nbsp;$T_{G_1}$ and&lt;/font&gt;&lt;/span&gt;&lt;/p&gt; &lt;p&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;$T_{G_2}$ are isomorphic if and only if their canonical stochastic graphs&amp;nbsp;$H(G_1)$ and&amp;nbsp;$H(G_2)$ are&lt;/span&gt;&lt;span style="COLOR: black"&gt;&lt;/span&gt;&lt;/font&gt;&lt;/p&gt; &lt;p&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;isomorphic. The stochastic graph&amp;nbsp;$H(G)$ is constructed by a special semigroup&amp;nbsp;$S(G)$ acting&lt;/span&gt;&lt;span style="COLOR: black"&gt;&lt;/span&gt;&lt;/font&gt;&lt;/p&gt; &lt;p&gt;&lt;font face="Arial, Helvetica, sans-serif"&gt;&lt;span style="FONT-FAMILY: &amp;#39;Arial&amp;#39;, &amp;#39;sans-serif&amp;#39;; COLOR: black; FONT-SIZE: 14px"&gt;on the vertexes of $G$.&lt;/span&gt;&lt;span style="COLOR: black"&gt;&lt;/span&gt;&lt;/font&gt;&lt;/p&gt;

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Stochastic graphs, completely simple semigroups and one-sided Markov shifts