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

00:00 22-12-2010
<p><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">We study stochastic graphs and one-sided Markov shifts corresponding to positively recurrent Markov chains with countable (finite or infinite) state spaces. </span><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: navy; FONT-SIZE: 14px"></span></font></p> <p><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">The following classification problem is considered: </span><span style="COLOR: black"></span></font></p> <p><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">when two one-sided Markov shifts are isomorphic up to a measure preserving isomorphism?</span><span style="COLOR: black"></span></font></p> <p><span style="COLOR: black"><font face="Arial, Helvetica, sans-serif"><font size="2"> </font></font></span></p> <p><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">We show that every ergodic one-sided Markov shift $T=T_G$<i> </i>corresponding to stochastic</span><span style="COLOR: black"></span></font></p> <p><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">graph $G$<i> </i>can be represented in a canonical form $T=T_{H(G)}$ by means of a canonical </span><span style="COLOR: black"></span></font></p> <p><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px"><font face="Arial, Helvetica, sans-serif">(uniquely determined by $T$) stochastic graph $H(G)$. In the canonical form, two such shifts $T_{G_1}$ and</font></span></p> <p><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">$T_{G_2}$ are isomorphic if and only if their canonical stochastic graphs $H(G_1)$ and $H(G_2)$ are</span><span style="COLOR: black"></span></font></p> <p><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">isomorphic. The stochastic graph $H(G)$ is constructed by a special semigroup $S(G)$ acting</span><span style="COLOR: black"></span></font></p> <p><font face="Arial, Helvetica, sans-serif"><span style="FONT-FAMILY: 'Arial', 'sans-serif'; COLOR: black; FONT-SIZE: 14px">on the vertexes of $G$.</span><span style="COLOR: black"></span></font></p>