We consider the directed graph $G = ( S , A )$ where $$\left\{ \begin{array} { l } S = \{ 1,2,3,4 \} \\ A = \{ ( 1,2 ) , ( 2,1 ) , ( 1,3 ) , ( 3,1 ) , ( 1,4 ) , ( 4,1 ) , ( 2,3 ) , ( 3,2 ) , ( 2,4 ) , ( 4,2 ) , ( 3,4 ) , ( 4,3 ) \} \end{array} \right.$$ We assume that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the three other vertices of the graph. Show that the sequence of matrices $\left( T ^ { k } \right) _ { k \in \mathbb { N } }$ converges and identify geometrically the endomorphism canonically associated with the limit matrix.
We consider the directed graph $G = ( S , A )$ where
$$\left\{ \begin{array} { l } S = \{ 1,2,3,4 \} \\ A = \{ ( 1,2 ) , ( 2,1 ) , ( 1,3 ) , ( 3,1 ) , ( 1,4 ) , ( 4,1 ) , ( 2,3 ) , ( 3,2 ) , ( 2,4 ) , ( 4,2 ) , ( 3,4 ) , ( 4,3 ) \} \end{array} \right.$$
We assume that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the three other vertices of the graph. Show that the sequence of matrices $\left( T ^ { k } \right) _ { k \in \mathbb { N } }$ converges and identify geometrically the endomorphism canonically associated with the limit matrix.