Long run behavior of a dynamical system I have to deal with a dynamical system that looks as follows, with $H$ being the initial state (parameters next to arrows denote transition probabilities between the states $H$, $L_1$ and $L_2$):

I must admit that I have not much clue how to analyze such a system. I would be interested in two basic questions:
(1) If we let the dynamical system run for a long while (as the number of periods $T=1,2,3,...$ goes to infinity), which fraction of the time will be spent in each of the 3 states?
(2) Conditional on having a switch from either $L_1$ or $L_2$ to $H$, which fraction of the time do we experience a switch from $L_2$ to $H$?
Any answers how to deal with that problem would be most welcome. Thanks in advance!
 A: A general result says that (under technical hypotheses which are met in the present case) the proportion of time the system spends in each state $H$, $L_1$ and $L_2$, in the long run, corresponds to the stationary distribution of this state, that is, its weight with respect to the so-called stationary distribution $\pi$. Furthermore, $\pi$ can be computed by solving the so-called balance equations, which roughly assert that, at equilibrium, as there is as much mass leaving each state than arriving on this same state. 
For example, at state $H$, in one step, the mass that leaves is $\varrho_H\pi(H)+(1-\varrho_H)\pi(H)$ and the mass that arrives is $\varrho_H\pi(H)+(1-\varrho_L)\pi(L_1)+(1-\varrho_L)\pi(L_2)$. The total mass $\pi(H)+\pi(L_1)+\pi(L_2)$ is $1$, hence one gets $(1-\varrho_H)\pi(H)=(1-\varrho_L)(1-\pi(H))$, from which the value of $\pi(H)$ follows.
The same approach applied at states $L_1$ and $L_2$ yields the values of $\pi(L_1)$ and $\pi(L_2)$.
Re (2), the transition probabilities from $L_1$ to $H$ and from $L_2$ to $H$ coincide hence the proportions of times when one arrives at $H$ coming from $L_1$ and when one arrives at $H$ coming from $L_2$ are in the ratio $\pi(L_1)/\pi(L_2)$.
A: I think(!) you can reason as follows:
time in H: Th
time in L1: Tl1
time in L2: Tl2
time in H is equal to the number of times a transition to H is made:
Th=Th*pH+Tl1*(1-pL)
Tl1=Th*(1-pH)+Tl1*(plx)
Tl2=tl1*(.....
etc.
And than you can solve it for Th, Tl1 and Tl2. proportion of time spend in Th=Th/(Th+Tl1+Tl2)
Again, I'm not sure this is correct
