If you are asking what P^n looks like for large n, the answer is:

A + B (b^n) + C (c^n), where:

A = {1 0 0 0 | 0 1 0 0 | 16/45 29/45 0 0 | 19/45 26/45 0 0 }

and b and c are less than 1

I can produce exact values in about 6 hours, if you are interested, but this gives the long-run approximation.

It took less time than I thought

b = 0.7 (7/10)

B = {0 0 0 0 | 0 0 0 0 | -10/27 -17/27 7/9 2/9 | -10/27 -17/27 7/9 2/9}

c = 0.25 (1/4)

C = { 0 0 0 0 | 0 0 0 0 | 2/135 -2/135 2/9 -2/9 | -7/135 7/135 -7/9 7/9}

writing x = [0.1 0.2 0.7 0], we get P^n x is

[0.1 + (16/45)0.7 0.2+(29/45)0.7 0 0] + b^n 0.7 [-10/27 -17/27 7/9 2/9] + c^n 0.7 [2/135 -2/135 2/9 -2/9]