At each step the point moves 2/3 of the way toward a left or right boundary. Simultaneously it moves 2/3 of the way toward the top or bottom.
The result, surprisingly, is a fractal know as "Cantor dust". The same pattern could also be generated by repeatedly removing the "middle cross" (five squares) in each three by three square area.
The fractal dimension is log(4)/log(3) = about 1.26.