The previous answers correspond to the recursive model of the form $$[x_n,y_n]=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1})]$$. However, the state-space model of the question is of the form \begin{align} [x_n,y_n]&=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1},x_n)]\\ &=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1},f(x_{n-1},y_{n-1}))] \end{align} The latter equation is the correct form that should be applied with the methods proposed by the previous answers. Another method could be to use NestList
h[{x_, y_}] := {x + 1/2 (Sqrt[1 - y^2] - x), y + 1/2 (Sqrt[1 - (x + 1/2 (Sqrt[1 - y^2] - x))^2] - y)}; NestList[h, {N[0 + 1/2 (Sqrt[1 - 0^2] - 0)], N[0 + 1/2 (Sqrt[1 - (0 + 1/2 (Sqrt[1 - 0^2] - 0))^2] - 0)]}, 10] {{0.5, 0.433013}, {0.700694, 0.573237}, {0.760042, 0.611556}, {0.775621, 0.621377}, {0.779567, 0.623848}, {0.780556, 0.624467}, {0.780804, 0.624622}, {0.780865, 0.624661}, {0.780881, 0.62467}, {0.780885, 0.624673}, {0.780886, 0.624673}}