Dynamics of the Iteration x_n+1 = x_n² - 2.

This paper analyzes the real dynamical system generated by the quadratic iteration x_n+1 = x_n²- 2. with particular attention to the distribution and behavior of rational and irrational initial values. The map admits exactly two real fixed points, 2 and -1, and all orbits with x₀ outside [-2,2] diverge monotonically to +infinity . For x₀ inside [-2,2], the orbit remains confined to this interval, and its structure is examined through the full backward‑iteration tree defined by x= + or - √2. This construction yields two countable dense subsets of [-2,2] consisting of all rational and irrational preimages of the fixed points. Their forward orbits converge to 2 and -1, respectively, and the nested‑radical representation provides a complete ordering of these preimage sets. Beyond these convergent families, the backward‑iteration framework produces uncountably many additional dense subsets of [-2,2], each arising from a distinct irrational seed whose orbit is not a preimage of either fixed point. These sets are pairwise disjoint and contain both rational and irrational elements, yet none of their forward orbits converge or diverge; instead, they remain perpetually in [-2,2] while exhibiting non‑periodic, non‑convergent behavior. The resulting decomposition of [-2,2] into countably many convergent branches and uncountably many non‑convergent branches highlights the intricate, fractal‑like structure inherent in this quadratic map.