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2025 ISI Entrance UGB, 4
Tags: mapping , composition , complex number , periodicity , inductive definition , algebra
Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the [i]period[/i] of $z$. Determine the total number of points in $S^1$ of period $2025$. (Hint : $2025 = 3^4 \times 5^2$)