Iterates

Iterates in mathematics denote the successive outputs of repeatedly applying a function. Let f: X → X be a function. The nth iterate, denoted f^n, is defined by f^1 = f and f^{n+1} = f ∘ f^n for n ≥ 1. Often f^0 denotes the identity on X. For a point x in X, the orbit under f is the sequence x, f(x), f^2(x), f^3(x), and so on.

Periodic points arise when an iterate returns to the starting point. A point x is periodic of period p if f^p(x) = x for some p > 0; if p is the smallest such integer, the orbit forms a finite cycle. In dynamical systems, one studies properties of orbits, including convergence to fixed points, periodic orbits, attractors, and more complex behavior such as chaos.

Fixed-point iteration is a common method that uses iterates to approximate solutions of equations of the form f(x) = x. Starting from an initial guess x0, one forms x_{n+1} = f(x_n). Under favorable conditions, such as the contraction mapping principle on a complete metric space, the sequence of iterates converges to a fixed point.

Iterates also underpin various numerical algorithms. The power method uses repeated multiplication by a matrix to approximate an eigenvector associated with the dominant eigenvalue. Newton’s method can be viewed as an iteration for finding roots, and many root-finding algorithms can be framed as fixed-point iterations.

In dynamics and fractals, iterating simple functions can generate intricate structures, as seen in the logistic map or through the iteration of complex quadratic polynomials that define the Mandelbrot set’s boundary.