Orbit — the mathematics of iteration
Parent: math
Definition. Let be a set and a map. The (forward) orbit of a point is
If is invertible one also considers the full orbit .
One definition — but the mathematics of orbits depends entirely on what structure carries. Each row is a child note:
| structure on | field | core question about orbits |
|---|---|---|
| none (a set) | discrete-dynamical-systems | fixed points, periodic points, basins — where do orbits settle? |
| topology (interval) | topological-dynamics | which periods force which — Sharkovskii's order, period 3 ⟹ chaos |
| a measure | ergodic-theory-of-orbits | statistics along the orbit — time average = space average (Birkhoff) |
| a finite alphabet | symbolic-dynamics | orbits as sequences of symbols — coding, entropy, counting |
| complex-dynamics | boundedness of orbits — Julia sets, the Mandelbrot set | |
| + number theory | gauss-map | orbits are continued fractions — digit statistics of almost every real |
| a number field | arithmetic-dynamics | rational points with finite orbits — heights, Northcott |
| , piecewise affine | collatz-orbit-statistics | the 3n+1 orbit: everything above converges on one open problem |
The Collatz problem itself is home at collatz (number theory); generalized-collatz shows why the family of such orbit questions is undecidable. The orbit-statistics child here is the dynamics-side view of the same object.