Tent map

The tent map is a piecewise-linear map on the unit interval defined by T(x) = 2x for x < 1/2 and T(x) = 2(1−x) for x ≥ 1/2. Its graph resembles a tent, hence the name. Despite its simplicity — it is nothing more than two straight lines — the tent map exhibits fully developed chaos with a uniform invariant measure, making it one of the most analytically tractable chaotic systems known.

The tent map is dynamically conjugate to the logistic map at parameter r = 4 via a nonlinear change of coordinates. This conjugacy is profound: it proves that the logistic map's chaos is not a special property of quadratic functions but a structural feature shared by a broad class of unimodal maps. The tent map is also conjugate to the binary shift map, which means that its dynamics can be completely understood in terms of symbolic dynamics — sequences of 0s and 1s that encode whether an orbit falls in the left or right half of the interval.

This exact solvability makes the tent map a cornerstone of ergodic theory. Its uniform invariant measure allows explicit calculation of time averages, Lyapunov exponents, and correlation functions. In this sense, the tent map is to chaotic dynamics what the harmonic oscillator is to linear systems: the simplest exactly solvable case against which all others are compared.