Gauss Map

The Gauss map is the dynamical system T: (0,1] → (0,1] defined by T(x) = {1/x}, where {·} denotes the fractional part. Iterating the Gauss map on a number x ∈ (0,1) generates precisely the sequence of partial quotients in the continued fraction expansion of x. The Gauss map is therefore not merely a tool for computing continued fractions; it is their dynamical essence.

The Gauss map is a piecewise continuous transformation with infinitely many branches, each corresponding to a possible integer value of the partial quotient. Its ergodic properties are well-understood: it preserves the Gauss measure dμ = (1/ln 2) dx/(1+x), and with respect to this measure, the partial quotients are identically distributed random variables with distribution P(aₙ = k) = log₂((k+1)²/(k(k+2))).

The entropy of the Gauss map is log(π²/6) = log ζ(2), connecting the dynamical complexity of continued fractions to the Riemann zeta function. This constant appears again in the Khinchin's constant, the geometric mean of partial quotients for almost all real numbers. The Gauss map thus serves as a bridge between number theory, dynamical systems, and information theory — a single transformation whose orbits encode arithmetic structure.