Continued Fraction Gauss Map
Iterate T(x) = {1/x} — invariant measure dμ = dx/((1+x)ln2), convergents, and CF expansion
Controls
Starting value x₀
φ−1 = 0.618…
e−2 = 0.718…
π−3 = 0.1415…
√2−1 = 0.414…
Compute
Histogram bins:
80
Iterations:
2000
CF expansion [a₀; a₁, a₂, …]
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Best rational approx (p/q)
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Gauss map:
T(x) = {1/x} = 1/x − ⌊1/x⌋
The floor ⌊1/x⌋ gives the next CF coefficient aₙ.
Invariant measure:
μ(A) = ∫_A dx/((1+x)ln2) — the Gauss–Kuzmin–Wirsing distribution.
φ
has CF [1;1,1,1,…] — the "most irrational" number; worst rational approximation.