When are ϕ and 1/ϕ attractors? Projective dynamics and empirical meaning – A first approach by Stefan Geier et al.: Abstract
When are ϕ and 1/ϕ attractors? Projective dynamics and empirical meaning – A first approach by Stefan Geier et al. Abstract A number is an attractor only relative to a specified dynamics. This distinction is especially important for the golden ratio ϕ=(1+√5)/2 and its positive reciprocal 1/ϕ. We show that they are exact attracting fixed points of the reciprocally conjugate Möbius maps T(x)=1+1/x and U(y)=1/(1+y) on the positive half-line. Cross-ratio coordinates linearize both maps globally: the signed projective residual is multiplied at every step by q=−ϕ⁻², so its amplitude contracts by ϕ⁻² and its same-phase two-step or squared amplitude by ϕ⁻⁴. The Fibonacci matrix selects one Perron–Frobenius eigenray [ϕ:1]; ϕ and 1/ϕ are therefore reciprocal coordinate descriptions of one projective object, not independent attractors. Continued-fraction dynamics demonstrate map dependence, and a theorem for positive second-order recurrences shows that attraction is generic while the exact golden...
