Parameter scaling for the Fibonacci point

Abstract

We prove geometric and scaling results for the real Fibonacci parameter value in the quadratic family $f_c(z) = z^2+c$. The principal nest of the Yoccoz parapuzzle pieces has rescaled asymptotic geometry equal to the filled-in Julia set of $z^2-1$. The modulus of two such successive parapuzzle pieces increases at a linear rate. Finally, we prove a ``hairiness" theorem for the Mandelbrot set at the Fibonacci point when rescaling at this rate.