Parameter scaling for the Fibonacci point

Details Parameter scaling for the Fibonacci point Parameter scaling for the Fibonacci point

1996-06-15

arxiv.org/abs/math/9606218v1

Mathematics

Dynamical Systems

Scientific paper

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.

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