2019/04/20 by Graczyk, Jacek, Świątek, Grzegorz · 1 citation
#37F45 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1904.09434
We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci \cal Md for unicritical polynomials fc(z)=zd+c. It is known that these parameters are structurally unstable and have stochastic dynamics. We prove C^1+\fracαd-ε-conformality, α= 2-HD (\cal Jc0), of the parameter-phase space similarity maps Υc0(z):ℂ↦ ℂ at typical c0∈ ∂ \cal Md and establish that globally quasiconformal similarity maps Υc0(z), c0∈ ∂ \cal Md, are C1-conformal along external rays landing at c0 in ℂ∖ \cal Jc0 mapping onto the corresponding rays of \cal Md. This conformal equivalence leads to the proof that the z-derivative of the similarity map Υc0(z) at typical c0∈ ∂ \cal Md is equal to 1/\cal T'(c0), where \cal T(c0)=∑n=0∞(D(fc0n)(c0))-1 is the transversality function. The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on ∂ \cal Md. In particular, we will explain how a non-linear dynamics creates abundance of hedgehog neighborhoods in ∂ \cal Md effectively blocking a good access of ∂ \cal Md from the outside.