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- W4288365764 abstract "We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci ${cal M}_d$ for unicritical polynomials $f_c(z)=z^d+c$. It is known that these parameters are structurally unstable and have stochastic dynamics. We prove $C^{1+frac{alpha}{d}-epsilon}$-conformality, $alpha = 2-mbox{HD},({cal J}_{c_0})$, of the parameter-phase space similarity maps $Upsilon_{c_0}(z):mathbb{C}mapsto mathbb{C}$ at typical $c_0in partial {cal M}_d$ and establish that globally quasiconformal similarity maps $Upsilon_{c_0}(z)$, $c_0in partial {cal M}_d$, are $C^1$-conformal along external rays landing at $c_0$ in $mathbb{C}setminus {cal J}_{c_0}$ mapping onto the corresponding rays of ${cal M}_d$. This conformal equivalence leads to the proof that the $z$-derivative of the similarity map $Upsilon_{c_0}(z)$ at typical $c_0in partial {cal M}_d$ is equal to $1/{cal T}'(c_0)$, where ${cal T}(c_0)=sum_{n=0}^{infty}(D(f_{c_0}^n)(c_0))^{-1}$ is the transversality function. The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on $partial {cal M}_d$. In particular, we will explain how a non-linear dynamics creates abundance of hedgehog neighborhoods in $partial {cal M}_d$ effectively blocking a good access of $partial {cal M}_d $ from the outside." @default.
- W4288365764 created "2022-07-29" @default.
- W4288365764 creator A5083540686 @default.
- W4288365764 creator A5086286755 @default.
- W4288365764 date "2019-04-20" @default.
- W4288365764 modified "2023-10-18" @default.
- W4288365764 title "Analytic structures and harmonic measure at bifurcation locus" @default.
- W4288365764 doi "https://doi.org/10.48550/arxiv.1904.09434" @default.
- W4288365764 hasPublicationYear "2019" @default.
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