2017/07/12 by Eva Uhre, Uhre, Eva
Mathematics · #Mathematical Dynamics and Fractals #Mathematics and Applications #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.1707.03798
The Cantor locus is the unique hyperbolic component, in the moduli space of quadratic rational maps \bf rat2, consisting of maps with totally disconnected Julia sets. Whereas the geometry and dynamics of the Cantor locus is well understood, its boundary and the dynamics of the maps on the boundary are not. In this paper, we explore the dynamics near the parabolic parts of the boundary. We introduce the concept dynamical marking of a map g, relative to the quadratic, parabolic polynomial P\opq(z)=\opq z+z2, with \opq=e2πip/q. A dynamical marking (x,ψ) of g is a conjugacy ψ between P\opq (on its parabolic basin of 0) and g, which marks the dynamical position of the critical values v1=ψ(-(λ2)/(4)), v2=ψ(x) of g. We construct a local parametrization of the Cantor locus, which parametrizes by dynamical marking, and use it to prove a form of stability of dynamical marking. That is, for sequences in the Cantor locus, of fixed dynamical marking x and such that the eigenvalue λk of the attracting fixed point tends to \opq subhorocyclicly, either the sequence converges to the unique parabolic parameter in the boundary, which has a fixed point eigenvalue \opq and which is marked by x relative to P\opq. Or, the sequence tends to infinity in \bf rat2, and certain representatives Gλk,ak have rescaled limits in the boundary of the Cantor locus within \bf rat2.