Julia sets as buried Julia components

Abstract

Let f f be a rational map with degree d ≥ 2 d\geq 2 whose Julia set is connected but not equal to the whole Riemann sphere. It is proved that there exists a rational map g g such that g g contains a buried Julia component on which the dynamics is quasiconformally conjugate to that of f f on the Julia set if and only if f f does not have parabolic basins and Siegel disks. If such g g exists, then the degree can be chosen such that deg ⁡ ( g ) ≤ 7 d − 2 \deg (g)\leq 7d-2 . In particular, if f f is a polynomial, then g g can be chosen such that deg ⁡ ( g ) ≤ 4 d + 4 \deg (g)\leq 4d+4 . Moreover, some quartic and cubic rational maps whose Julia sets contain buried Jordan curves are also constructed.