arXiv subjects
Publications and source records attributed to Misha Lyubich.
In this paper we prove a priori bounds for an ``elephant eye'' combinatorics. Little $M$-copies specifying these combinatorics are allowed to converge to the cusp of the Mandelbrot set. To handle it, we develope a new geometric tool: uniform thin-thick decompositions for bordered Riemann surfaces.
We study Henon maps which are perturbations of a hyperbolic polynomial p with connected Julia set. We give a complete description of the critical locus of these maps. In particular, we show that for each critical point c of p, there is a primary component of the critical locus asymptotic to the line y = c. Moreover, primary components are conformally equivalent to the punctured disk, and their orbits cover the whole critical set. We also describe the holonomy maps from such a component to itself along the leaves of two natural foliations. Finally, we show that a quadratic Henon map taken along with the natural pair of foliations, is a rigid object, in the sense that a conjugacy between two such maps respecting the foliations is a holomorphic or antiholomorphic affine map.
We consider the structure of substantially dissipative complex Hénon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycles.
We prove that if $ξ$ is a quasisymmetric homeomorphism between Sierpiński carpets that are the Julia sets of postcritically-finite rational maps, then $ξ$ is the restriction of a Möbius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpiński carpet Julia set of a postcritically-finite rational map is finite.