Extensions of Homeomorphisms between Limbs of the Mandelbrot Set

Homeomorphisms between Limbs of the Mandelbrot Set

Using a family of higher degree polynomials as a bridge, together with complex surgery techniques, we construct a homeomorphism between any two limbs of the Mandelbrot set of common denominator. Induced by these homeomorphisms and complex conjugation we obtain an involution between each limb and itself, whose fixed points form a topological arc. All these maps have counterparts at the combinato...

Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials

Homeomorphisms of the Mandelbrot Set

On subsets of the Mandelbrot set, EM ⊂M, homeomorphisms are constructed by quasi-conformal surgery. When the dynamics of quadratic polynomials is changed piecewise by a combinatorial construction, a general theorem yields the corresponding homeomorphism h : EM → EM in the parameter plane. Each h has two fixed points in EM , and a countable family of mutually homeomorphic fundamental domains. Po...

On Fibers and Renormalization of Julia Sets and Multibrot Sets

We continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in [S3] and [S4]. The question of local connectivity of these sets is discussed in terms of fibers and becomes the question of triviality of fibers. In this paper, the focus is on the behavior of fibers under renormalization and other surgery procedures. We show that triviality of f...

A cottage industry of lax extensions

In this work, we describe an adjunction between the comma category of Set-based monads under the V -powerset monad and the category of associative lax extensions of Set-based monads to the category of V -relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.