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...

We study homeomorphisms h : X onto −→ Y between two bounded domains in Rn having finite conformal energy E[h] = ∫ X ||Dh(x) || n dx < ∞ , h ∈ W (X,Y). We consider the behavior of such mappings, including continuous extension to the closure of X and injectivity of h : X → Y. In general, passing to the weak W 1,n-limit of a sequence of homeomorphisms hj : X → Y one loses injectivity. However, if ...

The classical Tarski theorem on topological semantics of modal logic states that the logic S4 is complete in R for each n. Recently several authors have considered logics of dynamic topological systems, which is a topological space and a function on it. In [1] a bimodal logic S4C was introduced and proven to be complete with respect to the class of all continuous dynamic systems. A number of po...

For a compact metric space X let G = H(X) denote the group of self homeomorphisms with the topology of uniform convergence. The group G acts on itself by conjugation and we say that X satisfies the topological Rohlin property if this action has dense orbits. We show that the Hilbert cube, the Cantor set and, with a slight modification, also even dimensional spheres, satisfy this property. We al...