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Abstract Let $$\Omega $$ be a connected open set in the plane and $$\gamma : [0,1] \to \overline {\Omega }$$ a path such that $$\gamma ((0,1)) \subset \Omega $$ . We show that the path $$\gamma $$ can be “pulled tight” to a unique shortest path which is homotopic to $$\gamma $$ , via a homotopy h with endpoints fixed whose intermediate paths $$h_t$$ , for $$t \in [0,1)$$ , satisfy $$h_t((0,1)) \subset \Omega $$ . We prove this result even in the case when there is no path of finite Euclidean length homotopic to $$\gamma $$ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a “shortest” path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.more » « less
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Abstract. Local similarity between the Mandelbrot set and quadratic Julia sets manifests itself in a variety of ways. We discuss a combinatorial one, in the language of geodesic laminations. More precisely, we compare quadratic invariant laminations representing Julia sets with the so-called Quadratic Minor Lamination (QML) representing a locally connected model of the Mandelbrot set. Similarly to the construction of an invariant lamination by pullbacks of certain leaves, we describe how QML can be generated by properly understood pullbacks of certain minors. In particular, we show that the minors of all non-renormalizable quadratic laminations can be obtained by taking limits of “pullbacks” of minors from the main cardioidmore » « less
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Abstract. It is known that a holomorphic motion (an analytic version of an isotopy) of a set X in the complex plane C always extends to a holomorphic motion of the entire plane. In the topological category, it was recently shown that an isotopy h : X [0; 1] -> C, starting at the identity, of a plane continuum X also always extends to an isotopy of the entire plane. Easy examples show that this result does not generalize to all plane compacta. In this paper we will provide a characterization of isotopies of uniformly perfect plane compacta X which extend to an isotopy of the entire plane. Using this characterization, we prove that such an extension is always possible provided the diameters of all components of X are uniformly bounded away from zero.more » « less
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