The deep significance of the question of the Mandelbrot set's local connectedness? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:39:25Z http://mathoverflow.net/feeds/question/95701 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95701/the-deep-significance-of-the-question-of-the-mandelbrot-sets-local-connectedness The deep significance of the question of the Mandelbrot set's local connectedness? David Feldman 2012-05-01T22:03:41Z 2012-10-07T03:02:07Z <p>I am given to understand that the celebrated open problem (MLC) of the Mandelbrot set's local connectness has broader and deeper significance deeper than some mere curiosity of point-set topology. </p> <p>From <a href="http://en.wikipedia.org/wiki/Mandelbrot_set" rel="nofollow">http://en.wikipedia.org/wiki/Mandelbrot_set</a> I see that conjecture has implications concerning the structure of the Mandelbrot set itself, but I don't think I grasp its broadest implications for complex dynamics and matters beyond.</p> <p>Request: Could someone explain the proper current context in which to view MLC and/or how the world would look it its full glory if MLC has a positive answer? Alternatively, please give a pointer to somewhere in the literature that does the same.</p> http://mathoverflow.net/questions/95701/the-deep-significance-of-the-question-of-the-mandelbrot-sets-local-connectedness/95704#95704 Answer by Tom Leinster for The deep significance of the question of the Mandelbrot set's local connectedness? Tom Leinster 2012-05-01T22:43:38Z 2012-05-01T22:43:38Z <p>I don't know enough to give a detailed answer, but I can at least give a reference: Milnor's book <em>Dynamics in One Complex Variable</em> (Vieweg). You can find <a href="http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims90-5" rel="nofollow">an early draft here</a>. Appendix F mentions a couple of conjectures that would follow from MLC: see the footnote on page F-2 in the linked pdf file.</p> <p>While I'm at it, I can't resist repeating one of my favourite mathematical stories ever, which concerns the <em>connectedness</em> of the Mandelbrot set. It <em>is</em> connected, as you'd guess from the picture, but because of the thin filaments involved, this is not obvious if your image is too low-resolution. So there was some confusion in the early days. I'll let Milnor (Appendix F) take up the story:</p> <blockquote> <p>Mandelbrot made quite good computer pictures, which seemed to show a number of isolated "islands". Therefore, he conjectured that [the Mandelbrot set] has many distinct connected components. (The editors of the journal thought that his islands were specks of dirt, and carefully removed them from the pictures.)</p> </blockquote> http://mathoverflow.net/questions/95701/the-deep-significance-of-the-question-of-the-mandelbrot-sets-local-connectedness/105524#105524 Answer by Alexandre Eremenko for The deep significance of the question of the Mandelbrot set's local connectedness? Alexandre Eremenko 2012-08-26T12:32:00Z 2012-08-26T12:32:00Z <p>MLC is indeed a very technical and complicated counterpart of a simple question which arises from the general theory of dynamical systems. For a generic system (in a given finitely-parametric family) can one describe generic behavior of trajectories?</p> <p>In our case "generic" means "open dense set". Our system is $z^2+c$ the simplest one parametric family of one-dimensional rational (polynomial) maps. For this case, the main question is: is it true, that there is an open dense set of parameters $c$, such that for $c$ in this set, every trajectory that begins on an open dense set is converging to an attractive cycle.</p> <p>This is usually called the Density of Hyperbolicity Conjecture. For our family it is equivalent to the MLC, but the equivalence is highly non-trivial. Density of hyperbolicity is known if one restricts to real $c$. </p> http://mathoverflow.net/questions/95701/the-deep-significance-of-the-question-of-the-mandelbrot-sets-local-connectedness/109036#109036 Answer by Jeremy Kahn for The deep significance of the question of the Mandelbrot set's local connectedness? Jeremy Kahn 2012-10-07T03:02:07Z 2012-10-07T03:02:07Z <p>If a connected compact $K \subset C$ is locally connected then the Riemann map $h\colon C \setminus \Delta \to C \setminus K$ extends continuously to $\partial \Delta$. For each $z \in K$, the boundary of the convex hull of $h^{-1}(\{z\})$ is the union of a set $\Lambda_z$ of chords; the union of these $\Lambda_z$ over all $z \in K$ is a closed set $\Lambda_K$ of disjoint chords; it is called a <em>lamination</em> of $\Delta$. We can reconstruct the convex hulls of each $h^{-1}(\{z\})$ from $\Lambda_K$, and when we collapse every convex hull to a point, we obtain a topological model for $K$. </p> <p>In the case where $K$ is the Mandelbrot set $M$, the lamination $\Lambda_M$ can be described combinatorially, so MLC would mean that we know the topology of $M$. </p> <p>There is a second answer which is more subtle and more important. For each $c \in M$, the filled Julia set $K_c$ of $z \mapsto z^2 + c$ is compact and connected; if it is locally connected the resulting lamination $\Lambda_c \equiv \Lambda_{K_c}$ is, in the right sense, invariant under $z \mapsto z^2$ on $\partial \Delta$. Even if $K_c$ is not locally connected, there is a way of defining what the lamination <em>would</em> be if $K_c$ were locally connected. Every invariant lamination appears as $\Lambda_c$ for <em>some</em> c, and MLC is equivalent to the statement there is a <em>unique</em> $c$ with a given lamination. We think of $\Lambda_c$ as describing the <em>combinatorics</em> of $K_c$, and we think of this uniqueness conjecture as "combinatorial rigidity"---two maps of the form $z \mapsto z^2 + c$ are conformally conjugate (and hence equal) if they are "combinatorially equivalent". </p> <p>(Actually, if $z \mapsto z^2 + c$ has an attracting periodic cycle, then the set of combinatorial equivalent parameters form an open subset of $C$, so the statement of combinatorial rigidity must be suitably modified in that case. It is known that structural stability is open and dense in the family of maps $z \mapsto z^2 + c$, so combinatorial rigidity implies that every $z \mapsto z^2 + c$ in this open and dense set must have an attracting periodic cycle; this is the implication that Eremenko alluded to in his answer. )</p> <p>In this sense MLC is closely analogous to Thurston's Ending Lamination Conjecture (proven by Brock, Canary, and Minsky), which says, broadly speaking, that a finitely generated Kleinian group is determined by the topology of its quotient and the ending laminations of its ends, which are also, when viewed appropriately, invariant laminations of the disk. </p> <p>There is a third answer which is more historical and empirical. We can prove MLC and combinatorial rigidity "pointwise" (or "laminationwise") by proving that for a given invariant lamination $\Lambda$, it appears as the lamination $\Lambda_c$ for a single $c$. This has been done in great many cases, first by Jean-Christophe Yoccoz, and then by Mikhail Lyubich, the author of this post, Genadi Levin, and Mitsuhira Shishikura. To prove this combinatorial rigidity for a given $c$ seems to require a detailed understanding of the geometry of the associated dynamical system, and this almost always leads to further results. So proving MLC would most likely mean having a thorough understanding of the geometry and dynamics of every map $z \mapsto z^2 + c$. </p>