François-René Rideau (fare) wrote,
François-René Rideau

  • Music:

The Garden of Forking Paths

The linux audio driver for the integrated nForce chipset of my old desktop machine often badly accelerates the speed at which sound playback happens, consequently distorting frequencies upward. My working hypothesis is that there is some bad interaction with some power saving feature that dynamically slows down the motherboard clocks without the audio driver being notified. Consequently, I've taken the habit of launching a CPU-eating application in the background (and sometimes two instances of it, for surer effect), so as to regulate CPU usage and keep the CPU clock to the max. I suppose I should try grab the latest kernel, compile power management as a module and disable it while playing music. But since the machine is doesn't have Internet access, I procrastinate trying out a new kernel on it; moreover, I can't run a distributed computing project on it; hence, the application I've chosen to burn CPU cycles is Xaos yet another popular program to draw fractals, by the omnipresent über-hacker Jan Hubicka of AA fame.

Thus, just like everyone else a few decades ago, I've finally come to fascinatedly watch the Mandelbrot set for hours, and I have tried to understand the patterns or its self-similarity. Below are the conjectures that I've made while watching Xaos draw the Mandelbrot set in high resolution, letting it zoom in and out at random, or zooming manually to check out some idea. I suppose these conjectures are all well-known by now, and probably also proven already, at least based upon some other more interesting mathematical conjectures. But I really don't feel like digging the mathematical litterature about it, so instead I'll bleg those of you who know to point me to proper documents that summarize the knowledge about these patterns and more. (If I were web connected while writing this, I'd Google for it; but then again, if I had an Internet connection, I wouldn't be writing this to begin with.)

  • Let M be the Mandelbrot set and O its interior. O's closure is M.
  • Let B be the border of M: B=M\O. M is simply connected, and so is B. [Update: that was proven]
  • M is path-connected.
  • We can equip M with a partial order structure. Let SEP(x,y,z) hold if x and z are in different connected components of M\{y}. (Assuming M is arc-connected, this means that all paths from x to z go through y before reaching y, i.e. if all continuous functions f from [0,1] to M such that f(0)=x and f(1)=z are such that f(t)=y for some t such that 0<t<1.) Let x<y if SEP(0,x,y). Let x<=y if x<y or x=y. Then < is a strict partial order relation on M. < is a closed subset of M*M.
  • Any non-empty subset S of M has a lower bound b in M, such that forall x in S, b <= x, and for any b' in M such that forall x in S, b' <= x, then b' <= b. Unless S is singleton, b in is B.
  • The set L of lower bounds of bigger-than-singletons subsets is a dense countable subset of B.
  • For x in M, we define the uppercut of x as U(x)={y : x < y}, the path to x as P(x)={y : y < x}, and the cell of x as C(x) = U(x) \ Union{ U(y) : x < y }. C(x) is also { y : x < y, there is a path from x to y in the interior of M }.
  • The set W of w such that C(w) is not empty is countable and dense in L. (Proof: the sum of their areas is finite.) We will call such points whole nodes, and I(w) the area of C(w).
  • If w is in W, C(w) is diffeomorphic to a closed disk minus a point (w).
  • <= defines a tree: for any x in M, P(x) is an closed countable totally ordered set, and W is dense in it.
  • Any w in W there exists a countable set S(w) of direct successors, such that for any x in U(w)\C(w), there exists a unique s in S(w) such that s<=x. [On the structure of that, see for instance the use of "Farey Addition" :)]
  • Each s in S(w) has a rank, a positive integer that determines the branching pattern at which further copies of M are found at branching points beyond s.
  • The pattern of ranks, and the order of point of given ranks along the circumference of C(w) is always the same; there with exactly one successor of rank 1, and finitely many successors of each rank. There must be an algorithm to recursively enumerate them.
  • If l is in L, then U(l) has a finite number of connected components, called the branching factor of l.
  • The branching factor of an element w of W is 1.
  • The branching pattern at an element l of L is actually characterized by some branching structure BS(l) that is determined by the structure of ranks of elements of P(l)/\W.
  • If l is in L is, then BS(l) is a finite list, and the branching factor of l is the top of BS(l).
  • BS(l) is actually the finite list of ranks greater than 1 in P(l)/\W.
  • For every w in W, we denote SS(w) the recursive set made of successors of successors of w, and K(w) the union of the SS(w) and C(x) for all x in SS(w). K(w) is diffeomorphic to K(0), in a way that preserves the rank structure. K(w) can be seen as the copy of kernel of M at w. We denote PW set elements of W that are not successors; they introduce primary copies of the kernel of M.
  • Except for obvious symmetry, there are no two points x and y in W that have the same area. (This conjecture seems obviously probable, but I have not the slightest idea how to prove something like that.)
  • To each w in W we can associate the tip t(w) as the limit of the iterative sequence starting from w and applying the function that to w in W associates its unique successor of rank 1. The tips of w is the set T(w) mapping t to SS(w). For each x in U(w)\K(w), there is a unique y in T(w) such that y<x. t(0)=-1.401155189...
  • For each tip t(w) there is a top point tt(w) such that there is a stream between t(w) and tt(w) of empty branching pattern (see below). tt(0)=-2.
  • A stream between two points a and b of branching pattern BP, is a continuous path between those points, made of copies K(w) of M reached in their singularity w, and of branching points with branching factors according to the pattern. We can recursively enumate the copies of M and branching points along such a stream, using a dichotomy process: if there is a stream between a and b, then there is point g(a,b) in PW whose K that has the greatest area of all (in the improbable case, that we conjectured never happened, that there are many of the same size, then there are finitely many so and we pick the first along the line); and we can iterate the process, with a stream from a to g(a,b) and from tt(g(a,b) to b. Actually, for the sake of univocal recursive enumeration, we must interleave the use of g with the finding of the branching point of greatest interest according to each of the ranks in BP, with the interest of a branching point being the least of the total areas of the connected components beyond the point. Any point in W or any branching point along the stream can be found in a finite number of such dichotomic operations of considering the greatest copy within an interval.
  • For each v and w in W such that BS(w) is h::BS(v), there is a diffeomorphism f that maps U(v) into U(w), v into w, and such that at each top point m in U(v) of stream of branching pattern BS(v), then f(m) is in L and has branching factor of h, with outgoing branches each having a branching pattern BS(w). This process can be used to recursively enumerate branches of increasing complexity that grow madly at each extremity of a copy of M, depending on the non-trivial ranks chosen along the path to the copy.
  • All elements of L can be enumerated by recursively applying one of the above constructions, and that this faithfully describes the branching topology of M. [Update: compare to the R2 notation]
  • The branching structure BS(l) somehow also describes the branching structure of Julia sets around l.
  • Problem is, when the product of the numbers in the branching pattern of a stream is larger than 10 or so, the branching pattern is soon beyond human visualization.
  • Elements of M\L are the dead-ends, maximal elements for <=. By the way we constructed <=, elements in O are dead-ends. We could have had it differently on O by using the preorder structure on M that matches <= on W but equates elements of C(w) for each w (that we can systematically represent by the successor of rank one of w): instead of AP(x,y,z), use EP(x,y,z) that holds when there is an injective continuous path (function of [0,1] to M) from x to z that contains y, which is not x or z.
  • Except for the stream from t(0) to tt(0), streams do not follow differentiable paths, and paths between two points outside of a same K(w) are of infinite length. Still, we can define the distance between two points x and y as the total area of U(z)\U(x)\U(y) where z is the lower bound of {x,y}. It would be interesting to compare this distance to the usual distance in the complex plane, and to see how fast the area of U(w) decreases as BS(w) grows.
  • There must be some not-too-inefficient algorithm to compute exactly the Hausdorff dimension of B. [Update: it was proven to be 2]
  • What would be interesting would be a way to trace the shortest path between two points, and/or to enumerate the points of the smallest enumerative index in a given area, particularly in copies of M that are surrounded by lots of paths and patterns that seem to radiate out of it, so as to understand where these paths all come from. We conjecture that this is feasible, and that most of the things that radiate.
  • It seems to me that often (always?), the radiating patterns are paths that are forked before (or once, after) the copy was reached -- otherwise their branching pattern would be quite different; and indeed, when the branching is different, this can be seen, the most obvious difference always being that with lowest branching factor, i.e. 2, which provides for a lot of kidneys -- just zoom around the small copies of M that approach the singularity of a bigger copy of M along a stream.

This is it. That's what I've conjectured while watching the Mandelbrot set. So, which of these are already conjectured, proved, disproved, maybe in corrected variants?

Update: since I (obviously) got a net connection after I wrote this article, I added a few web links.

Tags: en, fractal, mandelbrot, mathematics

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