Connectedness Proof

Robert P. Munafo, 1996 Oct 30.

The proof that the Mandelbrot Set is connected was first developed by Douady and Hubbard.

This outline of the proof is the work of David Ben-zvi; it was published in the sci.fractals newsgroup in 1993 April.

The idea of the proof (due to Douady and Hubbard of course) that the M-set is connected, is a very beautiful one. It's very closely linked to that for the Julia sets, exploiting a beautiful kind of "duality" between the parameter space and dynamic plane, which D&H later used for lots of other results.

Consider a polynomial map (or in general a complex analytic map) which has a superattractive fixed point, i.e. a critical point which coincides with a fixed point. What this means is that (at least) two sheets of the mapping come together at a fixed point. For example, consider Z->Z² near zero or infinity; the map is a two-to-one covering map except near those two "branch points" (it's worthwhile really trying to understand the geometry of Z->Zⁿ near the origin; this branch point geometry is essential to all of complex dynamics, and geometry.) In particular, any polynomial has a superattractive fixed point at infinity.

Now if P is a superattractive fixed point where exactly n sheets come together, then there is a local analytic change of coordinates that conjugates our map to Z->Zⁿ in a neighbourhood of zero. What this means is that a map with an order n superattractive fixed point has the same local geometry and dynamics as the map Z->Zⁿ near the origin (or near infinity, for that matter.) This new local coordinate near the fixed point is called the Botkher coordinate.

The domain of the Botkher coordinate, as "defined" above, is a small neighborhood of the superattractive point, contained in its basin of attraction of course. The range is inside the unit disc, since the basin of 0 for Z->Zⁿ is precisely the unit disc. When we try to extend the map to larger and larger domains by analytic continuation, the only obstruction we can encounter is another critical point contained in the basin of attraction of our fixed point (or of course the Julia set, i.e. the basin boundary.) The essential reason for this is that a complex analytic map is very simple geometrically away from the critical points (branch points). (Yes I know that's not a full explanation...)

If there is no other critical point in the basin of our superattractive orbit (for example, if our map is quadratic hence doesn't have any more critical points), then the Botkher coordinate extends to the entire immediate basin of attraction, mapping it into the unit disc. In fact as we approach the Julia set of our original map, the Botkher coordinate approaches the Julia set of Zⁿ, i.e. the unit circle. In fact the map is onto the unit disc. But we defined the Botkher coordinate as a one-to-one map (since it is a change of coordinates), so it follows that this map establishes an analytic isomorphism (i.e. one-to-one analytic map with analytic inverse) of the immediate basin of attraction with the unit disc. (For those who care, this gives us a nice canonical dynamical choice of a Riemann mapping.)

This in fact gives us a proof of the connectivity of the filled-in Julia set. Assume Z²+c is in the Mandelbrot set, in other words the critical point Z=0 doesn't escape to infinity. Now as noted before, infinity is a superattractive fixed point, and since 0 is not in its basin, its immediate basin of attraction (which is by definition the complement of the filled-in Julia set) contains no critical points besides infinity. So we can apply the above argument to conclude that this immediate basin is topologically a disc, hence its complement in the Riemann sphere, which is the filled-in Julia set, is connected and simply-connected. The log of the absolute value of the Botkher coordinate is called the Green's function of the Julia set. The equipotential lines around the Julia set are just the lines abs(Botkher coord)=const, and the field lines are the lines arg(Botkher coordinate)=const.

But what if there is another critical point in the immediate basin? (don't worry, we'll get to the connectivity of the Mandelbrot set soon now.. I'll assume there's only one other critical point in the basin, which is all you need for quadratics.)

Then we can't extend the Botkher coordinate to the entire basin. But as we said, the only obstruction to extending it is this other critical point, so we can extend the Botkher coordinate all the way up to that critical point, and get a number for it, it's "Botkher coordinate". This will be a number in the unit disc. How is this useful?

Well assume Z²+c is NOT in the Mandelbrot set, in other words 0 is in the immediate basin of infinity. We construct a new function B(c), where we assign to a polynomial Z²+c outside the M-set the Botkher coordinate of 0 (the critical point) in the immediate basin of the superattractive point infinity!! (Take time to absorb this definition.) What we're doing (after Douady and Hubbard of course) is using this "duality" between the dynamic plane and parameter space - a number associated to some chosen point for each dynamical system becomes a function on the parameter space! We call this the Botkher function, and note that it varies in the unit disc.

The connectivity of the M-set follows from the stronger statement proved by Douady and Hubbard, that this function B(c) is a complex analytic isomorphism between the complement of the Mandelbrot set (in the Riemann sphere) and the unit disc. This does a lot more! As byproducts it actually implies that the Mandelbrot set is simply-connected, and it gives us the Green's function for the M-set which is again just log of the absolute value.. the potential lines and field lines are defined just as for the Julia set above, again emphasizing the duality between the two. D&H then go on to prove lots of neat theorems using the local similarity between the M-set and Julia sets.

(Just a brief indication of the idea of the proof - it's not hard to show that this function B(c) is going to be holomorphic, since it is definable by a simple limit formula. Also, as long as B(c)<1, 0 is far away from the Julia set so it's not hard to imagine that this map is actually onto the unit disc, and as B(c) approaches the unit circle 0 gets closer to the Julia set of Z²+c. So now it's mainly left to show this map is not branched, so that it is globally one-to-one.)

Anyway, I really like this proof, and the ideas used in it are really universal techniques in the field. The proof of this theorem signaled the rebirth of complex dynamics of the '80s - except for the crucial work of Siegel, not much attention was given to coomplex dynamics between the pioneering work of Fatou and Julia and this work of Douady and Hubbard.

References : I have too many, so I'll be a miser... For information about this all-powerful Botkher coordinate, the best place to start is Milnor's wonderful book on complex dynamics (a must for anyone interested in the field), available as IMS preprint ims90-5 from math.sunysb.edu (anonymous ftp.) Of course check out Douady&Hubbard's original 1982 (?) paper, and also Branner's survey, The Mandelbrot Set, in Proceedings of Symposia in Applied Math, #39, 1989.

Original USENET article:

-------R-P-M----N-e-w-s-b-l-i-n-k------------------------------ Xref: world sci.fractals:1322 Newsgroups: sci.fractals Path: world!uunet!noc.near.net!howland.reston.ans.net!darwin.sura.net!news-feed-1.peachnet.edu!umn.edu!news From: benzvi@peano.geom.umn.edu (David Ben-zvi) Subject: Connectivity of Julia and Mandelbrot sets Message-ID: <C6298v.IG@news2.cis.umn.edu> Keywords: Botkher coordinate Sender: news@news2.cis.umn.edu (Usenet News Administration) Nntp-Posting-Host: peano.geom.umn.edu Organization: University of Minnesota Date: Sun, 25 Apr 1993 22:31:41 GMT Lines: 147 >The Mandelbrot connectedness proof is magic to me; it uses a Greens function >conformal automorphism. The proof that the Julia set is connected if the >corresponding Mandelbrot iteration doesn't diverge is simpler, but I don't >know how you'd extend it to random planes in 4-space. The idea of the proof (due to Douady and Hubbard of course) that the M-set is connected, is a very beautiful one. It's very closely linked to that for the Julia sets, exploiting a beautiful kind of "duality" between the parameter space and dynamic plane, which D&H later used for lots of other results. Consider a polynomial map (or in general a complex analytic map) which has a superattractive fixed point, i.e. a critical point which coincides with a fixed point. What this means is that (at least) two sheets of the mapping come together at a fixed point. For example, consider z->z^2 near zero or infinity; the map is a two-to-one covering map except near those two "branch points" (it's worthwhile REALLY trying to understand the geometry of z->z^n near the origin; this branch point geometry is essential to all of complex dynamics, and geometry.) In particular, any polynomial has a superattractive fixed point at infinity. Now if P is a superattractive fixed point where exactly n sheets come together, then there is a local analytic change of coordinates that conjugates our map to z->z^n in a neighbourhood of zero. What this means is that a map with an order n superattractive fixed point has the same local geometry and dynamics as the map z->z^n near the origin (or near infinity, for that matter.) This new local coordinate near the fixed point is called the Botkher coordinate. The domain of the Botkher coordinate, as "defined" above, is a small neighborhood of the superattractive point, contained in its basin of attraction of course. The range is inside the unit disc, since the basin of 0 for z->z^n is precisely the unit disc. When we try to extend the map to larger and larger domains by analytic continuation, the only obstruction we can encounter is another critical point contained in the basin of attraction of our fixed point (or of course the Julia set, i.e. the basin boundary.) The essential reason for this is that a complex analytic map is very simple geometrically away from the critical points (branch points). (Yes I know that's not a full explanation...) If there is no other critical point in the basin of our superattractive orbit (for example, if our map is quadratic hence doesn't have any more critical points), then the Botkher coordinate extends to the entire immediate basin of attraction, mapping it into the unit disc. In fact as we approach the Julia set of our original map, the Botkher coordinate approaches the Julia set of z^n, i.e. the unit circle. In fact the map is onto the unit disc. But we defined the Botkher coordinate as a one-to-one map (since it is a change of coordinates), so it follows that this map establishes an analytic isomorphism (i.e. one-to-one analytic map with analytic inverse) of the immediate basin of attraction with the unit disc. (For those who care, this gives us a nice canonical dynamical choice of a Riemann mapping.) This in fact gives us a proof of the connectivity of the filled-in Julia set. Assume z^2+c is in the Mandelbrot set, in other words the critical point z=0 doesn't escape to infinity. Now as noted before, infinity is a superattractive fixed point, and since 0 is not in its basin, its immediate basin of attraction (which is by definition the complement of the filled-in Julia set) contains no critical points besides infinity. So we can apply the above argument to conclude that this immediate basin is topologically a disc, hence its complement in the Riemann sphere, which is the filled-in Julia set, is connected and simply-connected. The log of the absolute value of the Botkher coordinate is called the Green's function of the Julia set. The equipotential lines around the Julia set are just the lines abs(Botkher coord)=const, and the "field lines" are the lines arg(Botkher coordinate)=const. But what if there is another critical point in the immediate basin? (don't worry, we'll get to the connectivity of the Mandelbrot set soon now.. I'll assume there's only one other critical point in the basin, which is all you need for quadratics.) Then we can't extend the Botkher coordinate to the entire basin. But as we said, the only obstruction to extending it is this other critical point, so we can extend the Botkher coordinate all the way up to that critical point, and get a number for it, it's "Botkher coordinate". This will be a number in the unit disc. How is this useful? Well assume z^2+c is NOT in the Mandelbrot set, in other words 0 is in the immediate basin of infinity. We construct a new function B(c), where we assign to a polynomial z^2+c outside the M-set the Botkher coordinate of 0 (the critical point) in the immediate basin of the superattractive point infinity!! (Take time to absorb this definition.) What we're doing (after Douady and Hubbard of course) is using this "duality" between the dynamic plane and parameter space - a number associated to some chosen point for each dynamical system becomes a function on the parameter space! We call this the Botkher function, and note that it varies in the unit disc. The connectivity of the M-set follows from the stronger statement proved by Douady and Hubbard, that this function B(c) is a complex analytic isomorphism between the complement of the Mandelbrot set (in the Riemann sphere) and the unit disc. This does a lot more! As byproducts it actually implies that the Mandelbrot set is simply-connected, and it gives us the Green's function for the M-set which is again just log of the absolute value.. the potential lines and field lines are defined just as for the Julia set above, again emphasizing the duality between the two. D&H then go on to prove lots of neat theorems using the local similarity between the M-set and Julia sets. (Just a brief indication of the idea of the proof - it's not hard to show that this function B(c) is going to be holomorphic, since it is definable by a simple limit formula. Also, as long as |B(c)|<1, 0 is far away from the Julia set so it's not hard to imagine that this map is actually onto the unit disc, and as B(c) approaches the unit circle 0 gets closer to the Julia set of z^2+c. So now it's mainly left to show this map is not branched, so that it is globally one-to-one.) Anyway, I really like this proof, and the ideas used in it are really universal techniques in the field. The proof of this theorem signaled the rebirth of complex dynamics of the '80s - except for the crucial work of Siegel, not much attention was given to coomplex dynamics between the pioneering work of Fatou and Julia and this work of Douady&Hubbard. References: I have too many, so I'll be a miser... For information about this all-powerful Botkher coordinate, the best place to start is Milnor's wonderful book on complex dynamics ( a must for anyone interested in the field), available as IMS preprint ims90-5 from math.sunysb.edu (anonymous ftp.) Of course check out Douady&Hubbard's original 1982 (?) paper, and also Branner's survey, The Mandelbrot Set, in Proceedings of Symposia in Applied Math, #39, 1989. Enjoy! David. ---------------------------------------------------------------------- Geometry is the User Interface to Mathematics - Bill Thurston God is always doing Geometry - Plato (the Geometry Center slogan) The Surface was invented by the Devil - Wolfgang Pauli