Welcome to Lecture 5 in our second week of the course, Analysis of a Complex Kind. Today, I'm going to introduce the Mandelbrot set to you and show you some beautiful pictures and also explain to you how to generate these beautiful pictures. Recall that the Mandelbrot set is the set of all those parameters c in the parameter plane for which the corresponding Julia set is connected. So you pick a c in the complex plane, you consider the quadratic polynomial f(z)=z squared + c. You calculate its Julia set, and you look at whether this Julia set is connected. And if so, then the parameter c belongs to the Mandelbrot set. So how could a computer check such a condition? Thankfully there's a theorem, which says the following. If f(z) is a quadratic polynomial of the form z squared + c, then the corresponding Julia set is connected if and only if 0 does not belong to A(infinity). In other words, if and only if the orbit of 0 remains bounded under iteration of this function, f(z). So, what do you do now? You pick a complex number, c. You consider the corresponding quadratic polynomial, f(z) = z squared + c. You plug in 0, and you follow the point 0 around. Where does 0 get mapped under f? It's mapped to f(0), which is 0 squared + c, so that's c. Where does c get mapped under f? It gets mapped to f(c) which is f(f(0)) which is c squared + c. And where does that get mapped under f? So if you look at the orbit of 0 under the function z squared + c and you find out, does it get attracted to infinity, or does it remain bounded? If it remains bounded under iteration, then this parameter c belongs to the Julia set. In fact, it is possible to show the following. A complex number c belongs to M if and only if the orbit of 0 remains bounded by the number 2 and absolute value for all n. And this tells us a great computer algorithm to check whether a parameter c belongs to the Mandelbrot set. Here it is. As always, you need to choose a window, a window of c values, parameter values that you want to display. We're no longer in the z plane. We're in the c plane, so we want to display certain parameter values. Typically, if you want to see the entire Mandelbrot set, you choose something like the set of all c values whose real part is between -2 and 0.75, and whose imaginary part is between -1.5 and 1.5. That will give you a large enough window to see the entire Mandelbrot set. As before, you're going to pick a largest number of the durations that you'll allow the computer to check. The computer can only check finitely many iterations. At some point, it has to stop. The larger you choose this number max iteration, the more accurate your picture will be, but also the slower the calculation will be. You can start with something like maximum number of iterations equal 100. Especially if you start with a large window like the one given above here. Once you start zooming into the Mandelbrot set, you choose windows that only show portions of the Mandelbrot set, or proportions of the boundary where behavior is even more interesting. You want to choose a larger and larger value for the maximum number of iterations. Now for each pixel on your screen, you're going to choose a parameter that corresponds to that pixel. This is very similar to when we were calculating Julia sets. If this is your screen, And here's this window that you pick, the window of parameter c, you're going to have to make the pixels correspond to the parameter values in your window. Once you have figured out what parameter c you're looking at, we'll be looking at the corresponding quadratic polynomial z squared + c. You need to next calculate the iterates of 0 under this polynomial. So f(0) = c, f(f(0)) = c squared + c, f(f(f(0))), = (c squared + c) quantity squared + c, and so forth. So whenever you have calculated one iterate, to get the next iterate, you square the number you obtained from the previous step and add c to it. If one of these iterates satisfies that it is bigger than 2 in absolute value, then you know this point does not belong to the Mandelbrot set. And you can color the corresponding initial pixel white. If you reach the maximum number of iterations, maxiter, without having left this ball of radius 2 centered at 0, then there's a good chance that the parameter c does belong to the Mandelbrot set. And you color the corresponding pixel that you began with black. This is the picture that you get this way. It's a black and white image of the Mandelbrot set. All the points that are colored white out here are points that are not in the Mandelbrot set. And the black points are points that are in the Mandelbrot Set. Again, you can use different colors for those parameters c for which 0 does escape to infinity under iteration, depending on how quickly the escape happens. So, for example, what you could do is, if f(0), which is c, is greater than 2 in absolute value, then color of the corresponding pixel in a zero color you chose. Otherwise, if f(f(0)) is the first one to get bigger than 2 in absolute value, color the corresponding pixel in color one. Otherwise if f(f(f(0))) is the first one that is greater than 2 in absolute value, use color two, and so forth. And only if all fn(0) remain less than or equal to 2 in absolute value, then color the corresponding pixel black, or whatever other color you choose for your Mandelbrot set. Zooming into the Mandelbrot set and coloring parameters by escape time yields beautiful pictures as we see in these following examples. [MUSIC] Here are some properties of the Mandelbrot set. The Mandelbrot set is a connected set. It's all in one piece. It took a while for this to be actually proved. The first images of the Mandelbrot set were not as accurate as the ones that we can calculate with today's powerful computers. And it didn't necessarily look connected at first. But it has been proven to be connected by Douady and Hubbard in 1982. The Mandelbrot set is contained in the disk of radius 2 centered at the origin. The boundary of the Mandelbrot set is very intricate. This is where you will find the most beautiful zooms. Zooming into locations close to the boundary gives you more and more beautiful pictures. Moreover, for c values near the boundary of the Mandelbrot set, their Julia sets have many different patterns. And here are some examples of what patterns you might see. The boundary of the main cardioid is given by the following equation. You see a picture right here of what that looks like. Writing theta, this exponent theta over here, this angle theta, as 2 pi alpha, we can distinguish whether alpha is a rational number or an irrational number. And it turns out, whether alpha is rational or irrational makes a big difference in what the Julia set looks like. Let's start with a c value of the form one-half e to the 2 pi i alpha- one-fourth e to the 4 pi i alpha. So on the main boundary of the Mandelbrot set where alpha is a rational number. So if it's a rational, it means it's a quotient. It's a fraction of the form p over q, where p and q are simply natural numbers. The parameter c is then an attachment point of another bud of the Mandelbrot set. And the Julia set looks similar to the Julia set for parameters within that bud. Let's look at an example. Let's look at alpha = one-half. Then c, if you plug in alpha = one-half, c becomes one-half times e to the 2 pi i times one-half. So one-half e to the pi i- one-fourth e to the 4 pi i times one-half. So one-fourth e to the 2 pi i. E to the pi i is -1, so that gives us a- one-half and even the 2 pi i is 1, so- one-fourth. So, we get -0.75. And that's this point right here that I colored in red. And as you know, in the Mandelbrot set, there's another bud here. And then there's another here, and another here. But this entire bud is attached at this attachment point and the Julia set looks very similar if you move within this bud. If you move parameter values within this bud and calculate the Julia set, they all look something like this picture that you can see here on the right. Here you can see the Mandelbrot set. And we're going to zoom in around the parameter value -0.75, this value that we just looked at before and whose Julia set we just looked at. You can see successive zooms here, and see how beautiful and intricate the pictures get if you zoom in around the boundary of the Mandelbrot set. Let's next look at a c value of the same form, but alpha is an irrational number. So there are no values p and q for which you could find alpha is p over q. So in your sets for such values, look more intricate and they come in several flavors. Here's a first example. For alpha = 1 + root 5 over 2, that's an irrational number. Then you can calculate the corresponding c value just by evaluating this equation up here. And for the corresponding f(z) = z squared + c, the interior of K has a so-called Siegel disk, in which iteration looks like a rotation by angle alpha. The filled in Julia set has these little disks. My picture's not entirely accurate, but you can see that there's rotations going on here. So under iteration, the iterates move along circles right there. Many points in the boundary of the Mandelbrot set are so-called Misiurewicz points. A point c is called a Misiurewicz point if the orbit of 0 under f(z) = z squared + c is pre-periodic but not periodic. We had seen before that the point is pre-periodic if it moves around for a while under f. But eventually it enters a periodic orbit. And a periodic orbit could be of order 2, like the one I drew here, or It could be something like this, Of a higher order. But the point is pre-periodic if it is not itself periodic. So the orbit does not return to this point itself, but it enters the periodic orbit after a number of iterations. An example is c = i. Then f(z) = z squared + i, and the orbit of 0 under f is given by 0 f(0) is i. f(i) is i squared + i, so that's -1 + i. f(-i) + i is -1 + i quantity squared + i. And you can check that that equals -i. f(-i) is -i squared + i, which is -1 + i. And we're back to this point that we were earlier inside the iteration here. But we did not repeat the point 0 in i. We started repeating at -1 + i. Clearly, Misiurewicz points must belong to the Mandelbrot set because the orbit of 0 under the function f(z) = z squared + c remains bounded because it's periodic. Here are some properties of Misiurewicz points. The Julia set is equal to the filled-in Julia set. In other words, the filled-in Julia set has no interior. Misiurewicz points are dense in the boundary of the Mandelbrot set. That means no matter where you look, you'll find them. You can zoom into the boundary of the Mandelbrot set wherever you want to and you'll find tons and tons of Misiurewicz points. In fact, you can zoom in as far as you want to and there are still infinitely many, many Misiurewicz points in any window that you zoom in to. And, the Mandelbrot set is self-similar under magnification near Misiurewicz points. Now, note the Mandelbrot set is quasi-self-similar everywhere. Small, slightly different versions of itself can be found at arbitrary small scales. But near Misiurewicz points, one can actually prove that it is self-similar. You zoom in, and you see exact replicas of the big picture in small scales. Here's the Julia set for c = i. It's called a dendrite, and it looks very much like the Mandelbrot set looks near z = i. We'll look at that in the next pictures. Here you can see the Mandelbrot set, and I've highlighted the point z = i, so the parameter value i. And we zoom into the Mandelbrot set around this point. And you can see that the Mandelbrot set looks like a dendrite. And it is very self-similar. We keep zooming in and zooming in, and it keeps looking exactly the same. And it looks just like the Julia set looked for c = i. Let's finish with one of the big outstanding conjectures in the field of complex dynamics. It says the following, the Mandelbrot set is locally connected. What that means is, it's a term we have not discussed before. It means that for every parameter c in the Mandelbrot set, so, for example, this one right here. And every open set v that contains c. So some open set around it, there exists an open set U, possibly smaller than the original open set that you picked, so that still contains your parameter such that the intersection of U and M is connected. Now this boundary of the Mandelbrot set is rather intricate. So you could imagine that you, by choosing a small, open set, accidentally pick two parts that are not really connected looking, because the Mandelbrot set wiggles so much. It goes in and out and in and out. So you could imagine accidentally picking a set like this, where you just get disconnected pieces. But this conjecture says, no matter where you are, how far you zoom in at every level, around every parameter, you're going to find a little open set that contains a connected piece of the Mandelbrot set. It will never happen to you that your Mandelbrot set falls apart. This is an outstanding conjecture which means it has not been proved to be true. It is known that the Mandelbrot set is connected, but it is not known that it is locally connected. This is considered one of the big outstanding conjectures in the field of complex dynamics. The reason why this is such an important conjecture goes beyond the scope of this course.