10 - The Mandelbrot set
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welcome to lecture five in their second
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week of the course analysis of a complex
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kind today I'm going to introduce the
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Mandelbrot set to you and show you some
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beautiful pictures and also explain to
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you how to generate these beautiful
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pictures we call that the Mandelbrot set
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is the set of all those parameters see
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in the parameter plane for which the
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corresponding juliette said it's
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connected so you pick a c in the complex
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plane you consider the quadratic
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polynomial f of Z equals Z squared plus
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C you calculate its Julia set and you
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look at whether this Julia set is
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connected and if so then the parameters
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C belongs to the Mandelbrot set so how
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could a computer check such a condition
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thankfully there's a theorem which says
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the following if f of Z is a quadratic
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polynomial of the form Z squared plus C
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then the corresponding Julia set is
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connected if and only if zero does not
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belong to a of infinity in other words
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if and only if the orbit of zero remains
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bound that under iteration of this
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function f of Z so what do you do now
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you pick a complex number C you consider
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the corresponding quadratic polynomial f
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of Z equals Z squared plus C you plug in
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0 and you follow the point zero around
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where does zero get mapped under F it's
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Mel to F of zero which is zero squared
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plus C so that's C where does C get
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mapped under F it's mapped to F of C
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which is F of F of zero which is C
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squared plus C and where does that get
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mapped under Epps if you look at the
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orbit of zero under the function Z
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squared plus C and you find out does it
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get attracted to infinity or does it
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remain bounded if it remains bounded
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under iteration then this parameter C
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belongs to the Julia set in fact it is
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possible to
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the following a complex number C belongs
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to M if and only if the orbit of zero
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remains bounded by the number to an
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absolute value for all N and this tells
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us a great computer algorithm to check
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whether a parameter C belongs to the
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Mandelbrot set here it is
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as always you need to choose a window a
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window of C values parameter value says
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you want to display we're no longer in
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the Z plane we're in the C plant so we
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want to display certain parameter values
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typically if you want to see the entire
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Mandelbrot set you choose something like
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the set of both C values whose real part
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is between negative 2 and point 7 5 and
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whose imaginary part is between negative
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1.5 and 1.5 that will give you a large
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enough window to see the entire
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Mandelbrot set as before you're gonna
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pick a largest number of iterations that
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you'll allow the computer to check the
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computer can only check finally many
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iterations at some point it has to stop
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the larger you choose this number max
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iteration the more accurate your picture
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will be but also the slower the
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calculation will be you can start with
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something like maximum number of
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iterations equal 100 especially if you
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start with a large window like the one
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given above here once you start zooming
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into the Mandelbrot set you choose
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windows that only show portions of the
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Mandelbrot set or portions of the
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boundary where behavior is even more
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interesting you want to choose a larger
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and larger value for the maximum number
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of iterations now for each pixel on your
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screen you're gonna choose a parameter
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that corresponds to that pixel this is
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very similar to when we were calculating
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Julia sets this is your screen and
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here's this window that you paid the
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window of parameter C you're gonna have
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to make the pixels correspond to the
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parameter values in your window once you
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have figured out what parameters C
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you're looking at we'll be looking at
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the corresponding quadratic polynomials
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u squared plus C you need
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calculate the iterance of zero under
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this polynomial so f of zero which is C
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f of f of zero which is C squared plus C
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F of F of F of zero which is C squared
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plus C quantity squared plus C and so
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forth so whenever you have calculated
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one iterate to get the next iterate you
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square the number you obtain from the
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previous step and add C to it if one of
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these either it satisfies that it is
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bigger than to an absolute value then
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you know this point does not belong to
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the Mandelbrot set and you can color the
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corresponding initial pixel white if you
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reach the maximum number of iterations
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max itter without having left this ball
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of radius 2 centered at zero and there's
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a good chance that the parameter C does
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belong to the Mandelbrot set and you
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color the corresponding pixel Icahn
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began with black
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this is the picture that you get this
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way it's a black-and-white image off the
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Mandelbrot set all the points that are
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colored white out here are points that
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are not in the Mandelbrot set and the
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black points are points that are in the
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Mandelbrot set again you can use
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different colors for those parameters C
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for which 0 dot Saskatchewan Finiti
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under iteration depending on how quickly
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the escape happens so for example what
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you could do is if F of 0 which is C is
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greater than 2 an absolute value then
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color the corresponding pixel in the 0th
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color you chose otherwise if F of F of 0
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is the first one to get bigger than 2 an
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absolute value color the corresponding
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pixel in color 1 otherwise if F of F of
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F of 0 is the first one that is greater
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than 2 an absolute value use color 2 and
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so forth and only if all F n of zero
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remain less than or equal to 2 and
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absolute value then color the
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corresponding pixel black or whatever
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other color you choose for your
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Mandelbrot set zooming into the
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Mandelbrot set and coloring parameters
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by escape time yields beautiful pictures
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as we see in these following examples
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here are some properties of the
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Mandelbrot set the man abroad set is a
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connected set it's all in one piece it
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took a while for this to be actually
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proved the first images of the man who
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brought said were not as accurate as the
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ones that we can calculate with today's
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powerful computers and it didn't
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necessarily look connected at first but
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it has been proven to be connected by
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duet Ian Hubbard in 1982 the Mandelbrot
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set is contained in the disc of radius 2
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centered at the origin the boundary of
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the Mandelbrot set is very intricate
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this is where you'll find the most
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beautiful zoo so zooming into locations
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close to the boundary gives you more and
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more beautiful pictures and moreover 4c
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values near the boundary of the
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Mandelbrot set there Julia sets have
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many different patterns and here are
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some examples of what patterns you might
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see the boundary of the mean cardioid is
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given by the following equation you see
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a picture right here of what that looks
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like
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writing theta this exponent theta right
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here this angle theta as chupa I alpha
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we can distinguish whether alpha is a
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rational number or an irrational number
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and it turns out whether alphas rational
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or irrational makes a big difference in
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what the Julia said looks like let's
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start with a C value of the form 1/2 e
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to the 2 pi alpha minus 1/4 e to the 4
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pi alpha so on the main boundary of the
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Mandelbrot set where alpha is a rational
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number so if it's irrational it means
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it's a quotient it's a fraction of the
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form P over Q where P and Q are simply
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natural numbers the parameter sees then
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an attachment point of another bud of
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the Mandelbrot set and the Julia said
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looks similar to the Julia set for
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parameters within that bud let's look at
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an example let's look at alpha equals
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1/2 then see if you plug in alpha equals
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1/2 C becomes 1/2 times e to the 2 pi I
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times 1/2 so 1/2 e to the PI I minus 1
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for e to the four pi times 1/2 so 1/4 e
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to the 2 pi even the PI is negative 1 so
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that gives us a minus 1 happened e to
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the 2 pi is 1 so minus 1/4 so we get
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minus 0.75 and that's this point right
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here that I colored in red and as you
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know and the Manuel Brad said there is
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another bud here and then there's
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another here and another here but this
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entire butt is attached at this
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attachment point and that Julie I said
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looks very similar if you move within
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this but if you move parameter values in
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within this button calculate that
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Julianne's if they all look something
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like this picture that you can see here
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on the right here you can see the
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Mandelbrot set and we're gonna zoom in
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around the parameter value minus 0.75
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this value that we just looked at before
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and who's Julia said we just looked at
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you can see successive tsums here and
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see how beautiful and intricate the
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pictures get if you zoom in around the
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boundary of the Mandelbrot set
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let's next look at a sea value of the
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same form but alpha is an irrational
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number so there are no values P and Q
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for which you could fight alpha s P over
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Q jr. sets where such values look more
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intricate and then come in several
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flavors here's the first example
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for alpha equals 1 plus root 5 over 2
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that's an irrational number then you can
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calculate the corresponding C value
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despite you know evaluating this
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equation up here and for the
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corresponding f of Z equals Z squared
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plus C the interior of K has a so-called
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Siegel disk in which iteration looks
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like a rotation by angle alpha the filth
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and Julius it has these little discs my
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picture is not entirely accurate but you
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can see that there's rotations going on
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here so under iteration iterates move
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along circles right there many points in
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the boundary of the Mandelbrot set are
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so-called misery edge points a Point C
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is called a misery image point if the
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orbit of zero under F of Z equals Z
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squared plus C is pre periodic but not
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periodic we had seen before that a point
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is pretty periodic if it moves around
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for a while under F but eventually it
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enters a periodic orbit in a periodic
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orbit could be afforded to like the one
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entry here or it could be something like
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this for higher order but the point is
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pre periodic if it is not itself
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periodic so the orbit does not return to
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this point itself but it enters a
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periodic orbit after a number of
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iterations an example is C equals I then
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f of Z is Z squared plus I then the
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orbit of 0 under F is given by 0 f of 0
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is AI F of I is I squared plus I so
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that's minus 1 plus I F of minus 1 plus
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I is minus 1 plus I
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one of these squared plus I'd you can
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check that that equals minus I F of
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minus I is minus I squared plus I which
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is minus one plus I and we're back to
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this point that we were earlier inside
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the iteration here but we did not repeat
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the point 0 and I we started repeating
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at minus one plus I clearly missing
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average points must belong to the
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Mandelbrot set because the orbit of 0
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under the function f of Z equals Z
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squared plus C remains bounded because
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it's periodic here are some properties
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of middle of each points the Julia set
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is equal to the field and Julia set in
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other words the filth and Julia set has
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no interior these are huge points are
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dense in the boundary of the Mandelbrot
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set that means no matter where you look
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you'll find them you can zoom into the
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boundary of the Mandelbrot set
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wherever you want to and you'll find
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tons and tons of music image points in
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fact you can zoom in as far as you want
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to and there are still infinitely many
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museums ones in any window that you zoom
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into and the Mandelbrot set is
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self-similar under magnification neomu
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so you [ ] points now note the
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Mandelbrot set this quasi salt similar
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everywhere
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smalls like the different versions of
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itself can be found at arbitrary small
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scales but near Miller give each point
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one can actually prove that it is
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self-similar you zoom in and you see
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exact replicas of the big picture in
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small scales
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here's the Julia said 4c equals AI it's
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called the dendrite and it looks very
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much like the man who brought Seth looks
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near C equals I will look at that in the
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next pictures here you can see the
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Mandelbrot set and I've highlighted the
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point C equals I so the parameter value
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I and we zoom into the Mandelbrot set
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around this point and you can see that
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the Mandelbrot set looks like a dendrite
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and it is very self similar we keep
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zooming in and zooming in and it keeps
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looking exactly the same and it looks
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just like the Julia set looked for C
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equals I
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let's finish with one of the big
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outstanding conjectures in the field of
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complex dynamics it says the following
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the Mandelbrot set is locally connected
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what that means is it's a term we have
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not discussed before it means that for
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every parameter C in the Mandelbrot set
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so for example this one right here and
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every open set Veeck that contains C so
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some open set around it there exists an
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open side you possibly smaller than the
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original open set you picked so that
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still contains your parameter such that
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the intersection of U and M is connected
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now this boundary of the Mandelbrot set
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is rather intricate so you could imagine
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that you by choosing a small open set
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accidentally pick two parts that are not
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really connected looking because the
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manler had said we told so much it goes
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in out in and out so you could imagine
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accidentally picking a set like this
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will you just get disconnected pieces
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but this conjecture says no matter where
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you are how far you zoom in at every
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level around every parameter you can
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find the little open set that contains a
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connected piece of the Mandelbrot set
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it'll never happen to you that
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Mandelbrot set falls apart this is an
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outstanding conjecture which means it
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has not been proved to be true it is
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known that the Mandelbrot set is
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connected but it is not known that it is
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locally connected this is considered one
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of the big outstanding conjectures in
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the field of complex dynamics the reason
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why this is such an important conjecture
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goes beyond the scope of this course