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Without getting too bogged down in details, my Ph.D. Research was in Fractals and Chaos in Complex Polynomials. A “real” polynomial is a sum of powers of x, the variable for real numbers. Similarly, a “complex” polynomial is a sum of powers of z, the “complex variable” z = x + yi, where x and y are real variables and “i” is the square root of minus one.
My field of study was “Dynamical Systems”. In a dynamical system, one has a function or mapping of a geometric space and one is interested in iterating the function, or applying it over and over, and following the time evolution orbit of points in the space. In our case, the space is the two-dimensional complex plane and the complex polynomial is just the quadratic polynomial P(z) = z^2 + c, where z is a complex variable z = x + yi, with x and y real variables and “i” is the square root of minus one, and the “^” carat is raising to the power:
e.g., x^3 = x * x * x.
The iterations of P(z) are the second iterate,
P(z)^2 = P( P(z) ), i.e., apply polynomial twice;
The third iterate:
P(z)^3 = P( P( P(z) ) ), i.e., apply P(z) thrice;
And so on:
The n-th iterate is applying P(z) n times.
One is interested in the over the long-term time evolution dynamical behavior of orbits. It turns out that complex polynomials have a chaotic geometrically fractal set where all the randomness lives, the so-called “Julia Set”. These Julia Sets were studied in the 1920’s by two French Mathematicians, Farou and Julia. This was before the advent of computers so it was all paper and pencil! In the 1970’s another French Mathematician, Benoit Mandelbrot, revisited Julia Sets using IBM (where he worked) computers. These pictures were crude. With modern powerful computers very detailed geometric pictures of the Julia Set Fractals are now available, with “Deep Zooms”, where one magnifies the Julia Set more and more revealing the complicated geometry. Here is a link to a five-minute Julia Set Deep Zoom:
The computer study of the Julia Sets led Mandelbrot to discover the grand-daddy of all fractals, called the Mandelbrot Set, a “Map” of the chaotic Julia Sets. Here is a two-minute Deep Zoom:
A twenty-six minute video explaining the Mandelbrot Set follows:
A fourteen minute video explaining how the Julia Sets are related to the Mandelbrot Set.
Lastly, FWIW in a concise statement is the result of my research which was detailed in my Ph.D. Dissertation:
For a complex polynomial P(z), the Komolgorov Entropy measure of randomness h(P) is equal to the Hausdorff Dimension of the Fractal Julia Set J(P) times the Lyapunov Exponent L(P) (which measures the long-term exponential expansiveness of the derivative of the iterates P(z)^n: in symbols:
H(P) = J(P) * L(P)
Slainte! (Irish for “Cheers!”)
Proinseas (Irish for “Frank”) O’Suilaebhain (Irish for “Sullivan”, aka “Eagle-Eyed”)
