SC 2018 Semilab: Fractals

by Misha Lyubich, Sabya Mukherjee

Ancient mathematicians believed that the straight line and the round circle are perfect shapes, to which everything else in this World can be reduced. Two millennia later this paradigm changed with the development of mathematical analysis, which allowed scientists to deal with quite general ``smooth" objects. It looked that it indeed provided a key to all conceivable problems. But then another shift of the paradigm occured, in the late XIX-th -- XX-th centuries, with realization that ``rough" objects naturally appear all over the place, and are actually beautiful if treated without prejudice. These days, such objects are often called ``fractals".

We will start by playing with various examples of fractals: Koch snowflakes (closed curves that would require an infinite speed to run around), Cantor dust (a set which have holes everywhere, but still contain as many points as a continuous curve), Peano curves (that fill in the whole square), Sierpinski carpets and gaskets (that also have so many holes that they would be useless in protecting your floor from dirt). All these sets are ``self-similar": if you look at any of them into a microscope then you will see exactly the same pattern as if you look with bare eyes. These examples will prompt a discussion of the meaning of mathematical concepts of ``smoothness", ``length/area", and ``dimension". A famous question ``What is the length of the British coastline?" can be addressed after this conceptual preparation.

More general fractals can be produced by ``non-linear dynamical systems" by applying the same operation over and over again. We will take a closer look at the simplest family of such systems generated by quadratic maps on the real line. Some key ideas of the theory of ``chaos" can be easily understood at this example. In particular, we will explore ``periodic orbits" for these systems, their ``stability" and ``bifurcations", as well as a remarkable ``Sharkovskii order" for their periods.

Then we will pass to even more intricate objects that can be generated by repeating the same non-linear operation over complex numbers, like ``Julia sets" and ``Apollonian gaskets". These sets are only approximately self-similar. They will motivate a discussion of ``complex analytic functions", ``groups", and ``Mobius maps". Moreover, such sets may ``bifurcate" in an explosive way as you start changing parameters, producing even more intriguing creatures like the ``Mandelbrot set".

We will finish with a discussion of ``determinism" vs ``randomness". Why the weather forecast is so unreliable though the underlying physical system seems to be completely deterministic (that is, its knowledge at some moment determines uniquely the future development)?. The answer is that randomness is actually built into such a system due to its ``instability": small errors in measurements can lead quickly to big inaccuracy of the forecast. At the same time, one can also consider intrinsically random processes (like tossing the coin). Such systems can generate beautiful fractals as well, like a trajectory of a ``classical Brownian particle" (or, of an ``elementary quantum particle" in the ``imaginary time"). In conclusion, we will try to explore some features of such random processes. Everything will be done in an informal visual way using computer simulation.

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