Fractal Geometry
                                             Special Topics in Dynamical Systems — WS2020
                                                         Sascha Troscheit
                        Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
                                                   sascha.troscheit@univie.ac.at
                                                            March 8, 2022
                                                              Abstract
                               Classical shapes in geometry – such as lines, spheres, and rectangles – are only rarely
                            found in nature. More common are shapes that share some sort of “self-similarity”. For
                            example, a mountain is not a pyramid, but rather a collection of “mountain-shaped”
                            rocks of various sizes down to the size of a gain of sand. Without any sort of scale
                            reference, it is difficult to distinguish a mountain from a ragged hill, a boulder, or ever
                            a small uneven pebble. These shapes are ubiquitous in the natural world from clouds
                            to lightning strikes or even trees. What is a tree but a collection of “tree-shaped”
                            branches?
                               Acentral component of fractal geometry is the description of how various properties
                            of geometric objects scale with size. Dimension theory in particular studies scalings
                            by means of various dimensions, each capturing a different characteristic. The most
                            frequent scaling encountered in geometry is exponential scaling (e.g. surface area and
                            volume of cubes and spheres) but even natural measures can simultaneously exhibit
                            very different behaviour on an average scale, fine scale, and coarse scale. Dimensions
                            are used to classify these objects and distinguish them when traditional means, such as
                            cardinality, area, and volume, are no longer appropriate. We shall establish fundamen-
                            tal results in dimension theory which in turn influences research in diverse subject areas
                            such as combinatorics, group theory, number theory, coding theory, data processing,
                            and financial mathematics. Some connections of which we shall explore. 1
                         1I have no doubt that there are many typos and inaccuracies in this manuscript. If you find anything
                       that would need correction, please let me know at saschatro@gmail.com. Thank you!
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                          Contents
                          1 Introduction                                                                                        3
                          2 Classes of “fractal” sets and measures                                                              6
                              2.1   middle-α Cantor sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        6
                              2.2   Moran sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        7
                              2.3   Invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      8
                              2.4   Quasi self-similar sets    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   16
                              2.5   The universe of “fractal sets” . . . . . . . . . . . . . . . . . . . . . . . . . . .       17
                          3 Dimension Theory                                                                                   17
                              3.1   Box-counting dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         17
                              3.2   Hausdorff dimension and Hausdorff measure . . . . . . . . . . . . . . . . . . .              22
                              3.3   Packing dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        30
                              3.4   Assouad dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        33
                              3.5   Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       35
                              3.6   Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        36
                          4 Iterated Function Systems                                                                          37
                              4.1   Two important lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           37
                              4.2   Ahlfors regular spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       38
                              4.3   Iterated function systems      . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   39
                              4.4   Self-similar sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    40
                              4.5   The weak separation condition & regularity of quasi self-similar sets . . . . .            48
                              4.6   Exercises    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   52
                          5 Self-similar multifractals                                                                         52
                              5.1   Frostman’s lemma and local dimension . . . . . . . . . . . . . . . . . . . . . .           52
                              5.2   Invariant and self-similar measures . . . . . . . . . . . . . . . . . . . . . . . .        54
                              5.3   Multifractal Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        55
                              5.4   Exercises    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   61
                          6 Projections of sets                                                                                61
                              6.1   Other dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       62
                              6.2   An application to self-similar sets . . . . . . . . . . . . . . . . . . . . . . . . .      63
                              6.3   Digital sundial    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   64
                              6.4   More recent results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      64
                              6.5   Exercises    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   65
                          7 Bounded distortion and pressure                                                                    65
                              7.1   Bounded distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       65
                              7.2   Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     66
                              7.3   The Hausdorff dimension of self-conformal sets . . . . . . . . . . . . . . . . .            67
                              7.4   Aprimer to thermodynamic formalism . . . . . . . . . . . . . . . . . . . . . .             68
                              7.5   Where it all fails: self-affine sets (optional)       . . . . . . . . . . . . . . . . . . .  69
                          8 Bonus: A fractal proof of the infinitude of primes                                                  69
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                                            Figure 1: The construction of the Cantor middle-third set.
                          1     Introduction
                          Fractal geometry is a relatively young field of mathematics that studies geometric properties
                          of sets, measures, and mother structures by identifying recurring patterns at different scales.
                          These objects appear in a great host of settings and fractal geometry links with many other
                          fields such as geometric group theory, geometric measure theory, metric number theory,
                          probability, amongst others. Invariably linked with fractal geometry is dimension theory,
                          which studies the scaling exponents of properties.
                             In this course we will investigate sets and measures, usually living in Rd, that have these
                          repeating patterns and provide applications to other fields. While we will predominately
                                    d
                          workinR ,manyresultseasilyextendtomuchmoregeneralmetricspaces. Inseveralplaces,
                          especially in the beginning we give an indication of how far it can be generalised. Another
                          reason to restrict oneself to Euclidean space is visualisation. Fractal geometry is particularly
                          suited for providing “proof by pictures” as a shortcut to understanding geometric relations.
                          For example, it is easy to see that the shapes in Figures 2, 3, 4, and 5 (the Sierpinski´    gasket,
                          Cantor four corner dust, von Koch curve, and Menger sponge, respectively) are composed
                          of a finite number of similar copies of itself. Many of these objects can be constructed by
                          successively deleting subsets. The Cantor middle-third set is constructed from the unit line
                          by successively removing the middle-third of remaining construction intervals, see Figure 1.
                          The Menger sponge and Sierpinski carpet can be constructed in a very similar manner.
                             Some of the fundamental questions investigated by fractal geometry are:
                             • How can we describe and formalise “self-similarity”?
                             • How “big” are irregular sets?
                             • How “smooth” are singular measures?
                             Ana¨ıve approach to determining the size of sets, and one that works well in (axiomatic)
                          set theory is their cardinality. Clearly,
                                      {A⊆Rd:Afinite}⊂{A⊆Rd:Acountable}⊂{A⊆Rd}=:P(Rd)
                          but this begs the question of how we differentiate within classes. For finite sets cardinality
                          works well enough, whereas we will need a much more geometric approach to differentiate
                          between sets such as Q and {1/n : n ∈ N}. One such way is to take local densities to
                          account.
                             For subsets of Rd, the d-dimensional Lebesgue measure provides a first approach. This
                          limits one to measurable sets (say Borel sets), which we are fine with. However, this classifi-
                          cation makes many Lebesgue null sets equivalent. For our purposes, the Lebesgue measure
                          is not fine enough as it does not provide a good “measuring stick” for highly irregular sets
                          such as the Sierpinski´  gasket (Figure 2) and the Cantor middle-third set (Figure 1). The
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                                         Figure 2: Sierpinski´    gasket (or triangle). A set exhibiting self-similarity.
                                                                Figure 3: Cantor four corner dust.
                                                                  Figure 4: The von Koch curve.
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