The Modular Group and Fractals

Linas' Mathematical Art Gallery has been running for fifteen or twenty years while being silent about the underlying math. I suppose its high time to make amends. The core idea of the dissertation is that the shapes of fractals are describable through Farey Fractions, which appear naturally through continued fractions, which have the symmetry of the Modular Group SL(2,Z), which is a subgroup of the Fuchsian group SL(2,R), in turn a subgroup of the Kleinian group SL(2,C), all of which are inter-twined with the Riemann Zeta and the structure of the set of rational numbers. Besides the four basic operations on the real numbers (addition, subtraction, multiplication, division), there is a fifth basic operation which is rarely taught in primary school and under-appreciated at higher levels, namely, "Farey Addition" or, expressed correctly, group multiplication in SL(2,Z). The modular group doesn't just lead to Pellian equations and algebraic numbers, it in fact intertwines all rational numbers (and their extensions to reals and p-adics) in crazy, fractal ways. This is why, for example, one sees Farey Fractions in the Mandelbrot Set. In number theory, the structure of the Modular Group provides a unifying theme for understanding the nature of factorization and primality. This is why, for example, power series and Dirichlet series (such as the Riemann Zeta) exhibit such crazy fractal Cantor-Set type patterns. Despite this connection being seen by Weierstrass as early as 1872, its more-or-less entirely ignored in standard textbooks on Analysis and Number Theory. The series of articles below tries to provide some of the underpinnings for the above breathless assertions.

• Chapter 1: Distributions of Rationals on the Unit Interval (or, How to (mis)-Count Rationals) (PDF) (19 pages) is a high-giggle-factor review of some so-called "facts" about fractions that you learned as a child, and which every math teacher ever since has repeated, but which are simply not true. I used to think Number Theory was boring until I saw this. Reducing a fraction to a relatively prime numerator and denominator isn't as boring as I used to think.

• Chapter 2: Continued Fractions and Gaps (PDF)(34 pages) provides a very curious function that is discontinuous on the rationals and whose discontinuities seem to be perfectly randomly distributed. I find this to be a rather dramatic result, possibly because I've never heard of such a thing before. I've stared at a lot of fractals and space-filling curves, but nothing like this.

• Chapter 3: The Minkowski Question Mark and the Modular Group SL(2,Z) (40 pages) shows that the distribution of Farey Fractions transforms under the dyadic representation of the Modular Group. It constructs the Minkowski Question Mark Function as a set-theoretic representation of (a subset of) the Modular Group. Reviews the hyperbolic rotations of binary trees. This paper provides the core background material for the structure of the modular group that is used in the other papers of this series.

• Chapter 6: Symmetries of Period-Doubling Maps (54 pages) identifies a certain period-doubling monoid subset of the Modular Group PSL(2,Z) as the basic symmetry of a large class of fractals. Although this relationship is obvious and not very deep, it never ceases to amaze me that books on modular forms never mention fractals, and that books on fractals never mention modular forms. Even Mandelbrot's most recent book, published in 2004, doesn't breath a word of this. This paper develops the Takagi or Blancmange Curve as a fractal curve that transforms under the three-dimensional matrix representation of this monoid. It then shows how to build higher-dimensional representations out of the Bernoulli polynomials. The Koch snowflake, the Peano space-filling curve and the Levy C-curve all appear as special cases of the de Rham curve construction.
• Chapter 6.5: A Gallery of de Rham curves (29 pages) A breif definition of de Rham curves, followed by a gallary of almost 50 images exploring the four-dimensional space of such curves.

• Chapter 7: The Bernoulli Map (PDF, 42 pages) applies Transfer Operator techniques to the Bernoulli Map. Much of the material presented is "well-known" in that Bernoulli processes are commonly studied in many areas, from probability theory, (the simplest Markov chains), the subshifts of finite type, the one-dimensional Potts model, the dyadic Wavelet transforms, the Cantor set, etc: all these are connected via a dyadic, representation, the set of strings in two letters.

  Perhaps the only new result here is the discussion of the continuous spectrum of the Bernoulli operator. It is given by the Hurwitz zeta function, which can be written as a linear combination of the Takagi curves.

• Chapter 8: The Gauss-Kuzmin-Wirsing Operator (PDF, 21 pages) is the Transfer Operator of the Gauss Map. A brief but incomplete review of some of the facts concerning this operator is presented.

  The new results here are the presentation of a "topologically equivalent" map, which is exactly solvable. However, the topological equivalence does not preserve the spectrum of the GKW, so this cannot be considered to be a solution of the GKW. This failure is attributed to the fact that the conjugating function is the Minkowski Question Mark function. The Jacobian of the transform is the infamous, prototypical "multi-fractal measure" discussed previously.

• Chapter 9: Yet Another Riemann Hypotheis (PDF, 10 pages) considers the action of the permutation group on the continued fraction expansion of the real numbers. This action generalizes a certain integral representation the Riemann zeta function, and leads to a set of functions that resemble the zeta in that they appear to have thier zeros in the critical strip and probably on the critical line. The exploration is purely numerical.

• Notes Relating to Newton Series for the Riemann Zeta Function (PDF, 34 pages) (2006) Extended working notes and observations made during the development of a joint paper with Philippe Flajolet. An evaluation of the asymptotic form for the Newton series of the Riemann zeta function and the Dirichlet L-functions are given. The asymptotic form is obtained by performing a saddle-point analysis of the Norlund-Rice integrals that correspond to the series. Similar series for other number-theoretic functions, such as the Mobius, Liouville and Euler Totient are explored. This extends results previously given by Lagarias, Coffey, Baez-Duarte and Maslanka.

• An efficient algorithm for computing the polylogarithm and the Hurwitz zeta functions (PDF, 33 pages) This paper develops an extension of the techniques given by Borwein's paper "An efficient algorithm for computing the Riemann zeta function", to the polylogarithm and the Hurwitz zeta function. The algorithm provides a rapid means of evaluating Li_s(z) for general values of complex s and the region of complex z values given by |z²/(z-1)|<3.3. This region includes the the Hurwitz zeta ζ(s,q) for general complex s and real 1/4≤ q ≤3/4. By using the duplication formula, the range of convergence for the Hurwitz zeta can be extended to the whole real interval 0<q<1, although the algorithm does run logarithmically slower as it approaches the endpoints. In particular, this algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. Includes a discussion of the monodromy group of the polylogarithm.

• On Plouffe's Ramanujan Identities (arxiv Math.NT/0609775) (16 pages). Simon Plouffe gave a series of identities discovered numerically (Part I and Part II) for the Riemann zeta at odd integer values, inspired by an identity for Apery's constant zeta(3) in the Ramanujan notebooks. This text presents an analytic derivation of these identities (both those from 1998, and the new ones from April 2006), showing their full generality.

• Graphs of Famous Number Theoretic Functions. This one is another tease; and worse than the first. I just don't get the underpinnings yet, so there is no explanation to go with these pretty pictures. There is something one can conclude, though: Graphing the Maclaurin series for random arithmetic function on the unit disk will reveal hyperbolic Riemann surfaces with visually evident Fuchsian group symmetries. Any random function will do. I suspect this is at the heart of the Riemann zeta and the Dirichlet L-functions: the actual series don't matter. Almost any randomly generated series will be hyperbolic, and exhibit a Fuchsian-group symmetry. I used to believe I knew what an analytic function was; I now realize how nearly total my ignorance is of such matters. Equally shameful is a prevailing academic attitude attitude that a few semesters of undergraduate analysis is all one will ever need to know about analytic functions; clearly, there is more to it than just that.

• Lattice models and fractal measures (33 pages) points out that the prototypical "multifractal measure", namely, the derivative of the Minkowski Question Mark function. is given by the distribution of the Farey fractions on the real number line. Some crude attempts are made to explore alternative topologies, including topologies associated with one-dimensional lattice models, such as the Ising model or the Potts model. It is conjectured that the mutli-fractal measures correspond to smooth, differentiable Hamlitonians on one-dimensional lattice models.

• Measure of the Very Fat Cantor Set (16 pages) This brief note defines the idea of a "very fat" Cantor set, and briefly examines the measure associated with such a very fat Cantor set. The canonical Cantor set is "thin" in that it has a measure of zero. There are a variety of methods by which on can construct "fat" Cantor sets (also known as Smith-Volterra-Cantor sets) which have a measure greater than zero. One of the commonest constructions, based on the dyadic numbers, has a continuously-varying parameter that is associated with the measure. The Smith-Volterra-Cantor set attains a measure of one for a finite value of the parameter; this paper then explores what happens when the parameter is pushed beyond this value. These are the "very fat" Cantor sets referred to in the title. The results consist almost entirely of a set of graphs showing this behavior.

• The Mandelbrot Set and Modular Forms (PDF) (27 pages) examines the limit cycles of iterated points in the interior of the Mandelbrot set, and discovers that these limit cycles seem to be some sort of modular form. The interior is compared visually to the Dedekind Eta (a modular form of weight 12) and the Weierstrass elliptic invariant g2 (a modular form of weight 2), as well as to sums built from the number-theoretic divisor function. The visual resemblance is remarkable, but an explicit expression for the modular form is not obtained. The statement here is that this provides an even more direct link between modular forms and fractals, exhibiting explicitly a modular group symmetry of the Mandelbrot Set. This is an expansion and revision of the old draft (2000) in the art gallery.

• Annotations to Abramowitz & Stegun (DVI) (PDF) (13 pages) includes a set of annotations to the classic Handbook of Mathematical Functions, including new integrals over Bessel functions, and some sums over the Riemann Zeta function.

• Gap Theory develops some basic relationships between continued fractions, fractals and Farey Numbers.

Obsolete or Less Interesting

• The Newton Series Representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator (2004/2005) (PDF, 14 pages). It is well known that the Riemann zeta function is the Mellin Transform of the Gauss Map. As shown above, the GKW operator is the Transfer operator of the Gauss Map. Putting these together leads to a curious representation of the Riemann zeta function as a Newton Series (a finite difference series) of the Riemann zeta. One of the peculiar and interesting results is that the coefficients of this expansion are exponentially small.