# The Modular Group and Fractals

An exposition of the relationship between Fractals, the Riemann Zeta, the Modular Group Gamma, the Farey Fractions and the Minkowski Question Mark. Well, all that, and more.

Linas' Mathematical Art Gallery has been running for over thirty years while being silent about the underlying math. At some point, this became untenable, and this page attempts 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. These have the same symmetry of the infinite binary tree. This is termed the "dyadic monoid" below; but it is known by other names. It is the self-similarity (symmetry) monoid of the Cantor set, aka Cantor space. Via a run-length encoding, aka the "continued fraction encoding", aka "the Minkowski question mark function", it is the similarity monoid of Baire space.

The dyadic monoid is a certain subset 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 (via the continued-fraction mapping).

This provides an interesting knot of ideas to study, in part because they relate together so many different things. Most obviously, it offers some hints as to why Farey Fractions are prominent in the Mandelbrot Set and other fractals.

In number theory, the structure of the Modular Group provides a unifying theme for understanding the nature of factorization and primality. This offers glimmers as to why, for example, power series and Dirichlet series (such as the Riemann Zeta), and even the prime numbers exhibit such crazy fractal Cantor set style patterns.

The Cantor space space and Baire space are both examples of "Polish spaces", along with the real numbers (including Euclidean space) and seperable Banach spaces (including Hilber spaces). Large parts of topology, including explorations of the Borel hierarchy and even subsets of descritive set theory, happen on Polish spaces, if not directly on Cantor or Baire space. Those analyses rarely (if ever?), explore the self-similarity underlying these spaces, and yet its an underlying property. So one can go explore. Yummy! Like a candy store!

Last but not least, the Bernoulli shift on Cantor space provides one of the simplest available measure-preserving dynamical systems that exhibits many of the characteristic features of more complex dynamical systems, including periodic orbits, chaotic orbits, ergodicity and an invariant measure. It is simple enough to be exactly solvable, yet rich enough to be foundational.

Any one of the above topics seems worthy of examination. An exquisite tangle of all of them, intertwined, is too much to be spurned.

This page provides several dozen tracts on these topics. They are presented in a sequence of increasing sophistication and complexity. Many, perhaps most of these require little or no formal education in mathematics. They can be read by anyone exicted by the topics being presented. There are almost no theorems or proofs at all; instead, there are a lot of images, figures and illustrations.

Despite this attempt at general readability, a lot of complicated material does show up in here. In the end, I wrote all this for myself, and not for you, dear reader. Alas. My favorites? Hard to say. The newer stuff in the middle. I saved the worst for last; the bottom of this list is populated by assorted dregs that didn't quite work out. I've attempted to mark the latest stuff with a bright red (New!) marker.

• 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 its made out to be.

• 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 2.1: Entropy of Continued Fractions (Gauss-Kuzmin Entropy) (6 pages) computes the entropy of the Gauss-Kuzmin distribution, and finds it to be approximately 3.432527514776... By contrast, the defacto entropy of small rationals is considerably smaller, rising slowly and barely getting as large as H=3 when computed for all rationals with denominators smaller than 100,000.

• Chapter 3: The Minkowski Question Mark and the Modular Group PSL(2,Z) (51 pages) shows that the distribution of Farey Fractions transforms under a certain subset of the Modular Group, the dyadic monoid. This monoid is defined, it's action on the infinite binary tree (the dyadic tree) is explored. The relationship between these ideas and the Cantor set is reviewed. The Minkowski Question Mark Function is then constructed, and it is shown how the self-similarities of this fractal curve are given by the dyadic monoid. Also reviews the hyperbolic rotations of binary trees; defines and reviews the dyadic lattice. This paper provides the core background material for the structure of the dyadic monoid that is used in the other papers of this series.

• On the Minkowski measure (2008) (27 pages) (See also arXiv:0810.1265v2 [math.DS]) points out that the derivative of the Minkowski Question Mark function is given by the distribution of the Farey fractions on the real number line. This is anchored by a foundationally sound derivation of the Minkowski measure, set in the terms of measure theory. This allows an exact result to be presented for the measure, as the infinite product of a set of piece-wise differentiable functions, each piece being in the form of a Mobius transform. Additional theoretical machinery is developed to express transfer functions as push-forwards on Banach spaces; this is used to demonstrate that the Minkowski measure is an invariant measure, a Haar measure, induced by a certain twisted Bernoulli operator. The theoretical machinery allows the discussion of some of the eigenvectors of the transfer operator. It is pointed out that the Minkowski measure is also an eigenvector of the Gauss-Kuzmin-Wirsing operator.

• Modular fractal measures (23 pages) A working diary: a numerical exploration of the Fourier transform of the Minkowski measure. Assorted odds and ends, half-baked, poorly-expressed ideas.

• Chapter 6: Symmetries of Period-Doubling Maps (54 pages) identifies a certain period-doubling monoid subset of the Modular Group PSL(2,Z), the dyadic monoid, 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, who put fractals into both the popular and scientific limelight, doesn't breath even a word of this in his most recent (2004) book. This paper develops the Takagi or Blancmange Curve as a fractal curve that transforms under the three-dimensional matrix representation of the dyadic 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 brief definition of de Rham curves, followed by a gallery of almost 50 images exploring the four-dimensional space of such curves.

• Chapter 7: The Bernoulli Map (PDF, 50 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, 47 pages) is the Transfer Operator of the Gauss Map. An incomplete review of some of the facts concerning this operator is presented; some results are new.

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: Spectrum of the Beta Transform (62 pages) and On the Beta Transform - Diary and Notes (123 pages). (2017) The beta transformation is the iterated map βx mod 1. This text explores the transfer operator of the beta transform and discovers that it has eigenvalues resting on the unit circle of radius 1/β in the complex plane.

For certain values of β, these are finite in number; these correspond to a generalization of the Golden ratio, and are the zeros of a generalized series of "golden polynomials". These can be counted by Moreau's necklace-counting function, but they do not appear to be related to other structures that the necklace-counting function counts(!)

These special values of β are in one-to-one correspondence with the famous "Islands of Stability" (the period-doubling regions) in the logistic map. In this sense, they "explain" why the period-doubling regions are where they are, and how to control them.

The paper on the spectrum is condensed and readable; the diary contains additional material that is scattered, incomplete and poorly organized. (Some of the material is boring, it's just dead-ends.) The diary does contain some good stuff, though. This includes:

• Connections to Bergman space and Bergman polynomials, via the Hessenberg matrix operator form are sketched, as well as a move towards Jacobi polynomials.
• The complicated structure of the iterated logistic map, tent map, etc. is entirely due to the chaotic dynamics of the carry bit in multiplication. If the carry bit is suppressed, then one obtains only reshufflings, which have a completely uniform distribution lacking in structure.

• Jimm, a Fundamental Involution by A. Muhammed Uludağ, Hakan Ayral (2015) I didn't write this, but I've posted it here as it it is an involution of the real numbers, and it's singular on the "nobel numbers" (the orbits of the golen mean under the Beta transformation, explored in the paper immediately above.) I have not explored as to whether there is a direct connection and whether Jimm can be used to obtain insight. Still, it seems like a promising, unfinished, strange corner to explore.

• Chapter 10: Linas' Art Gallery Unlike everything else on this page, this points not at some PDF's filled with ... stuff, but rather at an Art Gallery filled with pretty pictures. The PDF's use fancy words like "subshifts of finite type" but this animated gif actually shows what some random typically atypical shift really looks like: in this case, the shift applied to the greatest prime factor exponential generating function. (Right before the end of the strip, observe how two zeros, left of center, merge and then split! Cool, huh? What's going on here is not quite as simple as you might first think.)

• The Newton Series Representation for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing Operator (2004/2005) (PDF, 18 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.

This paper is cited by:

• On Differences of Zeta Values is a cleaned-up, expanded, published variant of the above, co-authored with Philippe Flajolet. The second half of this paper gives several statements that are equivalent to the Riemann hypothesis. (Unfortunately, neither the abstract nor the introduction make this clear, which is an unfortunate oversight).

• Notes Relating to Newton Series for the Riemann Zeta Function (PDF, 34 pages) (2006) Extended working notes and observations made during the development of the above-mentioned 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.

• Yet Another Riemann Hypothesis (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 their zeros in the critical strip and probably on the critical line. The exploration is purely numerical.

• An efficient algorithm for computing the polylogarithm and the Hurwitz zeta functions (2007) (PDF, 33 pages) (See also: arXiv:math/0702243v4 [math.CA].) 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 Lis(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.

• The Polylog and the Riemann Hypothesis (New!) (PDF, 9 pages) (October 2023) Supplements the above by providing a more detailed review the connection between the RH and the polylog. It allows the refinement of RH into a strong and a weak version, one of which must hold true for RH to hold. The breakdown of RH would suggest some pretty weird behaviors in the polylog that would run counter to conventional ideas about holomorphic functions. Perhaps this opens up a new route for searching for RH proofs. Things still look hard: one would have to prove that certain polylog "varieties" are "entire", in a certain sense. But I don't know how to do that. See the Polylog Movie Page for animations of the polylog function.

• Eventually Periodic Orbits and the Vitali Set (New!) (PDF, 18 pages) (November 2023) The Vitali set is a canonical example of a non-measurable set. It's description requires the use of the Axiom of Choice an uncountable number of times. The classical approach is to demonstrate the quotient R/Q of the reals by the rationals, and then chose an uncountable number of points from the quotient, resulting in a set that does not have a representation in any sigma algebra, and thus cannot be assigned a size. A sophisticated approach is to employ the language of Borel sigma-algebras, demonstrate a comparable quotient via a Borel equivalence relation, and arrive at a set that does not admit a measure. A middle ground is to work with the Bernoulli shift on the Cantor space. This provides a concrete, direct and easy-to-describe system to work with, while retaining the essential properties of the abstract approach. This approach indicates exactly why the use of the Axiom of Choice is forced: the cosets of the quotient space are not well-founded and thus do not have any unique, distinct element that can be described. Lacking the ability to isolate a specific element, one is instead forced to declare that surely, there must be some element and that it can be chosen. There are uncountably many such cosets, uncountably many choices, and as a result, a set without any representation in a sigma algebra.

• On Plouffe's Ramanujan Identities (Springer The Ramanujan Journal: Volume 27, Issue 3 (2012), Page 387-408) (arxiv arXiv:math/0609775v3 [math.NT]) (2012) (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. Turns out that these identities are "well known", and there have been over half-a-dozen independent, published re-discoveries of these identities in the century since Ramanujan's time. Add my name to the list!

• 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.

• Divisor Operator (New!) (PDF, 12 pages) (2020) The “divisor operator” is defined as an infinite-dimensional matrix operator, encoding Dirichlet convolution in a linear algebra setting. The finite-dimensional variant is known as the Redheffer matrix. As a matrix operator, it naturally acts on the Banach space l1 of summable sequences. On this space, it is not a bounded operator. It's point spectrum consists of all completely multiplicative arithmetic series (that are l1-summable).

• Euler Re-summation of Multiplicative Series (PDF, 6 pages) ...doesn't work. Which is a surprising result. Because it works great on the Riemann zeta. The Euler transformation of alternating series is known to improve numeric convergence. Sometimes. Applied to a zeta-like series constructed from a completely multiplicative arithmetic function, it fails. The intended question to be posed is: what classes of completely multiplicative arithmetic functions result in sums obeying the Riemann hypothesis? A numerical survey addressing this question seems straightforward, if only the summation can be re-written to converge quickly in the critical strip. Euler re-summation is a basic, simple trick for achieving this. It works like a charm, for the Riemann zeta, and utterly fails otherwise. Hmmm. What does tht mean?

• 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.

• The Simple Harmonic Oscillator (November 2006) (9 pages.) A glance at the non-square-integrable eigenfunctions of the quantum simple harmonic oscillator. Unlike the square-integrable eigenfunctions, these form a continuous spectrum. The eigenfunctions themselves are given by the confluent hypergeometric series (Kummer’s function). Some pretty pictures are graphed.

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

• Algorithms in Analytic Number Theory is the set of routines I wrote to explore some of the math above. Written for the Gnu MP arbitrary-precision math library, these implement the Hurwitz zeta, Riemann zeta, polylogarithm, Minkowski question mark, confluent hypergeometric function, as well as assorted trig functions (sine, cosine, exp, log), the gamma function, binomial coefficients, a complex number type, and assorted high-precision constants. All written in C for the GMP math library.

• 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.

• Why Humans Hallucinate Fractals A mathematical, neuroscientific hypothesis about why humans hallucinate fractals and hyperbolic space artifacts, when under the influence of DMT, LSD, etc. In short: visual processing is done by cortical columns in the visual cortex. The perception of 3D arises as "constraints" between cortical columns, and these constraints are exactly those that convert free groups (free modules) into Cartesian spaces. (The "constraints" are the presentations, the quotients of the modulo quotienting operation.) This is a special case of the history monoid/trace monoid of parallel computing, where mutexes/semaphores give rise to a semi-commutative monoids. In the visual cortex, they are fully abelianized, giving rise to 3D cartesian space. If these constraints are lossened, one gets not cartesian space, but e.g. elliptic curves, assorted hyperbolic spaces, hyperbolic tilings, etc. These are assorted half-way points between free groups (free modules) and flat Cartesian space. The hypothesis is that hallucinogens disrupt communications between cortical columns, loosening the constraints that project onto pure Cartesian 3D space, and instead generate the various intermediary group/module structures, which are all fratcal-ish to varying degrees.

I am also intrigued by another rather more complex connection: the chaotic dynamics of a pendulum is described by the KAM torus. The classical theory of the pendulum involves elliptic integrals. But elliptic integrals are closely related to the Jacobi theta functions, and thence to modular forms. But modular forms have the modular group symmetry; this can be explicitly seen in some of the elliptic functions. Does this mean that if we root around a bit, that we will find some modular group symmetry in the KAM torus? I don't know, but I'll bet that it is there; it can't be just a pure "accident" that this model of chaotic dynamics just happens to be so close to modular forms.

Of course, everything is even crazier than it all seems at first. Elliptic integrals show up in a number of rather distinct, seemingly unrelated places. One is in a class of (exact) solutions for the three-wave equation. This is interesting, because the three-wave equation provides one of the simplest systems in which to study chaotic dynamics and ergodicity in wave mechanics. In short: three (or more) waves can mix and exchange energy between modes whenever their dispersion equation allows conservation of both energy and momentum. This is the primary avenue from turbulence in high-dimensional systems to thermodynamic equilibrium.

A second, utterly unrelated appearence of the elliptic functions are in providing exact solutions for geodesics in the Schwarzschild vacuum. This is not odd at all; this is merely just some geometric coincidence, a side effect of the particular form that the Hamilton-Jacobi equations just happen to have in this spacetime metric. This has nothing to do with anything else, whatsoever, at all. Nada, Zero. Zippo, Zilch. Not in the slightest, not in the least. Just some unrelated phenomenon. Right, guys? Right? Guys? Uh ... guys? Hey, where are you?

Created in 2004
Last updated December 2017
Last updated December 2023
linasvepstas@gmail.com