An exploration of the interior or the Mandelbrot Set and the appearance
of functions appearing to be similar to modular forms. This provides
yet another example of the close interconnection between the structure
of the Modular Group SL(2,Z) and fractals. The relationship is demonstrated
computationally and visually, and not from first principles; visually,
the interior resembles the Weierstrass elliptic invariant .
However, it is a resemblance only; the various explicit expressions
that can be found are shown to not actually be modular forms. It is
hypothesized that some simple but currently unknown transformation
will convert them into modular forms.

The construction of the interior is based on averaging together iterated values with a spectral-type summation, and then analyzing the asymptotic behavior of the sum. Leading divergence are easy to explain and remove; the remaining finite parts hint at modular symmetry.

This is a work in progress. A final conclusion and analysis has not been reached.

This paper is part of a set of chapters that explore the relationship between the real numbers, the modular group, and fractals. Updated and revised versions of this monograph can be found at http://www.linas.org/math/sl2z.html

XXX This paper may be subject to occasional revision. XXX

Modular forms are a particular kind of function on the complex upper
half-plane studied in analytic number theory and the theory of elliptic
curves. A precise definition of a modular form[WMF] will be
given later in this paper. As a simple example, consider the Euler
function

A rendition of the absolute value of the Euler function on the -disk. Note the readily apparent fractal self-similarity. This type of self-similarity is explicitly associated with the properties of the modular group . A crude general resemblance to the structure of the Mandelbrot set should be equally evident. This paper is devoted to making this vague resemblance into a relationship as concrete as possible. |

Now consider the Mandelbrot set. By means of a sequence of figures below, we shall uncover a structure inside the Mandelbrot set that appears to be some kind of modular form. The general development will be as follows: the first section develops a set of series that capture the asymptotic behavior of an iterated function. In the next section, these series are then applied to the Mandelbrot set iterator, where they are found to contain divergent and finite terms. The next section develops explicit, exact expressions for the divergent terms. The remainder of the paper is devoted to an exploration of the finite terms, and attempts to draw analogies to such modular forms as the Weierstrass elliptic invariant and to series involving the divisor function. Both the main cardioid and the large western bulb are explored. The paper concludes with an appendix reviewing the numeric techniques of series acceleration.

This paper is an expansion and revision of an earlier paper posted at http://www.linas.org/art-gallery/spectral/spectral.html.

This section reviews the construction of a regulated series. These series will be used to perform a kind of averaging over the values of an iterated function; the asymptotic behavior of an iterated function may be studied in terms of these series.

Consider the sequence
. Then for small
positive , construct the sum

This sum participates in some interesting number-theoretic relationships when the are considered to be the spectrum of an operator. In the following, we will be considering the not as a spectrum, but instead as the iterates of the Mandelbrot Set. Before doing so, lets quickly review some basic properties.

One can define a Dirichlet series

The spectral analysis consists of exploring the behavior of the sum
in the limit of . Depending on the series, it may
diverge. For example, if we take all to be one, the sum
diverges as 1/t, while
diverges
as . The corresponding Dirichlet series exhibit poles at
and , respectively, for these sums. The core idea behind
spectral analysis is that in general, one can gain insight into the
structure of the series by understanding the analytic
structure of the related series. In other words, instead of studying
directly, we study the expansion

When engaging in numerical calculations, the Dirichlet series is nearly
numerically intractable, because of its painfully slow convergence.
Thus,one is instantly motivated to use the exponential series instead.
However, one gets an even more stable and numerically well-behaved
series by considering the Gaussian regulator, namely

Now consider the standard Mandelbrot set iteration

where we've added the subscript to remind us that this series takes on distinct values for every point . For most of the interior of the Mandelbrot set, this sum diverges as . To normalize this sum, let us define

which also diverges as . Figure shows the nature of the divergence by normalizing against this value. As a practical matter when performing numerical computations, it is more appropriate to let stand in the place of divergences, rather than to try to use directly. The utility of this procedure is discussed in detail in a later section on numerical methods.

Plot of the divergent term of . The figure shows , where is the ordinary complex modulus. Black represents a value of zero, and green a value of 1/2. Points outside of the M-set are explicitly excluded from this picture. An explicit expression for this divergent term is given in the text. |

The first order of business is to provide an explicit expression for
the divergent term. We can do this by considering a related and somewhat
more interesting sum, the sum over second derivatives of
with respect to . Realizing that each is parameterized
by , we can take its derivative:

Note that , and are well defined for all values of and finite : they are entire functions, 'merely' polynomials in . Note that these polynomials never involve , the complex conjugate of , and thus all derivatives with respect to are vanishing.

Let us then define the sum

holds for all positive . also diverges as . Figures and shows the magnitude and phase of that divergence. By comparing the figures, it is relatively straightforward to determine that inside of the main cardiod, the divergent term of is given by

This picture shows the divergent term of , that is, . Red denotes any value equal or greater than 1, black corresponds to a value of zero. The value of this limit in the largest bud to the left is precisely zero over the entire bud. For the next smallest buds (at the top, bottom, and the second to the left), the value seems to be uniformly 1/30 across the whole bud, although there does seem to be a slight gradation which is hard to distinguish from numerical errors. By looking at this image, we can see that this limit seems to take on other, constant, values in the progressively smaller buds. The color scheme here has black <= 0.0, blue ~= 0.2, green ~= 0.5, yellow ~= 0.75, red >= 1.0. If the values were indeed constant over the smaller buds, this would have some interesting implications on the limit-cycles for these buds, as discussed in the text. |

The phase of in the limit of . That is, it shows . The color scheme is such that black=, green=0, red=. The rays on the outside of the set correspond to Duoady-Hubbard rays. Note the first hint of a modular form-like structure in the largest bulb immediately to the left of the main cardiod. |

The divergent term term of can be immediately integrated
to obtain the divergent term in inside the cardiod:

Let us now turn to the finite remainders. By subtracting away the
divergent pieces, we are essentially subtracting away the contribution
of the asymptotic limit cycle. The remaining finite parts indicate
how the asymptotic behavior is approached. If the finite part is large,
this indicates that the iteration took a long time to approach the
asymptotic limit. If the finite part is small, then the series converged
to its limit cycle quickly. The figure
shows this rate of convergence. Curiously, we find that

This figure shows the rate of convergence, that is, . We take to be as given in the text for the main cardiod, and equal to precisely 1/2 in the largest bud. For this numerical calculation, we take it to be zero everywhere else, thus leading to artifacts outside the main cardiod and bulb, where in fact shouldn't be zero. The color ramp has been scaled by -1: i.e. black = 0, green ~= -0.5, red <= -1.0. There appear to be a number of poles arrayed along the perimeter of the cardiod, located at the tangent points of the bulbs. The pole at the unicorn horn and at the largest bulb is clearly visible. These poles indicate areas where the iterated series has a very difficult time converging to a limit cycle. There is considerably more structure inside of this image than is immediately evident. The structure is exhibited when one examines the non-divergent parts of instead. |

The overall structure at first doesn't look all that inspiring. As before, we can discern considerably more structure if we examine instead of . This reveals some of the true complexity in the system.

The finite term in the main cardiod is shown in figure and at least a superficial resemblance to the image of the Dedekind zeta/Euler function shown in figure should be immediately apparent.

The image above shows the finite piece in the main bulb, after the divergent piece has been subtracted. That is, it shows . It appears to have dipoles (saddles) arrayed along the perimeter. There don't seem to be any simple zeros. The color scheme has been adjusted so that black <= -10, green ~= 0 and red >= 10. These (multi-)poles visually indicate something that is commonly known: the series has a hard time converging near the tips of the horns. |

The sums on the main circular bud immediately to the west of the cardioid do not have a divergent parts; the sums appear to be finite. The bud interior is shown in the figure .

The sum doesn't have a divergent piece in the large bud. This image shows for the square area . As can be seen, there is a considerable amount of structure here. There seem to be poles located on the boundary, where-ever another bud is tangent to this one. This is of course everywhere, since a bud is tangent for every possible rational angle. The strength of the pole is somehow proportionate to the size of the bud; the residue of the poles all seem to be the same sign. Note there seems to be a sequences of zeros inside the bud. The color ramp has been logarithmically compressed to highlight the zeros: black = 0, green ~= 10, red >= 100. A very similar figure results if one graphs the finite part on the main cardiod, after removing the divergence. That is, the graph for in the main cardiod is essentially the same. |

The bud interior shows a remarkable visual resemblance to the divisor
series

The sum on the unit disk. The colormap is logarithmically compressed, so that blue represents ares with a value of less than one, and green represents areas with a value of more than 10. |

Re-expressing the coordinates on the interior of the bud as

The numbers in parenthesis give the uncertainty in the least significant digit. This is a fairly quick and rough fit; only the values of the first two terms seem certain, and that the coefficient of seems to vanish.

Figure shows the real part of in the bud.

This figure shows the real part of in the western bud. The color scheme is identical to that used to show the modulus. Black areas here represent negative values for the real part. A similar graph of the divisor series would have more or less a rather similar look. |

Explicit numerical work shows that it does not seem to be a modular form of integer weight. Nor does it seem to be a modular form of fractional weight. But it sure seems to ``come close''. Lets review what this means.

Modular symmetry on the -disk is best explored by mapping the
-disk to the Poincare upper half-plane, applying a Mobius transformation
there, and then mapping back. Given a point in the upper half-plane,
i.e. , one maps to the -disk with

and then map this back to the -disk coordinates.

A function on the upper half-plane is said to be a modular
form if

There is one interesting mapping whose properties are worth reviewing, and that is the mapping of the upper half-plane to the Poincare disk. This mapping is curious because it is not infrequent in the literature, and because a periodic function on the upper half-plane takes the appearance of a self-similar function on the disk.

The mapping if the upper half-plane to the Poincare disk is given
by

It is not hard to show that ; that is is on the edge of the disk. A function which is periodic in with integer periodicity will manifest itself with an M-set like periodicity on the circumference of the Poincare disk. Specifically, a feature located at integer values of will have a specific angular location on the Poincare disk, which is obtained by solving for the angle in

In particular, the region is mapped to the angular interval .

Images constructed by mapping the -disk to upper half-plane, via
equation , will be inherently periodic. The
Mobius transform

The figures and show a mapping of the western bud to the Poincare disk. More precisely, the mapping is actually a half-angle mapping, taking to instead of , and then re-mapping to the Poincare disk. The result of the half-angle mapping is that the figures do not have the left-right symmetry , but this is only an artifact of the construction.

This figure shows the absolute value of the imaginary part of in the western bud, remapped by means of the half-angle mapping, onto the Poincare disk. The color scheme is identical to that used in other graphs. As the values shown here are by definition positive, the color black corresponds to small but positive values. |

In order to proceed with the exploration of the interior of the Mandelbrot set as a modular form, we need to find a way of mapping the the cardioid to the complex upper half-plane. The most obvious mapping is to express the interior in terms of the coordinates and with the interior given by

The linearized coordinates can be immediately remapped to a circle
by using the coordinates

with the word ``puncture'' referring to the fact that never occurs for finite values of . This mapping maps the upper half-plane to values of . The orientation of this mapping is specifically picked in order to be consistent with standard definitions of modular functions on the punctured disk. For example, the Euler phi-function, expressed in coordinates, is

The cardioid interior should be compared to the image of the Weierstrass invariant , shown in figure .

An image of the the real part of the Weierstrass invariant
expressed in coordinates. This function can be written explicitly
as which can be re-expressed as a Lambert series This image uses a highly compressed logarithmic color scale adjusted to resemble that used for the Mandelbrot interior. Note that the modulus of does not show this lobe structure; the real and imaginary parts of this function have complementary values. Graphs of resemble this figure, as do those of higher terms in the Eisenstein series. As one goes up the series, the number of lobes increases arithmetically. For example, the above figure shows three red lobes; the comparable figure for shows four lobes. |

By comparing the figures for the interior of the Mandelbrot set and the Weierstrass elliptic invariant, a general resemblance becomes painfully apparent, even if not explicitly demonstrated.

By construction, the function just demonstrated on the interior of the Mandelbrot set is a real function. To more fully explore the modular symmetry, we really need a complex function, that is, one with real and imaginary parts. Such a function is provided by not working with the modulus, but subtracting the divergence directly; this is explored in the next section.

There is also a more subtle issue. It is not clear that the simple cardiod mapping is the correct mapping. If one examines the figures, one can note a subtle, ever-so-slightly visible feature. Each of the ``flames'' in the figure lean slightly over to the right. Under Mobius transforms, this tilt is preserved, destroying the naive symmetry. Its possible that the mapping from cardioid to coordinates is not the right mapping, and that some other, more complex mapping is required. What this mapping may be is not clear at this point.

This problem is presumably related to the fact that the buds on the exterior of the M-set are almost circles, but not quite (with the exception of the main bud to the west). If one could find a suitable remapping on the exterior, that mapping might presumably carry over into the interior as well, and vice-versa.

Lets revisit, this time exploring the function

This figure shows the modulus of the finite part . Some of the rest of the M-set is visible, but for the most part is blanked out by the subtraction of the divergent term in the main cardioid. Since this divergent term is inappropriate for the other parts of the M-set, these other features wash out. The same compressed logarithmic color scheme is used as in the other illustrations. |

Despite the remarkably suggestive graphics, it seems that is
not a modular form either; in particular, I was unable to find a real-valued
number for which even the less-demanding relation

This section reviews the numeric techniques applied to perform the series sums. Specifically, some well-known techniques for series acceleration are applied; but these are not so well known as not to merit review.

Note that the series explored on these pages can be slow to converge, especially near the 'horns' of the Mandelbrot set. There are several well-known and established techniques of series acceleration that can improve the convergence. This section quickly reviews the technique used in this paper.

Consider the sum

It is clear that if the limit exists and is finite, then one has

and furthermore, for small one has

The correctness of equation at first seems naively 'obvious' but is in fact quite subtle, and depends on having a series that is 'well-behaved' in certain ways.

The study of such series and the numerical techniques to sum them falls under the name of 'series acceleration' and is a well-developed branch of mathematics in its own right. It is outside of the scope of this section to review any deeper results. Suffice it to say that this entire paper is predicated on the assumption that the equation does hold for the sums encountered. This does seem to be the case, but is hardly obvious from first principles, especially for points in troublesome sections of the M-set. By contrast, in the well-behaved areas of the M-set, it is straightforward to verify that equation holds, and that the resulting sums are accurate for five to ten decimal places, corresponding to values in the range of 0.01 to 0.001 for sums with 2 to 50 thousand terms.

Some of the sums in encountered in this paper are divergent. The simplest
such sums have a limit point, namely

The extra factor of comes from the behavior of the Gaussian regulator; that is,

In order to correctly subtract this linear divergence from a divergent sum, one is advised to subtract rather than directly. This advice comes from the need to subtract the divergence so that the equation is not violated. In general, will have , and terms as well, any one of which will mess up equation if not properly accounted for. Thus, in general, the correct way to subtract a divergent term is to form

and then form from to obtain the finite part. One can perform the subtraction under the summation, or outside of it. Performing it under the summation potentially minimizes round-off errors. Note that these considerations apply to series with limit cycles as well as limit points. That is, the need not converge to a point; as long as they do converge to a limit cycle, this mechanism of subtracting the divergent piece will work.

One final remark: note that, in general, after removing a linear divergence in a summation, the next leading order need not be finite, but may be a weaker divergence, such as a logarithmic divergence. This is presumably the nature of the divergences seen at the horns of the M-set, for example. On the complex plane, finite sums grow until they hit a pole. At the pole, the sums are logarithmically divergent.

This paper reprises and revises an earlier draft from November 2000, located at http://www.linas.org/art-gallery/spectral/spectral.html.

Although an explicit expression for the apparent modular symmetry was not found, it is believed that a convincing argument has been made that such a symmetry lurks within the asymptotic limits of the Mandelbrot iterator. Specifically, the actual symmetry appears to most closely resemble that of sums involving the number-theoretic divisor function. Obtaining an explicit form will open up additional avenues of research, possibly shedding light on the maddening contour of the Mandelbrot Set.

What more can we say? This is wild stuff.

- Apo90
- Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer Verlag, New York, 1990.
- WMF
- Wikipedia, Modular forms, http://en.wikipedia.org/wiki/Modular_form
- Lin00
- Linas Vepstas, Spectral Analysis of Mandelbrot Interior, (2000) located at http://www.linas.org/art-gallery/spectral/spectral.html.

Linas Vepstas 2005-05-30