Symmetries of Period-Doubling Maps

The concept of self-similarity is central to the notion of a fractal, but the actual symmetry group that generates that self-similarity is rarely named, and it seems fractals are almost never studied according to their symmetries. Yet, in other branches of mathematics and physics, it is well understood that symmetry provides a powerful mechanism for understanding systems. In this paper, we identify the symmetry group of period-doubling maps as being a monoid (semigroup) of the modular group PSL(2,Z). To anchor this assertion, we work out an explicit, exactly-solvable fractal curve, the Takagi or Blancmange Curve, as transforming under the three-dimensional representation of the (monoid of the) modular group. By replacing the triangular shape that generates the Blancmange curve with a polynomial, we find that the resulting curve transforms under the n + 2 dimensional representation of the monoid, where n is the degree of the polynomial. We also find that the (ill-defined) derivative of the Blancmange curve is essentially the (inverse of the) Cantor function, thus demonstrating the semigroup symmetry on the Cantor Set as well. In fact, any topologically conjugate map will transform under the three-dimensional representation. We then show how all period-doubling maps can demonstrate the monoid symmetry, which is essentially an outcome of the dyadic representation of the monoid. This paper also includes a review of Georges deRham’s 1958 construction of the Koch snowflake, the Levy C-curve, the Peano space-filling curve and the Minkowski Question Mark function as special cases of a curve with the monoid symmetry. The left-right symmetric Levy C-curve and the left-right symmetric Koch curve are shown each belong to another, inequivalent three dimensional representation. This paper is part of a set of chapters that explore the relationship between the real numbers, the modular group, and fractals. Its also a somewhat poorly structured, written at least partly as a diary of research results. 1 Symmetries of Period-Doubling Maps It has been widely noticed that Farey numbers appear naturally in certain fractals, most famously in the Mandelbrot set. For example figure 1 shows how to count the buds of the Mandelbrot set by the Farey numbers. The reason why the Farey numbers are appropriate for such counting is somewhat more opaque. However, it can be said that many fractal phenomena, and in particular, period-doubling maps, have an