On the Beta Transformation

Abstract

The beta transformation is the iterated map β x mod 1. The special case of β = 2 is known as the Bernoulli map, and is exactly solvable. The Bernoulli map provides a model for pure, unrestrained chaotic (ergodic) behavior: it is the full invariant shift on the Cantor space {0, 1} ω . The Cantor space consists of infinite strings of binary digits; it is notable for many properties, including that it can represent the real number line.The beta transformation defines a subshift: iterated on the unit interval, it singles out a subspace of the Cantor space that is invariant under the action of the left-shift operator. That is, lopping off one bit at a time gives back the same subspace.The beta transform seems to capture something basic about the multiplication of two real numbers: β and x. It offers insight into the nature of multiplication. Iterating on multiplication, one would get β n x -that is, exponentiation; the mod 1 of the beta transform contorts this in strange ways.Analyzing the beta transform is difficult. The work presented here is more-orless a research diary: a pastiche of observations and some shallow insights. One is that chaos seems to be rooted in how the carry bit behaves during multiplication. Another is that one can surgically insert "islands of stability" into chaotic (ergodic) systems, and have some fair amount of control over how those islands of stability behave. One can have islands with, or without a period-doubling "route to chaos".The eigenvalues of the transfer operator seem to lie on a circle of radius 1/β in the complex plane. Given that the transfer operator is purely real, the appearance of such a quasi-unitary spectrum unexpected. The spectrum appears to be the limit of a dense set of quasi-cyclotomic polynomials, the positive real roots of which include the Golden and silver ratios, the Pisot numbers, the n-bonacci (tribonacci, tetranacci, etc.) numbers.