The Bernoulli Operator

This document explores the Bernoulli operator, giving it a variety of different definitions. In one definition, it is the shift operator acting on infinite strings of binary digits. In another definition, it is the transfer operator (the Frobenius-Perron operator) of the Bernoulli map, also variously known as the doubling map or the sawtooth map. The map is interesting for multiple reasons. One is that the set of infinite binary strings is the Cantor set; this implies that the Bernoulli operator has a set of fractal eigenfunctions. These are given by the Takagi (or Blancmange) curve. The set of all infinite binary strings can also be understood as the infinite binary tree. This binary tree has a large number of self-similarities, given by the dyadic monoid. The dyadic monoid has an extension to the modular group PSL(2,Z), which plays an important role in analytic number theory; and so there are many connections between the Bernoulli map and various number-theoretic functions, including the Moebius function and the Riemann zeta function. The Bernoulli map has been studied as a shift operator, in the context of functional analysis [24]. More recently, it has been studied in the physics community as an exactly solvable model for chaotic dynamics and the entropy-increaing thermodynamic arrow of time[26, 14, 6]. Some of what is presented here is review material. New content includes a full development of the continuous spectrum of this operator, including a presentation of the fractal eigenfunctions, and how these span the same space as the the Hurwitz-zeta function eigenfunctions. That is, both provide a basis for the operator, and the fractal basis is a linear combination of the smooth basis, and v.v. The connection to both classic and analytic number theory is also underscored. A simple proof that any totally multiplicative function is associated with a function obeying the multiplication theorem is provided. This document is meant to be read along with multiple companion papers which further expand the ties into number theory, and into fractal symmetries[30]. Most notable of these is that the shift operator acting on continued fractions gives the Gauss-Kuzmin-Wirising (GKW) operator[28]; the (iso-)morphism between the continued fractions and the real numbers given by the Minkowski Question Mark function[33, 29]. This, in turn, allows a novel reformulation of the Riemann Hypothesis as a bound on a rapdily-converging series representation for the Riemann zeta function[11, 31]. An effort is made to keep the presentation as simple as possible, making this document accesible to the general mathematical audience. This document is perennially in draft form, having multiple unfinished sections. It is occasionally updated with new results or clarifications. 1. THE BERNOULLI OPERATOR THIS IS A SET OF WORKING NOTES. It’s somewhat loosely structured, and sometimes messy. The presentation alternates between easy, and advanced: many parts of this paper are readable by students with modest mathematical knowledge, while other sections presume and require expert familiarity. In general, the harder topics appear later in the text. The intro hasn’t been written yet; but if it was, it would explain why this is an interesting topic. In short, having a clear understanding of transfer operators is vital to the branches of physics concerned with dynamical systems and the arrow of time. The Bernoulli operator Date: 2 January 2004 (revised 2008, 2010, 30 November 2014).