On the Beta Transformation

Abstract

The beta transformation is the iterated map $\beta x\,\mod1$; it generates the base-$\beta$ expansion of a real number x. Every iterated piece-wise monotonic map is topologically conjugate to the beta transformation. For all but a countable subset of $\beta$, the orbits of $x$ are ergodic; yet it is the finite orbits that determine overall behavior. This is a large text; it splits into four parts. The first part provides a review of general concepts and properties associated with the beta shift. The second part examines the spectrum of the Ruelle-Frobenius-Perron operator, and gives explicit expressions for a set of bounded eigenfunctions. These form a discrete spectrum, accumulating on a circle of radius $1/\beta$ in the complex plane. The third part examines the finite and the periodic orbits. These are in one-to-one correspondence with monic integer polynomials. They are "quasi-cyclotomic" and can be counted with Moreau's necklace-counting function; curiously, they do not have any obvious relation to other systems countable by the necklace function. The positive real roots are dense in the reals; they include the Golden and silver ratios, the Pisot numbers, the n-bonacci (tribonacci, tetranacci, etc.) numbers. The beta-polynomials yoke all of these together into a regular structure. An explicit bijection to the rationals is presented. The fourth part of this text examines small perturbations. These introduce Arnold tongues, which inflate the finite orbits, a set of measure zero, to finite size. This text assumes very little mathematical sophistication on the part of the reader, and should be approachable for any enthusiast with minimal or no prior experience in ergodic theory. Most of the development is casual. As a side effect, the introductory sections are perhaps a fair bit longer than strictly needed to present the new results.