Scaling by 5 on a 1/4-Cantor Measure

TL;DR

This paper investigates the ergodic properties of a unitary operator connecting two orthonormal bases of exponential functions in a specific Cantor measure, revealing that only constant functions are invariant under this operator.

Contribution

It establishes the ergodicity of the operator U linking two ONBs for a 1/4-Cantor measure, a novel insight into the spectral structure of such measures.

Findings

Operator U is ergodic, fixing only constant functions.

The specific ONBs are connected via a unitary operator U.

The result applies to the case where one ONB is 5 times the other.

Abstract

Each Cantor measure (\mu) with scaling factor 1/(2n) has at least one associated orthonormal basis of exponential functions (ONB) for L^2(\mu). In the particular case where the scaling constant for the Cantor measure is 1/4 and two specific ONBs are selected for L^2(\mu), there is a unitary operator U defined by mapping one ONB to the other. This paper focuses on the case in which one ONB (\Gamma) is the original Jorgensen-Pedersen ONB for the Cantor measure (\mu) and the other ONB is is 5\Gamma. The main theorem of the paper states that the corresponding operator U is ergodic in the sense that only the constant functions are fixed by U.