Scaling by $5$ on a $\frac{1}{4}$--Cantor measure

Abstract

Each Cantor measure $\mu$ with scaling factor $\frac{1}{2n}$ 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 $\frac14$ and two specific ONBs are selected for $L^2(\mu_{1/4})$, 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 $\mu_{{1}/{4}}$ and the other ONB 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 $U$.