  The quantum walk is a counterpart of the random walk. The 2-state quantum
walk in one dimension can be determined by a measure on the unit circle in the
complex plane. As for the singular continuous measure, results on the
corresponding quantum walk are limited. In this situation, we focus on a
quantum walk, called the Riesz walk, given by the Riesz measure which is one of
the famous singular continuous measures. The present paper is devoted to the
return probability of the Riesz walk. Furthermore, we present some conjectures
on the self-similarity of the walk.
