The Pascal automorphism has a purely continuous spectrum

Abstract

We give the detale description from various points of view of Pascal automorphism,--- a natural transformation of the space of paths in the Pascal graph (= infinite Pascal triangle), and describetha plan of the proof of continuiuty of its spectrum. If we realize this automorphism as the shift in the space of 0-1 sequences, we obtain a stationary measure, called the Pascal measure, whose properties we study. The transformations generated by classical graded graphs, such as the ordinary and multidimensional Pascal graphs, the Young graph, the graph of walks in Weyl chambers, etc., provide examples of combinatorial nature from a new and very interesting class of adic transformations introduced as early as in \cite{V81}; some considerations by V. I. Arnold also lead to such transformations. We discuss problems arising in this field. This is the first paper of the series of articles about adic transformations.