The Pascal automorphism has a purely continuous spectrum

TL;DR

This paper thoroughly analyzes the Pascal automorphism, demonstrating its purely continuous spectrum, and introduces a new class of adic transformations with potential applications in combinatorics and dynamical systems.

Contribution

It provides a detailed description of the Pascal automorphism, proves the continuity of its spectrum, and introduces a new class of adic transformations based on classical graded graphs.

Findings

Pascal automorphism has a purely continuous spectrum.

Introduces a new class of adic transformations from classical graded graphs.

Studies properties of the Pascal measure and related transformations.

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.