An adic dynamical system related to the Delannoy numbers

TL;DR

This paper introduces a new adic dynamical system linked to Delannoy numbers, analyzes its invariant measures, and explores its ergodic and algebraic properties.

Contribution

It presents the first construction of an adic system based on Delannoy numbers and studies its ergodic measures and dynamical characteristics.

Findings

Identified all ergodic invariant measures.

Proved total ergodicity for each measure.

Initiated analysis of the system's dimension group.

Abstract

We introduce an adic (Bratteli-Vershik) dynamical system based on a diagram whose path counts from the root are the Delannoy numbers. We identify the ergodic invariant measures, prove total ergodicity for each of them, and initiate the study of the dimension group and other dynamical properties.