Ergodic properties of {\beta}-adic Halton sequences

TL;DR

This paper studies the ergodic properties of a generalized Halton sequence with Pisot number bases, demonstrating unique ergodicity of the associated Kakutani-Fibonacci transformation using ergodic theory methods.

Contribution

It extends the analysis of Halton sequences to Pisot number bases and proves the unique ergodicity of the Kakutani-Fibonacci transformation.

Findings

Sequences are shown to be uniformly distributed.

Kakutani-Fibonacci transformation is uniquely ergodic.

Provides new ergodic theory insights for multidimensional sequences.

Abstract

We investigate a parametric extension of the classical s-dimensional Halton sequence, where the bases are special Pisot numbers. In a one- dimensional setting the properties of such sequences have already been in- vestigated by several authors [5, 8, 23, 28]. We use methods from ergodic theory to in order to investigate the distribution behavior of multidimen- sional versions of such sequences. As a consequence it is shown that the Kakutani-Fibonacci transformation is uniquely ergodic.