# The asymptotics of the partition of the cube into Weyl simplices, and an encoding of a Bernoulli scheme

## TL;DR

This paper introduces a combinatorial encoding method for continuous symbolic dynamical systems, demonstrating that partitioning the infinite-dimensional cube into Weyl simplices effectively distinguishes almost all points, advancing understanding of symbolic dynamics.

## Contribution

It presents a novel combinatorial encoding approach that transforms the infinite-dimensional cube into a path space, showing that Weyl simplices partition distinguishes almost all points.

## Key findings

- Partition into Weyl simplices is almost surely distinguishable.
- The encoding maps the shift to a transfer transformation.
- The method applies to Bernoulli schemes and graded graphs.

## Abstract

We suggest a combinatorial method of encoding continuous symbolic dynamical systems. A~continuous phase space, the infinite-dimensional cube, turns into the path space of a tree, and the shift is mapped to a transformation which was called a "transfer." The central problem is that of distinguishability: does the encoding separate almost all points of the space? The main result says that the partition of the cube into Weyl simplices satisfies this property.\footnote{{\it Keywords:} combinatorial encoding, transfer, Bernoulli scheme, graded graph.