# Substitution Markov chains and Martin boundaries

## TL;DR

This paper investigates the Martin boundaries of substitution Markov chains, revealing differences in metric structure despite topological similarities, and offers insights into their probabilistic and topological properties.

## Contribution

It provides a novel analysis of Martin boundaries for substitution Markov chains, highlighting differences between topological and metric descriptions.

## Key findings

- Martin boundaries can be topologically similar but metrically different
- An example shows boundary homeomorphic to coin tossing process with distinct metric properties
- Enhances understanding of probabilistic and topological aspects of substitution Markov chains

## Abstract

Substitution Markov chains have been introduced [7] as a new model to describe molecular evolution. In this note, we study the associated Martin boundaries from a probabilistic and topological viewpoint. An example is given that, although having a boundary homeomorphic to the well-known coin tossing process, has a metric description that differs significantly.

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Source: https://tomesphere.com/paper/1705.10830