Chains of infinite order, chains with memory of variable length, and maps of the interval

TL;DR

This paper presents a method to construct topological Markov maps of the interval that have invariant measures matching the stationary laws of stochastic chains of infinite order, including those with variable-length memory.

Contribution

It characterizes the maps corresponding to stochastic chains with variable-length memory and reverses the classical Gibbs formalism construction for Markov expanding maps.

Findings

Constructed topological Markov maps for chains of infinite order

Characterized maps for chains with variable-length memory

Reversed classical Gibbs formalism construction

Abstract

We show how to construct a topological Markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order. In particular we caracterize the maps corresponding to stochastic chains with memory of variable length. The problem treated here is the converse of the classical construction of the Gibbs formalism for Markov expanding maps of the interval.