A Combinatorial Characterization of the Critical Itineraries of a Uniform Dynamical System

TL;DR

This paper provides a combinatorial characterization of critical itineraries in a specific class of uniform dynamical systems, enhancing understanding of their role in $eta$-expansions and fractal transformations.

Contribution

It introduces a novel combinatorial framework to characterize critical itineraries of piecewise constant slope functions with a discontinuity.

Findings

Characterization of critical itineraries using combinatorial methods

Insights into the structure of $eta$-expansions

Applications to fractal transformations

Abstract

For a function from the unit interval to itself with constant slope and one discontinuity, the itineraries of the point of discontinuity are called the critical itineraries. These critical itineraries play a significant role in the study of $\beta$-expansions (with positive or negative $\beta$) and fractal transformations. A combinatorial characterization of the critical itineraries of such functions is provided.