# Conjugacy problem of strictly monotone maps with only one jump discontinuity

## TL;DR

This paper studies the conjugacy problem for strictly monotone maps with a single jump discontinuity, providing conditions for conjugacy, methods for construction, and criteria for smoothness.

## Contribution

It offers new necessary and sufficient conditions for conjugacy and methods to construct all conjugacies for these maps, including smoothness criteria.

## Key findings

- Established conditions for conjugacy of maps with one jump discontinuity.
- Developed methods to construct all conjugacies explicitly.
- Provided criteria for the $C^1$ smoothness of conjugacies.

## Abstract

The conjugacy problem is one of the central questions in iteration theory. As far as we, for discontinuous strictly monotone maps there is no complete result. In this paper, we investigate the conjugacy problem of strictly monotone maps with only one jump discontinuity. We give some sufficient and necessary conditions for the conjugacy relationship. And we present some methods to construct all conjugacies. Furthermore, we present the conditions to guarantee $C^1$ smoothness of these conjugacies.