# Generic point equivalence and Pisot numbers

## TL;DR

This paper investigates conditions under which the dynamics of a beta transformation and a related piecewise linear transformation share the same generic points, focusing on Pisot numbers and their properties.

## Contribution

It establishes sufficient conditions for the equivalence of generic points between beta transformations and certain piecewise linear maps involving Pisot numbers.

## Key findings

- Identifies conditions for point equivalence in beta and linear maps
- Extends understanding of dynamics involving Pisot numbers
- Provides criteria for generic point sharing

## Abstract

Let $\beta >1$ be an integer or generally a Pisot number. Put $T(x) = \{ \beta x \}$ on $[0,1]$ and let $S: [0,1]\to [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm \beta^m$ with positive integers $m$. We give sufficient conditions that $T$ and $S$ have the same generic points.