Renormalization and conjugacy of piecewise linear Lorenz maps

TL;DR

This paper proves that expanding piecewise linear Lorenz maps are either periodically renormalizable or prime, and are conjugate to β-transformations, clarifying their dynamical classification.

Contribution

It establishes a dichotomy for expanding piecewise linear Lorenz maps, showing they are either renormalizable or prime, and are conjugate to β-transformations, advancing understanding of their structure.

Findings

Expanding piecewise linear Lorenz maps are either periodic renormalizable or prime.

Such maps are conjugate to β-transformations.

The dichotomy clarifies the classification of these maps.

Abstract

For each piecewise linear Lorenz map that expand on average, we show that it admits a dichotomy: it is either periodic renormalizable or prime. As a result, such a map is conjugate to a $\beta$-transformation.