On the continuity of entropy of Lorenz maps

TL;DR

This paper investigates how the topological entropy of Lorenz maps, parameterized by their discontinuity point, varies continuously with the parameter, and explores related conjectures on entropy monotonicity.

Contribution

It proves the continuity of topological entropy for a family of Lorenz maps constructed from bilipschitz functions and discusses Milnor's monotonicity conjecture in this context.

Findings

Topological entropy varies continuously with the discontinuity point p.

The paper confirms the continuity property for the specific family of Lorenz maps.

Discussion on the validity of Milnor's monotonicity conjecture for these maps.

Abstract

We consider a one parameter family of Lorenz maps indexed by their point of discontinuity $p$ and constructed from a pair of bilipschitz functions. We prove that their topological entropies vary continuously as a function of $p$ and discuss Milnor's monotonicity conjecture in this setting.