Absolutely Continuous Invariant Measures of Piecewise Linear Lorenz Maps

TL;DR

This paper characterizes when piecewise linear Lorenz maps have an absolutely continuous invariant measure, showing it exists under a specific condition, and explores its uniqueness, ergodicity, and relation to Lebesgue measure through renormalization.

Contribution

It provides a complete criterion for the existence of an acim in piecewise linear Lorenz maps and analyzes its properties and relation to Lebesgue measure.

Findings

Existence of acim characterized by $ac+(1-c)b \,\ge\, 1$

Unique and ergodic acim unless conjugate to a rational rotation

Full investigation of acim and Lebesgue measure relation via renormalization

Abstract

Consider piecewise linear Lorenz maps on $[0, 1]$ of the following form \[ f_{a,b,c}(x)= {ll} ax+1-ac & x \in [0, c) b(x-c) & x \in (c, 1].\] We prove that $f_{a,b,c}$ admits an absolutely continuous invariant probability measure (acim) $\mu$ with respect to the Lebesgue measure if and only if $f_{a,b,c}(0) \le f_{a,b,c}(1)$, i.e. $ac+(1-c)b \ge 1$. The acim is unique and ergodic unless $f_{a,b,c}$ is conjugate to a rational rotation. The equivalence between the acim and the Lebesgue measure is also fully investigated via the renormalization theory.