Physical Measures for Infinitely Renormalizable Lorenz Maps

TL;DR

This paper investigates the existence of physical measures in infinitely renormalizable Lorenz maps, demonstrating that unlike unimodal maps, some Lorenz maps lack physical measures due to critical point behavior.

Contribution

It constructs examples of infinitely renormalizable Lorenz maps without physical measures, highlighting differences from unimodal dynamics and emphasizing the role of critical point control.

Findings

Some Lorenz maps lack physical measures.

Critical point position influences measure existence.

A priori bounds are not always present in Lorenz maps.

Abstract

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics. Namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure, is the control of the position of these critical points.