Renormalization in Lorenz maps -- completely invariant sets and periodic orbits

TL;DR

This paper investigates the dynamics of Lorenz maps, focusing on the relationship between renormalizations, invariant sets, and periodic points, providing new algorithms and insights into their structural properties.

Contribution

It introduces a novel analysis connecting renormalizations with invariant sets in Lorenz maps and offers an algorithm to identify recoverable renormalizations, filling gaps in existing literature.

Findings

Some renormalizations are linked with invariant sets, others are not.

An algorithm to detect recoverable renormalizations is provided.

Enhanced understanding of the structure of Lorenz map dynamics.

Abstract

The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincar\`e maps in geometric models of well-known Lorenz attractor. Using both analytical and symbolic approaches, we study connections between periodic points, completely invariant sets and renormalizations. We show that some renormalizations may be connected with completely invariant sets while some others don't. We provide an algorithm to detect the renormalizations that can be recovered from completely invariant sets. Furthermore, we discuss the importance of distinguish one-side and double-side preimage. This way we provide a better insight into the structure of renormalizations in Lorenz maps. These relations remained unnoticed in the literature, therefore we are correcting some gaps existing in the literature, improving and completing to some extent the description of possible dynamics in this important field of study.