Renormalization for Lorenz maps of monotone combinatorial types

TL;DR

This paper develops a renormalization framework for Lorenz maps with specific monotone combinatorics, establishing bounds and proving the existence of periodic points and Cantor attractors in their dynamics.

Contribution

It introduces real a priori bounds for renormalizable Lorenz maps and demonstrates the existence of periodic points and Cantor attractors for infinitely renormalizable cases.

Findings

Existence of periodic points of renormalization.

Presence of Cantor attractors in dynamics.

Construction of real a priori bounds for certain Lorenz maps.

Abstract

Lorenz maps are maps of the unit interval with one critical point of order rho>1, and a discontinuity at that point. They appear as return maps of leafs of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.