A renormalization fixed point for Lorenz maps

TL;DR

This paper establishes the existence of a hyperbolic fixed point in the renormalization operator for Lorenz maps with a critical point, using computer-assisted proofs to ensure rigorous estimates.

Contribution

It introduces a renormalization fixed point for Lorenz maps and provides a computer-assisted proof of its hyperbolicity, advancing the understanding of Lorenz flow dynamics.

Findings

Existence of a hyperbolic fixed point in the renormalization operator

Computer-assisted rigorous estimates for the fixed point

Detailed methodology for computer-based proofs in dynamical systems

Abstract

A Lorenz map is a Poincar\'e map for a three-dimensional Lorenz flow. We describe the theory of renormalization for Lorenz maps with a critical point and prove that a restriction of the renormalization operator acting on such maps has a hyperbolic fixed point. The proof is computer assisted and we include a detailed exposition on how to make rigorous estimates using a computer as well as the implementation of the estimates.