Lyapunov Exponent and Stochastic Stability for Infinitely Renormalizable Lorenz Maps

TL;DR

This paper proves that certain Lorenz maps with specific geometric bounds have zero Lyapunov exponents at critical points and are stochastically stable, advancing understanding of their dynamical behavior.

Contribution

It establishes the slow recurrence condition for infinitely renormalizable Lorenz maps and demonstrates their stochastic stability and zero Lyapunov exponents at critical points.

Findings

Lyapunov exponents at critical points are zero.

Maps satisfy slow recurrence condition.

Maps are stochastically stable.

Abstract

We prove that infinitely renormalizable contracting Lorenz maps with bounded geometry or the so-called {\it a priori bounds} satisfies the slow recurrence condition to the singular point $c$ at its two critical values $c_1^-$ and $c_1^+$. As the first application, we show that the pointwise Lyapunov exponent at $c_1^-$ and $c_1^+$ equals 0. As the second application, we show that such maps are stochastically stable.