Lyapunov Exponent and Stochastic Stability for Infinitely Renormalizable Lorenz Maps

The paper proves stochastic stability for infinitely renormalizable contracting Lorenz maps with bounded geometry by (i) establishing slow recurrence of the two critical values to the singularity, (ii) adapting Tsujii’s entropy-formula result to the Lorenz setting, and (iii) excluding absolutely continuous invariant measures to identify any zero-noise limit with the unique physical measure on the Cantor attractor. The candidate solution, however, hinges on building a uniformly expanding, full-branched Gibbs–Markov inducing scheme with exponential return-time tails and then projecting a Young tower—an approach that would force positive Lyapunov exponents and typically produce an absolutely continuous invariant measure, contradicting the paper’s Theorem 2 (χμ(f)=0) and the Cantor-attractor structure in this class. It also misapplies the Qian–Zhu SRB characterization in one dimension, which would imply absolute continuity of the limit measure, again contradicting the setting.