Lyapunov Exponent and Stochastic Stability for Infinitely Renormalizable Lorenz Maps

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.