  We study $2d$ conformal field theory (CFT) at large central charge $c$ and
finite temperature $T$ with heavy operators inserted at spatial infinity. The
heavy operators produce a nearly thermalized steady state at an effective
temperature $T_{\rm eff}\leq T$. Under some assumptions, we find an effective
Schwarzian-like description of these states and, when they exist, their gravity
duals. We use this description to compute the Lyapunov exponents for light
operators to be $2\pi T_{\rm eff}$, so that scrambling is suppressed by the
heavy insertions.
