  Recently the bound on the Lyapunov exponent $\lambda_L \le 2\pi T/ \hbar$ in
thermal quantum systems was conjectured by Maldacena, Shenker, and Stanford. If
we naively apply this bound to a system with a fixed Lyapunov exponent
$\lambda_L$, it might predict the existence of the lower bound on temperature
$T \ge \hbar \lambda_L/ 2\pi $. Particularly, it might mean that chaotic
systems cannot be zero temperature quantum mechanically. Even classical
dynamical systems, which are deterministic, might exhibit thermal behaviors
once we turn on quantum corrections. We elaborate this possibility by
investigating semi-classical particle motions near the hyperbolic fixed point
and show that indeed quantum corrections may induce energy emission which obeys
a Boltzmann distribution. We also argue that this emission is related to
acoustic Hawking radiation in quantum fluid. Besides, we discuss when the bound
is saturated and show that a particle motion in an inverse harmonic potential
and $c=1$ matrix model may saturate the bound although they are integrable.
