  If an invertible linear dynamical systems is Li-York chaotic or other
chaotic, what's about it's inverse dynamics? what's about it's adjoint
dynamics? With this unresolved but basic problems, this paper will give a
criterion for Lebesgue operator on separable Hilbert space. Also we give a
criterion for the adjoint multiplier of Cowen-Douglas functions on $2$-th Hardy
space. Last we give some chaos about scalars perturbation of operator and some
examples of invertible bounded linear operator such that $T$ is chaotic but
$T^{-1}$ is not.
