On the invariance of maximal distributional chaos under an annihilation operator

It is known that certain physical systems, which do not generate deterministic chaos under conventional frameworks, may generate such complex behavior in a quantum mechanical setting. In this paper, it is proved that the annihilation operator of an unforced quantum harmonic oscillator admits an invariant distributionally ϵ -scrambled set for any 0 < ϵ < 2 , showing that this operator can exhibit maximal distributional chaos on an uncountable invariant subset.