In the study of modern dynamical systems, an invariant measure is an important topic; and in the study of modern complex analysis, the quasisymmetric condition on a map is an important topic. In this talk, I will combine these two topics to introduce a new interesting topic, symmetric invariant measure. I will discuss the existence and the uniqueness of a symmetric invariant measure. I will also discuss some ergodicity problems for symmetric measures; and a related topic, smooth rigidity and symmetric rigidity.