In a topological dynamical system (X, T), a Borel probability measure is a distribution measure for a sequence \(\underline{x}=(x_i)_{i=0}^{\infty }\in X^{\infty }\) if it is a limit of some subsequence of \(\{\frac{1}{n}\sum _{i=0}^{n-1}\hat{\delta }(x_i)\}_{n=1}^{\infty }\) , where \(\hat{\delta }(x_i)\) is the Dirac measure at \(x_i\) . It is proved that all distribution measures of \(\underline{x}\) are T-invariant if and only if \(\underline{x}\) is an asymptotic quasi-orbit. Equivalent characterizations of proximal (distal) quasi-orbit are also given. In the pseudo-metric space \((X^{\infty },\bar{E})\) , topological properties of the set of all asymptotic (proximal, distal) quasi-orbits of T are discussed. Similar to generic point, we introduce equidistributed quasi-orbit for which the set of distribution measures is a singleton, and obtain several equivalent statements for uniquely ergodic systems. Finally, we extend Weyl’s Theorem on equidistribution modulo 1.
