Firstly, we introduce the definitions of
G
-asymptotic tracking property,
G
-asymptotic average tracking property, and
G
-quasi-weak almost-periodic point. Secondly, we study their dynamical properties and characteristics. The results obtained improve the conclusions of asymptotic tracking property, asymptotic average tracking property, and quasi-weak almost-periodic point in the inverse limit space and provide the theoretical basis and scientific foundation for the application of tracking property in computational mathematics, biological mathematics, and computer science.