A Note on D_lambda-n-Scrambled Sets

The \mathcalD_\lambda-scrambled sets are to be analysed in the aspect of whole space. By means of Furstenberg families, the definition of distributionally scrambled set is extended to define a \mathcalD_\lambda-n-scrambled set. Then the results on distributionally scrambled sets with the whole space obtained are generalized to the case of \mathcalD_\lambda-n-scrambled sets. For each real \lambda\in0,1 and each integer n\geqslant2, the main conclusions are as follows: (1) there is no compact dynamical system with the whole space being a \mathcalD_\lambda-n-scrambled set; (2) based on the example provided by WANG et al, an invertible noncompact dynamical system consisting of countable many points are constructed, whose \mathcalD_\lambda-n-scrambled set can be the whole space.