ℱℱ-mixing property and (ℱ1,ℱ2)(ℱ1,ℱ2)-everywhere chaos of inverse limit dynamical systems

This paper investigates some chaotic properties via Furstenberg families generated by inverse limit dynamical systems. It is proved that the inverse limit dynamical system (lim⟵(X,f),σf) of a dynamical system (X,f)(X,f) is ℱℱ-transitive (resp., ℱℱ-mixing, (ℱ1,ℱ2)(ℱ1,ℱ2)-everywhere chaotic) if and only if the system (∩n=0∞fn(X),f|∩n=0∞fn(X)) is ℱℱ-transitive (resp., ℱℱ-mixing, (ℱ1,ℱ2)(ℱ1,ℱ2)-everywhere chaotic), where ℱℱ, ℱ1ℱ1 and ℱ2ℱ2 are Furstenberg families.