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
)
,
σ
f
)
of a dynamical system
(
X
,
f
)
is
ℱ
-transitive (resp.,
ℱ
-mixing,
(
ℱ
1
,
ℱ
2
)
-everywhere chaotic) if and only if the system
(
∩
n
=
0
∞
f
n
(
X
)
,
f
|
∩
n
=
0
∞
f
n
(
X
)
)
is
ℱ
-transitive (resp.,
ℱ
-mixing,
(
ℱ
1
,
ℱ
2
)
-everywhere chaotic), where
ℱ
,
ℱ
1
and
ℱ
2
are Furstenberg families.
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