The measure of scrambled sets of interval self-maps
f
:
I
=
[
0
,
1
]
→
I
f:I=[0,1] \to I
was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of “
∗
\ast
-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map
f
:
I
k
→
I
k
(
k
≥
1
)
f: I^{k} \to I^{k}~(k\geq 1)
of the unit
k
k
-cube
I
k
I^k
is
∗
\ast
-chaotic on
I
k
I^{k}
, then for any
ϵ
>
0
\epsilon > 0
there is a map
g
:
I
k
→
I
k
g: I^{k} \to I^{k}
such that
f
f
and
g
g
are topologically conjugate,
d
(
f
,
g
)
>
ϵ
d(f,g) > \epsilon
and
g
g
has a scrambled set which has Lebesgue measure 1, and hence if
k
≥
2
k \geq 2
, then there is a homeomorphism
f
:
I
k
→
I
k
f: I^{k} \to I^{k}
with a scrambled set
S
S
satisfying that
S
S
is an
F
σ
F_{\sigma }
-set in
I
k
I^k
and
S
S
has Lebesgue measure 1.