Affiliation:
1. Faculty of Applied Sciences, Politehnica University of Bucharest, Splaiul Independenţei 313 , Bucharest , Romania
2. Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Iuliu Maniu Street, nr. 50, 500091 , Braşov , Romania
3. Faculty of Mathematics and Computer Science, Bucharest University, Romania, Str. Academiei 14, 010014 , Bucharest , Romania
Abstract
Abstract
In this article, we present a structure result concerning fuzzy fractals generated by an orbital fuzzy iterated function system
(
(
X
,
d
)
,
(
f
i
)
i
∈
I
,
(
ρ
i
)
i
∈
I
)
\left(\left(X,d),{({f}_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I})
. Our result involves the following two main ingredients: (a) the fuzzy fractal associated with the canonical iterated fuzzy function system
(
(
I
N
,
d
Λ
)
,
(
τ
i
)
i
∈
I
,
(
ρ
i
)
i
∈
I
)
\left(\left({I}^{{\mathbb{N}}},{d}_{\Lambda }),{\left({\tau }_{i})}_{i\in I},{\left({\rho }_{i})}_{i\in I})
, where
d
Λ
{d}_{\Lambda }
is Baire’s metric on the code space
I
N
{I}^{{\mathbb{N}}}
and
τ
i
:
I
N
→
I
N
{\tau }_{i}:{I}^{{\mathbb{N}}}\to {I}^{{\mathbb{N}}}
is given by
τ
i
(
(
ω
1
,
ω
2
,
…
)
)
≔
(
i
,
ω
1
,
ω
2
,
…
)
{\tau }_{i}\left(\left({\omega }_{1},{\omega }_{2},\ldots )):= \left(i,{\omega }_{1},{\omega }_{2},\ldots )
for every
(
ω
1
,
ω
2
,
…
)
∈
I
N
\left({\omega }_{1},{\omega }_{2},\ldots )\in {I}^{{\mathbb{N}}}
and every
i
∈
I
i\in I
; (b) the canonical projections of certain iterated function systems associated with the fuzzy fractal under consideration.