Fuzzy Iterated Function Systems

• Fuzzy IFS is an extension of classical iterated function systems that uses fuzzy sets and grey-level maps to model fractal dynamics.
• It replaces traditional set unions with fuzzy Hutchinson operators based on Zadeh’s extension principle, ensuring convergence to unique fuzzy attractors under contractive conditions.
• The framework broadens to include topological, orbital, generalized, and measure-theoretic extensions, supporting robust computational algorithms and interpolation methods.

A fuzzy iterated function system (fuzzy IFS) is an extension of Hutchinson–Barnsley iterated function system theory in which the state space is no longer a hyperspace of compact sets alone, but a space of fuzzy sets $u:X\to[0,1]$, and the set-valued union-of-images operator is replaced by a fuzzy Hutchinson operator built from Zadeh’s extension principle, admissible grey-level maps, and pointwise supremum. In the standard metric formulation, the underlying spatial maps remain continuous self-maps of a complete metric space, while the fuzzy dynamics acts on normal, upper semicontinuous, compactly supported fuzzy sets; in later developments, this framework was generalized to generalized IFSs, orbital systems, topological and multimetric settings, and measure-theoretic counterparts (Banakh et al., 28 Sep 2025).

1. Formal framework and terminology

The classical backbone of the subject is the ordinary IFS $\{\mathcal X; w_1,\dots,w_m\}$, where $(\mathcal X,\rho)$ is a complete metric space and each $w_k:\mathcal X\to\mathcal X$ is a strict contraction; the associated set operator on compact sets is

$w(X)=\bigcup_{k=1}^{m} w_k(X),$

with unique attractor $X^\star$ satisfying

$w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$

This fixed-point viewpoint is the crisp precursor of fuzzy IFS theory (Ghosh et al., 2022).

In the fuzzy setting, a fuzzy set is a function $u:X\to[0,1]$. The standard state spaces are built from normality, upper semicontinuity, and compact support. In the metric literature one frequently writes

$$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$

while in the topological formulation the notation

$$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$

is used for compact fuzzy sets. These notational differences matter because the later theory depends heavily on the exact hyperspace and topology under consideration (Miculescu et al., 2022).

A central auxiliary notion is that of an admissible system of grey-level maps. In the standard form, a family $\{\mathcal X; w_1,\dots,w_m\}$0 or $\{\mathcal X; w_1,\dots,w_m\}$1 of maps $\{\mathcal X; w_1,\dots,w_m\}$2 is admissible when each map is nondecreasing and right continuous, each satisfies $\{\mathcal X; w_1,\dots,w_m\}$3, and at least one satisfies $\{\mathcal X; w_1,\dots,w_m\}$4. These conditions guarantee preservation of normality and compatibility with cutwise analysis (Banakh et al., 28 Sep 2025).

The historical motivation most often cited in the literature goes back to the fuzzyfication of IFS theory associated with Cabrelli–Forte–Molter–Vrscay, while later work systematically merged that line with generalized IFS, orbital IFS, and topological IFS theory. This suggests that “fuzzy IFS” is not a single rigid formalism, but a family of closely related operator-theoretic constructions sharing the same basic architecture (Oliveira et al., 2016).

2. Fuzzy Hutchinson dynamics and classical attractor theory

The standard fuzzy Hutchinson–Barnsley operator has the form

$\{\mathcal X; w_1,\dots,w_m\}$5

or, in the topological notation,

$\{\mathcal X; w_1,\dots,w_m\}$6

Here $\{\mathcal X; w_1,\dots,w_m\}$7 is the fuzzy image of $\{\mathcal X; w_1,\dots,w_m\}$8 under Zadeh’s extension principle,

$\{\mathcal X; w_1,\dots,w_m\}$9

and $(\mathcal X,\rho)$0 is pointwise supremum (Mihail et al., 2021).

The natural metric in the metric theory is the levelwise Hausdorff metric

$(\mathcal X,\rho)$1

where $(\mathcal X,\rho)$2 and $(\mathcal X,\rho)$3 is the Hausdorff–Pompeiu metric on compact subsets. If $(\mathcal X,\rho)$4 is complete, then $(\mathcal X,\rho)$5 is complete. This is the exact analogue of the hyperspace metric used in ordinary IFS theory and underlies the standard contraction proof for fuzzy attractors (Miculescu et al., 2022).

In the ordinary fuzzy IFS setting, when the underlying maps $(\mathcal X,\rho)$6 are contractions, the fuzzy operator is a Banach contraction on $(\mathcal X,\rho)$7, hence Picard. Consequently there exists a unique fixed point, usually called the invariant fuzzy set or fuzzy fractal, and every iteration sequence converges to it. This is the direct fuzzy analogue of the classical attractor theorem (Miculescu et al., 2022).

Cutwise formulas are structurally decisive. In the topological theory one has

$(\mathcal X,\rho)$8

These identities reduce fuzzy dynamics to ordinary hyperspace dynamics on level cuts and are the main reason why Hausdorff-type techniques remain effective after fuzzification (Banakh et al., 28 Sep 2025).

A common misunderstanding is to identify fuzzy IFS with probabilistic IFS. The classical survey literature treats random map selection, Markov kernels, and invariant measures in depth, but does not explicitly define fuzzy IFS in the fuzzy-set-theoretic sense, and does not develop membership functions, $(\mathcal X,\rho)$9-cuts, fuzzy attractors, or fuzzy Hutchinson operators. The probabilistic and fuzzy frameworks are structurally adjacent, but not identical (Ghosh et al., 2022).

3. Generalized, topological, and orbital extensions

A major extension replaces self-maps $w_k:\mathcal X\to\mathcal X$0 by generalized maps $w_k:\mathcal X\to\mathcal X$1. In a generalized iterated fuzzy function system (GIFZS), the operator acts on $w_k:\mathcal X\to\mathcal X$2-tuples of fuzzy sets by first forming the product fuzzy set

$w_k:\mathcal X\to\mathcal X$3

and then applying

$w_k:\mathcal X\to\mathcal X$4

Under generalized Matkowski contractivity of the underlying $w_k:\mathcal X\to\mathcal X$5, the generalized fuzzy operator is itself a generalized Matkowski contraction and has a unique fuzzy attractor. The resulting hierarchy is strict in the sense that there exist complete metric spaces with

$w_k:\mathcal X\to\mathcal X$6

and, more generally,

$w_k:\mathcal X\to\mathcal X$7

This establishes that generalized fuzzy attractors form a strictly richer class than ordinary fuzzy IFS attractors (Oliveira et al., 2016).

A second extension removes the global metric-contraction requirement. In the topological approach, a fuzzy IFS is defined on a Hausdorff topological space $w_k:\mathcal X\to\mathcal X$8, and strong attractor theory is recovered when the underlying ordinary IFS is compactly dominating and Edelstein contracting with respect to some admissible multimetric. In that case the fuzzy Hutchinson operator has a unique fuzzy attractor, and every iterate converges to it in the canonical topology $w_k:\mathcal X\to\mathcal X$9, or equivalently the $w(X)=\bigcup_{k=1}^{m} w_k(X),$0-topology on Tychonoff spaces. On general Hausdorff spaces one still obtains a weak fuzzy attractor, but convergence is only guaranteed in the hypograph topology $w(X)=\bigcup_{k=1}^{m} w_k(X),$1. The paper also proves that pointwise and uniform topologies are not suitable in general for fuzzy attractor convergence (Banakh et al., 28 Sep 2025).

A third extension is orbital fuzzy IFS theory. Here the spatial maps are not assumed globally contractive; instead they satisfy

$w(X)=\bigcup_{k=1}^{m} w_k(X),$2

only for points $w(X)=\bigcup_{k=1}^{m} w_k(X),$3 lying in the same orbit $w(X)=\bigcup_{k=1}^{m} w_k(X),$4, with $w(X)=\bigcup_{k=1}^{m} w_k(X),$5. The associated fuzzy operator

$w(X)=\bigcup_{k=1}^{m} w_k(X),$6

is no longer generally Picard on the full fuzzy hyperspace, but it is weakly Picard on a special orbit-adapted class $w(X)=\bigcup_{k=1}^{m} w_k(X),$7. Thus every iterative orbit converges to a fixed point, but the limit may depend on the initial fuzzy set (Mihail et al., 2021).

This nonuniqueness is clarified by a structural characterization theorem. If $w(X)=\bigcup_{k=1}^{m} w_k(X),$8, then

$w(X)=\bigcup_{k=1}^{m} w_k(X),$9

where each $X^\star$0 is a local fuzzy fractal generated from an orbit-supported component. In particular, the limit is already determined by the core $X^\star$1, not by the full membership profile. This is one of the sharpest distinctions between globally contractive fuzzy IFS theory and orbitally contractive fuzzy IFS theory (Miculescu et al., 2022).

4. Measure-theoretic and hypograph formulations

An important measure-theoretic bridge identifies a substantial class of fuzzy IFS with idempotent-measure dynamics. For a compact metric space $X^\star$2, every idempotent measure $X^\star$3 has a unique upper semicontinuous density $X^\star$4, and an increasing homeomorphism $X^\star$5 yields a bijection

$X^\star$6

Under this correspondence, the support is preserved, and the idempotent Markov operator of a max-plus normalized IFS is conjugate to the fuzzy Hutchinson operator of the associated fuzzy IFS: $X^\star$7 This implies that invariant idempotent measures are exactly fuzzy attractors of a corresponding fuzzy IFS, with support equal to the classical attractor of the underlying IFS (Cunha et al., 2021).

The same paper induces on $X^\star$8 a Hausdorff-type metric

$X^\star$9

and proves that the induced topology is finer than the canonical pointwise topology. A further consequence is that contraction, convergence, and computational algorithms for invariant idempotent measures can be transferred from fuzzy IFS theory. In this formulation, grey-level maps arise from max-plus weights according to

$w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$0

This shows that the “grey-level” part of fuzzy IFS admits an exact nonadditive-measure reinterpretation for this special class of systems (Cunha et al., 2021).

A broader generalization replaces idempotent measures by $w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$1-measures associated with a continuous triangular norm $w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$2. These objects are functorially isomorphic to idempotent measures and admit a hypograph representation by saturated subsets of $w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$3. The paper explicitly states that these saturated sets are exactly the hypographs of normal upper semicontinuous fuzzy sets, and that in the language of fuzzy sets the scalar maps $w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$4 are grey-level maps. For generalized IFS on $w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$5-symmetric powers, the invariant $w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$6-measure equation

$w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$7

therefore defines a measure-theoretic companion of fuzzy GIFS theory rather than an unrelated formalism (Mazurenko et al., 1 Apr 2026).

These correspondences also explain why hypographs play a central role in the topological theory. For a compact fuzzy set $w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$8, one defines

$w(X^\star)=\bigcup_{k=1}^{m} w_k(X^\star).$9

Embedding $u:X\to[0,1]$0 into the Vietoris hyperspace yields the hypograph topology $u:X\to[0,1]$1, which is the appropriate fallback when the stronger multimetric topology is unavailable (Banakh et al., 28 Sep 2025).

5. Algorithms, interpolation constructions, and applications

The computational side of fuzzy IFS theory has been developed most explicitly through deterministic multiresolution schemes. For a Banach-contracting IFZS

$u:X\to[0,1]$2

one discretizes $u:X\to[0,1]$3 by a proper $u:X\to[0,1]$4-net $u:X\to[0,1]$5, uses an $u:X\to[0,1]$6-projection $u:X\to[0,1]$7, defines discretized maps $u:X\to[0,1]$8, and iterates the discrete fuzzy operator $u:X\to[0,1]$9 on fuzzy sets over $$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$0. The central approximation theorem states

$$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$1

where $$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$2. This yields a rigorous deterministic image-generation algorithm for fuzzy attractors, explicitly presented as FuzzyIFSDraw(\mathcal S) (Cunha et al., 2019).

The same paper emphasizes that this framework recovers the classical fuzzy IFS of Cabrelli–Forte–Molter–Vrscay as the case $$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$3, and extends seamlessly to generalized fuzzy attractors by diagonalizing the higher-order operator,

$$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$4

Its examples include one-dimensional and planar Cabrelli-type fuzzy systems, as well as fuzzy Barnsley fern and fuzzy maple leaf constructions in which nontrivial grey-level maps visibly modulate intensity across branches (Cunha et al., 2019).

A conceptually different but related line concerns fuzzy-number-valued interpolation. In the construction of fuzzy-valued fractal interpolation functions, the IFS is classical on a product space $$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$5, with maps

$$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$6

and the attractor is the graph of a fuzzy-number-valued function $$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$7. The paper proves existence, uniqueness, and Hölder continuity, and shows that every $$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$8-cut of the interpolant is determined by two ordinary fractal interpolation functions for the lower and upper endpoints of the cut (Yun et al., 17 Jul 2025).

The recurrent analogue on $$F_X^*=\{u\in F_X\mid u \text{ is normal and compactly supported}\},\qquad F_X^{**}=\{u\in F_X^*\mid u \text{ is upper semicontinuous}\},$$9 yields fuzzy-valued recurrent fractal interpolation functions (RFIFs). There the fixed-point equation takes the form

$$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$0

and the graph of $$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$1 is the attractor of a hyperbolic recurrent IFS. The paper proves Hölder continuity and stability under perturbations of the nodes, fuzzy data values, and vertical scaling factors (Choe et al., 17 Jul 2025).

These interpolation papers are closely related to fuzzy IFS in spirit, but they are not the same object as a fuzzy Hutchinson operator acting on fuzzy subsets of a space. The fuzzy aspect lies in the codomain $$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$2 or $$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$3, not in a fuzzification of the Hutchinson operator itself. This distinction is explicit in both papers and is often overlooked in secondary discussions (Yun et al., 17 Jul 2025).

6. Misconceptions, limitations, and research directions

Several interpretive points recur across the literature. First, fuzzy IFS should not be conflated with probabilistic IFS. The probabilistic survey framework provides the classical analytical scaffold—Hausdorff metric, Markov operators, state-dependent probabilities, invariant measures, and contraction-on-average—but it does not itself define fuzzy IFS, membership-function dynamics, grey-level maps, or fuzzy attractors (Ghosh et al., 2022).

Second, not every fuzzy-looking fractal construction is a fuzzy IFS in the narrow operator-theoretic sense. Fuzzy-valued fractal interpolation and fuzzy-valued recurrent interpolation use classical IFS or RIFS on spaces whose vertical coordinate is a space of fuzzy numbers; this is a distinct, though related, methodology (Choe et al., 17 Jul 2025).

Third, contractivity assumptions differ substantially between frameworks. In the standard metric theory the fuzzy operator is typically a Banach contraction and therefore Picard. In orbital fuzzy IFS theory, by contrast, the operator is weakly Picard rather than contractive on the full fuzzy hyperspace, so fixed points need not be globally unique and can depend on the initial fuzzy set (Mihail et al., 2021). In the topological theory, convergence is topology-dependent: the strong attractor lives in the $$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$4-topology, whereas on general Hausdorff spaces one only obtains a weak fuzzy attractor in $$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$5 (Banakh et al., 28 Sep 2025).

Fourth, the regularity assumptions on grey-level maps and the finiteness of the index set are not merely cosmetic. Finite families are repeatedly used for upper semicontinuity preservation and max-attainment arguments, especially in orbital theory. The authors of the orbital papers also note that if each grey-level map is continuous, some proofs simplify, which suggests that right continuity is mathematically sufficient but not always technically benign (Mihail et al., 2021).

There are also localized technical caveats. The orbital example on $$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$6 in the original paper appears to contain a slight inconsistency between the formula involving $$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$7 and the choice of identity grey-level maps $$K(X)=\{u:X\to[0,1] : u \text{ is normal, usc, compactly supported}\}$$8. The authors’ summary explicitly notes this discrepancy while treating the example’s intended purpose as illustrative rather than foundational (Mihail et al., 2021).

Current directions suggested by the cited literature include extension to infinite families of maps, weakening continuity or admissibility assumptions, uniqueness criteria for orbital fuzzy attractors, algorithmic generation and inverse problems for orbital fuzzy fractals, and further development of fuzzy-valued fractal interpolation, including stability, derivatives, and integrals in fuzzy and fractional calculus settings. A plausible implication is that future work will continue to bifurcate into two complementary streams: one centered on fuzzy-set Hutchinson operators and attractor theory, the other on classical IFS/RIFS mechanisms in spaces of fuzzy-valued states (Miculescu et al., 2022).