Invariant idempotent $\ast$-measures for generalized iterated function systems

Abstract: The notion of $\ast$-measure on a compact Hausdorff space can be defined for arbitrary continuous triangular norm $\ast$. The well-known Hutchinson-Barnsley theory deals with the iterated function systems (IFSs) of probability measures and establishes existence and uniqueness of invariant measures. In the previous paper, IFSs of $\ast$-measures were considered. In the present paper we deal with generalized invariant function systems (GIFSs) of $\ast$-measures, which are counterparts of GIFSs in the sense of Mihail and Miculescu. The notion of invariant $\ast$-measure is introduced for such GIFSs and we prove existence and uniqueness of such elements.

• The paper proves existence and uniqueness of invariant ∗-measures for GIFSs using a monad-based categorical framework.
• It extends classical IFS measure theory by incorporating generalized Matkowski contractions and t-norm induced measures.
• The hyperspace representation links ∗-measures to fuzzy attractors, providing a unified approach to non-additive and fuzzy dynamics.

Invariant Idempotent $\ast$-Measures for Generalized Iterated Function Systems

Overview

This paper introduces and rigorously develops the theory of invariant idempotent $\ast$-measures for generalized iterated function systems (GIFSs) defined over compact Hausdorff spaces. The main result establishes the existence and uniqueness of invariant $\ast$-measures for GIFSs, extending previous work from the setting of conventional IFSs to significantly more general functorial constructs. The authors make substantial use of monad-based category-theoretic formalism, allowing for a uniform treatment that encompasses various classes of measures, including probability, idempotent, and $\ast$-measures, and interfaces naturally with the theory of fuzzy sets.

Triangular Norms and $\ast$-Measures

The framework begins with the definition of a $\ast$-measure, which generalizes idempotent measures using an arbitrary continuous triangular norm (t-norm) $\ast$ on $[0,1]$. A $\ast$-measure is a functional $\mu: C(X,\mathbb{I}) \to \mathbb{I}$ satisfying normalization, $\ast$0-homogeneity, and maxitivity. This subsumes classical cases such as probability measures (additive), idempotent measures (based on $\ast$1), as well as other t-norm-induced functionals like those using multiplication or Łukasiewicz t-norm.

The set $\ast$2 of $\ast$3-measures is equipped with the weak* topology, is a functor in $\ast$4, and possesses a natural monad structure.

Hyperspace Representation and Categorical Equivalence

The authors detail the hyperspace construction $\ast$5, comprising compact, saturated subsets of $\ast$6 with natural topological properties. There exists a functorial isomorphism between the category of $\ast$7-measures and this hyperspace, linking the two via a representation in terms of hypographs of normal upper semicontinuous (usc) fuzzy sets.

This isomorphism not only elucidates the structure of $\ast$8-measures but also allows the transfer of results from the setting of fuzzy set attractors—such as those studied in GIFSs of fuzzy sets—back to $\ast$9-measure spaces. The connection is made precise through explicit isomorphisms of the associated monads.

Generalized Iterated Function Systems

GIFSs are generalizations of classical IFSs where the mapping domain is not singletons, but $\ast$0-th symmetric powers $\ast$1 for a subgroup $\ast$2 of the symmetric group $\ast$3. This encompasses symmetric and other functorially structured product actions, enabling further generality. Each map $\ast$4 in such a system is assumed to be a generalized Matkowski contraction, characterized by an appropriate contractivity function $\ast$5.

The functorial and monad-based approach allows the definition of the IFS operator $\ast$6 acting on $\ast$7 by combining the effect of the constituent maps with corresponding $\ast$8-weights. An invariant $\ast$9-measure $\ast$0 is then a fixed point of this operator applied to the $\ast$1-symmetrized tensor product of copies of $\ast$2.

Main Theorem: Existence and Uniqueness

The centerpiece is the existence and uniqueness theorem for invariant $\ast$3-measures associated with any $\ast$4-GIFS whose maps are generalized Matkowski contractions and where the measure on the index set is a t-norm-based convex combination. Uniqueness is established through a careful analysis of contraction properties lifted to measure hyperspaces, sidestepping the need for explicit metricization of the space of $\ast$5-measures. The existence of an invariant $\ast$6-measure is shown using monotone sequence construction and compactness arguments, transferring contraction properties from the space to its measure-theoretic or hyperspace analogs.

Connections and Extension

The developed framework subsumes classical IFS measure theory (e.g., Hutchinson measures) and its fuzzy set variants, providing a unifying categorical context. The results generalize previous work on IFSs of probability and idempotent measures (classical and t-norm based), probabilistic GIFSs, and GIFS attractors in fuzzy set theory.

The approach via functors and monads enables further extensions, such as replacing symmetric powers with arbitrary functorial constructions of finite degree (e.g., hypersymmetric powers), and considering infinite function systems or place-dependent systems. The machinery allows invariant objects (attractors) to be defined and analyzed across diverse categories, including compact subsets, probability measures, fuzzy measures, and beyond.

The link to fuzzy sets implies that the invariant $\ast$7-measures correspond to invariant fuzzy attractors, further demonstrating that the category-theoretic perspective not only unifies but also expands the theory of attractors in self-similarity and related fields.

Remarks, Open Problems, and Future Directions

• The hierarchy between $\ast$8-GIFSs and $\ast$9-GIFSs for $\ast$0 raises questions about completeness and strictness of the inclusion relations between invariant measure classes.
• The universal categorical formalism suggests that further generalizations could encompass new classes of attractors and invariant measures under broader contractivity or continuity assumptions.
• The natural extension to place-dependent systems and functors of finite degree offers a rich area for further investigation, including the interplay with the topology of compact spaces and the algebraic properties of t-norms.
• The hyperspace representation relates the invariant $\ast$1-measure theory to the geometry of upper semicontinuous fuzzy sets, suggesting applications to analysis on fuzzy metric spaces and the study of attractors in fuzzy dynamical systems.

Conclusion

The paper significantly advances the theory of invariant idempotent $\ast$2-measures by introducing a monad-based categorical approach to GIFSs, establishing strong existence and uniqueness results for invariant measures, and situating the theory in a framework that is simultaneously algebraic, topological, and analytic. The methods and results forge connections with fuzzy set theory, categorical topology, and the general theory of self-similarity, and provide the foundation for further expansions in non-additive measure theory and categorical fixed point analysis.