Uniqueness of the invariant probability measure

Establish uniqueness of the Borel probability measure supported on the attractor K of a finite family of weak contractions on a complete, regular, normal semimetric space that satisfies the invariance equation μ = ∑_{j=1}^{ℓ} p_j μ ∘ f_j^{-1} for prescribed positive weights p_1,…,p_ℓ summing to one.

Background

Proposition 2.1 constructs a Borel probability measure μ supported on the attractor K of a finite family of weak contractions f_1,…,f_ℓ, with weights p_1,…,p_ℓ, satisfying the invariance equation μ = ∑_{j=1}{ℓ} p_j μ ∘ f_j{-1}. The construction uses the coding space and the push-forward of the corresponding product measure.

The paper does not establish that this invariant measure is unique. Determining uniqueness would complete the measure-theoretic part of the attractor theory developed in the paper and clarify whether the coding construction yields the sole probability measure satisfying the stated invariance relation.

References

We have not shown the uniqueness of the invariant measure satisfying eq:invariance-measure.

eq:invariance-measure:

μ=∑j=1ℓpjμ∘fj−1.\mu = \sum_{j = 1}^{\ell} p_j \mu \circ f_j^{-1}.

— The Moran--Hutchinson formula in semimetric spaces  (2608.16817 - Okamura, 17 Aug 2026) in Remark following Proposition 2.1, Section 2 (Existence of an invariant measure)

Consequently, it remains an open problem whether the existence and uniqueness of an invariant probability measure can be established for IFSs generated by Matkowski contractions.

— Generalized multivariate Fractal Interpolation Function and $α$-Fractal Function  (2608.17991 - Pandey et al., 18 Aug 2026) in Remark following Theorem 5.1 (Theorem \ref{measureRakotch}), Section 5, “Invariant Measures for Rakotch Contractions”