Shaping of invariant measures of iterated function systems
Abstract
We study an inverse problem for iterated function systems: given maps w1,…,wm on a Polish space and a target probability measure ν, choose the law μ with which the maps are sampled so that the invariant measure πμ of the resulting Markov chain is as close as possible to ν in the Wasserstein-1 distance. The minimization runs over the admissible set A of sampling laws for which the chain has a unique invariant measure attracting every initial law; the maps need not be contractions. On bounded state spaces, regions certified by a Doeblin or splitting condition are convex unions of relatively open faces of the simplex, and they exhaust A when the state space is finite; for affine systems with nonnegative matrices A is a spectral sublevel set; in general A is neither open, closed, convex nor connected. Existence of an optimal law reduces to the invariant measures left behind at the missing boundary cl(A)∖ A: boundary points without invariant measures of finite first moment repel minimizing sequences, which yields existence for all systems of homotheties of ℝd, and for compact state spaces a solution exists for every target if and only if the set of reachable invariant measures is closed; a two-state example shows that existence can fail. A search over a δ-net returns an ε-optimal law over any region on which μ↦πμ is Lipschitz and the chains are uniformly attracting, with explicit constants and, on an interval, an explicit sample size. Under contraction on average, μ↦πμ is differentiable, its derivative is injective if and only if the push-forwards (wi)#πμ are linearly independent, and the objective is weakly convex, so that a projected subgradient method with inexact subgradients converges to near-stationary points.
Related
- On the Existence of Geometrically Attracting Measures for Iterated Function Systems with Varying Sets of Transformations
- Hausdorff and Packing Measures for β-L{ü}roth Infinite Iterated Function Systems
- Intrinsic Hutchinson measures and KMS states for iterated function systems with large overlaps
- Fractal transference principle for conformal iterated function systems
- Dimension properties of a family of random self-similar iterated function systems
BibTeX
@misc{shafiei2026shaping,
title = {{Shaping of invariant measures of iterated function systems}},
author = {Allahkaram Shafiei and Robert Shorten and Jakub Marecek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34172},
eprint = {2610.34172},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34172},
}
RIS
TY - UNPB
TI - Shaping of invariant measures of iterated function systems
AU - Allahkaram Shafiei
AU - Robert Shorten
AU - Jakub Marecek
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34172
UR - https://arxiv.org/abs/2610.34172
AB - We study an inverse problem for iterated function systems: given maps w_1,…,w_m on a Polish space and a target probability measure ν, choose the law μ with which the maps are sampled so that the invariant measure π_μ of the resulting Markov chain is as close as possible to ν in the Wasserstein-1 distance. The minimization runs over the admissible set A of sampling laws for which the chain has a unique invariant measure attracting every initial law; the maps need not be contractions. On bounded state spaces, regions certified by a Doeblin or splitting condition are convex unions of relatively open faces of the simplex, and they exhaust A when the state space is finite; for affine systems with nonnegative matrices A is a spectral sublevel set; in general A is neither open, closed, convex nor connected. Existence of an optimal law reduces to the invariant measures left behind at the missing boundary cl(A)∖ A: boundary points without invariant measures of finite first moment repel minimizing sequences, which yields existence for all systems of homotheties of ℝ^d, and for compact state spaces a solution exists for every target if and only if the set of reachable invariant measures is closed; a two-state example shows that existence can fail. A search over a δ-net returns an ε-optimal law over any region on which μ↦π_μ is Lipschitz and the chains are uniformly attracting, with explicit constants and, on an interval, an explicit sample size. Under contraction on average, μ↦π_μ is differentiable, its derivative is injective if and only if the push-forwards (w_i)_#π_μ are linearly independent, and the objective is weakly convex, so that a projected subgradient method with inexact subgradients converges to near-stationary points.
ER -