{
  "abstract": "We study an inverse problem for iterated function systems: given maps $w_1,\\dots,w_m$ 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 $\\mathrm{cl}(A)\\setminus A$: boundary points without invariant measures of finite first moment repel minimizing sequences, which yields existence for all systems of homotheties of $\\mathbb R^d$, 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 $μ\\mapstoπ_μ$ is Lipschitz and the chains are uniformly attracting, with explicit constants and, on an interval, an explicit sample size. Under contraction on average, $μ\\mapstoπ_μ$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34172",
  "author_total": 3,
  "authors": [
    {
      "name": "Allahkaram Shafiei"
    },
    {
      "name": "Robert Shorten"
    },
    {
      "name": "Jakub Marecek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4an3mrtwy8z2r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34172",
  "fulltext_url": null,
  "id": "4an3mrtwy8z2r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34172",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223918150",
  "pdf_url": "https://arxiv.org/pdf/2610.34172",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4an3mrtwy8z2r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34172",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34172"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Shaping of invariant measures of iterated function systems",
  "title_text": "Shaping of invariant measures of iterated function systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/4an3mrtwy8z2r",
  "venue": "arXiv",
  "versions": []
}