{
  "abstract": "Motivated by Keller's W-shaped maps and its variants, we introduce a general class of families of expanding maps such that the perturbed maps have a shrinking almost invariant neighborhood about a fixed point of the limit map. The unique absolutely continuous invariant measures (ACIM) of the perturbed maps can converge to limit measures of various types. A local quantity is identified which determines if the limit is absolutely continuous, singular, or a non-trivial convex combination of these. Furthermore, for the case of a nontrivial convex combination, we prove that the dynamics of the perturbed system, when viewed on an appropriate slow time scale, converges to a jump Markov process, a convergence that extends to the diffusion coefficients for observables of bounded variation. Compared to analogous results on expanding maps with metastable behavior, a new feature of our setting is the emergence of a localized state of the Markov process which corresponds to the shrinking almost invariant interval about the fixed point. Our approach provides a general framework which, in particular, accommodates, to the best of our knowledge, all previously studied families that limit to Keller's W-shaped map.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11704",
  "author_total": 2,
  "authors": [
    {
      "name": "Ábel Komálovics"
    },
    {
      "name": "Péter Bálint"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/59yracpkntfk9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11704",
  "fulltext_url": null,
  "id": "59yracpkntfk9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11704",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7212242871",
  "pdf_url": "https://arxiv.org/pdf/2609.11704",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/59yracpkntfk9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11704",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11704"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "ACIM instability of piecewise expanding maps through the lens of metastability",
  "title_text": "ACIM instability of piecewise expanding maps through the lens of metastability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/59yracpkntfk9",
  "venue": "arXiv",
  "versions": []
}