ACIM instability of piecewise expanding maps through the lens of metastability
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.
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BibTeX
@misc{komalovics2026acim,
title = {{ACIM instability of piecewise expanding maps through the lens of metastability}},
author = {Ábel Komálovics and Péter Bálint},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11704},
eprint = {2609.11704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11704},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - ACIM instability of piecewise expanding maps through the lens of metastability
AU - Ábel Komálovics
AU - Péter Bálint
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11704
UR - https://arxiv.org/abs/2609.11704
AB - 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.
ER -