Critical cusp cancellation and analytic response for a Bernoulli random interval map
Abstract
We show that the critical cusp exponent β=−1/2 admits W2,1 invariant densities in an explicit family of expanding full-branch interval maps. The usual estimates exclude this exponent because two zero-order coefficients in the second derivative of the transfer operator are individually non-integrable. Their signed combination, however, contains the factor β(2β+1) for a pure power cusp. We exploit its cancellation at β=−1/2 to prove a W2,1 Lasota--Yorke inequality and a spectral gap for the explicit family. We also identify a separate obstruction caused by an interior cusp value. As an application, a small Bernoulli probability perturbation between two fixed maps has a stationary density that depends real-analytically on the perturbation parameter in W2,1, with explicit formulas for all response derivatives.
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BibTeX
@misc{oltiboev2026critical,
title = {{Critical cusp cancellation and analytic response for a Bernoulli random interval map}},
author = {Davrbek Oltiboev and Furqat Vafoyev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23180},
eprint = {2610.23180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23180},
}
RIS
TY - UNPB
TI - Critical cusp cancellation and analytic response for a Bernoulli random interval map
AU - Davrbek Oltiboev
AU - Furqat Vafoyev
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23180
UR - https://arxiv.org/abs/2610.23180
AB - We show that the critical cusp exponent β=−1/2 admits W^2,1 invariant densities in an explicit family of expanding full-branch interval maps. The usual estimates exclude this exponent because two zero-order coefficients in the second derivative of the transfer operator are individually non-integrable. Their signed combination, however, contains the factor β(2β+1) for a pure power cusp. We exploit its cancellation at β=−1/2 to prove a W^2,1 Lasota--Yorke inequality and a spectral gap for the explicit family. We also identify a separate obstruction caused by an interior cusp value. As an application, a small Bernoulli probability perturbation between two fixed maps has a stationary density that depends real-analytically on the perturbation parameter in W^2,1, with explicit formulas for all response derivatives.
ER -