Sparse empirical measures for non-statistical interval maps with two neutral fixed points
Abstract
We consider balanced doubly intermittent full-branch interval maps with two neutral fixed points. Although the ordinary empirical measures fail to converge for Lebesgue-almost every initial point, we show that this non-statistical behavior persists under a broad class of deterministic observation schemes: if the sampling sequence is block-dominating, then both endpoint Dirac masses are accumulation points of the sparse empirical measures for Lebesgue-almost every orbit. This class includes sampling sequences of positive lower density, arithmetic and polynomial sequences, and finite-index regularly varying sequences. We also prove that every map in the balanced class has the strong natural-measure property, including at the boundary exponent β=1: the pushforwards of every absolutely continuous initial probability converge to a distinguished endpoint mixture ν*. Abstract consequences of this one-time limit include annealed convergence along every deterministic sampling sequence, the existence of arbitrarily sparse sequences with almost-sure convergence, and an almost-sure convergence criterion based on a power-saving variance bound. For the symmetric quadratic map g1∈F*, we verify the dyadic covariance condition on a countable uniformly dense endpoint-flat family: the double covariance sum is Oϕ(nlog(n+2)). Consequently its dyadic empirical measures converge for Lebesgue-almost every initial point to 1/2(δ−1+δ1).
Related
- Statistical Depth for Interval Data: A Closeness-Based Approach and Its Applications
- Reliability-Centered Evaluation of Sparse Longitudinal CT Lesion-Size Forecasting with Conformal Interval Calibration and Gompertz-Inspired Regularization
- A Randomness Test Formalism for Neutral Measures and Beyond
- Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions
- Dimension-free Convergence Rate in Sliced Wasserstein Distance for Empirical Measures of Markov Processes
BibTeX
@misc{mukhamedov2026sparse,
title = {{Sparse empirical measures for non-statistical interval maps with two neutral fixed points}},
author = {Farrukh Mukhamedov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24828},
eprint = {2609.24828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24828},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Sparse empirical measures for non-statistical interval maps with two neutral fixed points
AU - Farrukh Mukhamedov
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24828
UR - https://arxiv.org/abs/2609.24828
AB - We consider balanced doubly intermittent full-branch interval maps with two neutral fixed points. Although the ordinary empirical measures fail to converge for Lebesgue-almost every initial point, we show that this non-statistical behavior persists under a broad class of deterministic observation schemes: if the sampling sequence is block-dominating, then both endpoint Dirac masses are accumulation points of the sparse empirical measures for Lebesgue-almost every orbit. This class includes sampling sequences of positive lower density, arithmetic and polynomial sequences, and finite-index regularly varying sequences. We also prove that every map in the balanced class has the strong natural-measure property, including at the boundary exponent β=1: the pushforwards of every absolutely continuous initial probability converge to a distinguished endpoint mixture ν_*. Abstract consequences of this one-time limit include annealed convergence along every deterministic sampling sequence, the existence of arbitrarily sparse sequences with almost-sure convergence, and an almost-sure convergence criterion based on a power-saving variance bound. For the symmetric quadratic map g_1∈F_*, we verify the dyadic covariance condition on a countable uniformly dense endpoint-flat family: the double covariance sum is O_ϕ(nlog(n+2)). Consequently its dyadic empirical measures converge for Lebesgue-almost every initial point to 1/2(δ_−1+δ_1).
ER -