Full-measure historic behavior and emergence in simplex dynamics
Abstract
We give a finite-dimensional boundary-residence mechanism for Takens-type historic behavior. For a class of continuous self-maps of a simplex, a multiplicative Lyapunov drift toward the boundary, combined with cyclic residence near separated boundary regions, forces every non-fixed interior orbit to have non-convergent Cesàro averages and non-convergent empirical measures. Thus the historic set has full relative Lebesgue measure. The mechanism is independent of hyperbolic specification, wandering domains and positive entropy: in fact, for the maps considered here the topological entropy is carried entirely by the boundary restriction. The non-convergence persists under all positive powers and, more generally, under multiplicatively thick sampling times. The abstract criterion applies to Stein--Ulam and Lotka--Volterra type stochastic operators once the model-specific cyclic coding and logarithmic residence estimates are verified. In explicit cyclic-collar models we identify the full weak-* accumulation set of empirical measures; this accumulation set is a non-trivial circle of probability measures and yields polynomial pointwise emergence of order ε−1. We also obtain examples with positive-dimensional omega-limit sets by combining the mechanism with a Barański--Misiurewicz type boundary-tracing construction.
Related
- Reachable sets under Kolmogorov dynamics in the probability simplex
- Below-threshold Bistability and Implementation Lag in a Simplex Model of Radical Vote-Share Dynamics
- Mechanistic inhibition of herpes simplex virus-1 UL21 immune-evasion function by natural-product scaffolds: A multi-tier docking, dynamics, and energetic profiling approach
- Balance laws with measure valued source for commuting dynamics
- Classical and Quantum Dynamics in State Space: Projective Hilbert Space, Measurement, and Classical Emergence
BibTeX
@misc{mukhamedov2026fullmeasure,
title = {{Full-measure historic behavior and emergence in simplex dynamics}},
author = {Farrukh Mukhamedov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18979},
eprint = {2609.18979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18979},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Full-measure historic behavior and emergence in simplex dynamics
AU - Farrukh Mukhamedov
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.18979
UR - https://arxiv.org/abs/2609.18979
AB - We give a finite-dimensional boundary-residence mechanism for Takens-type historic behavior. For a class of continuous self-maps of a simplex, a multiplicative Lyapunov drift toward the boundary, combined with cyclic residence near separated boundary regions, forces every non-fixed interior orbit to have non-convergent Cesàro averages and non-convergent empirical measures. Thus the historic set has full relative Lebesgue measure. The mechanism is independent of hyperbolic specification, wandering domains and positive entropy: in fact, for the maps considered here the topological entropy is carried entirely by the boundary restriction. The non-convergence persists under all positive powers and, more generally, under multiplicatively thick sampling times. The abstract criterion applies to Stein--Ulam and Lotka--Volterra type stochastic operators once the model-specific cyclic coding and logarithmic residence estimates are verified. In explicit cyclic-collar models we identify the full weak-* accumulation set of empirical measures; this accumulation set is a non-trivial circle of probability measures and yields polynomial pointwise emergence of order ε^−1. We also obtain examples with positive-dimensional omega-limit sets by combining the mechanism with a Barański--Misiurewicz type boundary-tracing construction.
ER -