Research Article · Preprint Mathematics
Mean ergodic convergence from near-diagonal subadditivity
- Primary subject
- Mathematics
- Subject subfields
- Functional Analysis · Dynamical Systems
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
We prove a mean ergodic theorem under subadditivity required only for nearly equal block lengths. The operator is positive, unital and integral-preserving, without ergodicity or invertibility. Convergence of invariant projections of uniformly integrable normalized inputs, with both ordered adjacent-two inequalities, forces L1 convergence. An existing scalar shrinking-corridor theorem supplies invariant Lp convergence for every finite p when the nonincreasing relative width has divergent square integral and eventually admits length differences two. Nonnegative inputs require individual Lp membership; for signed inputs, uniform integrability of normalized negative-part pth powers is necessary and sufficient. Under the same scalar-admissible width and norm criterion, signed integrable inputs have almost-sure limsup equal to the invariant limit, uniformly over macroscopic operator powers and lengths. The corresponding lower-limit assertion remains open.
Keywords Functional Analysis · Dynamical Systems · mathematical research
Scope of the work
General nonergodic positive unital integral-preserving Markov norm theorem and exact negative-part UI criterion retained for every scalar-admissible width. Signed individual-L1 inputs with both eventual ordered adjacent-two inequalities and explicit finite almost-sure invariant projected slopes satisfy the conditional pointwise upper and uniform macroscopic operator-power upper laws; norm convergence gives almost-sure limsup equality. The existing scalar-width theorem supplies these projected slopes and norm convergence for every nonincreasing width with divergent square integral and eventual n*w(log n)>=2, including nonnegative individual-Lp inputs. No lower almost-sure bound, new scalar threshold, stochastic sample-path law, rate or diverging start budget is claimed.
Reproduction notes
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Scott Schubr. (2026). Mean ergodic convergence from near-diagonal subadditivity (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/mean-ergodic-convergence-from-near-diagonal-subadditivity/v/1
BibTeX
@techreport{luminary_mean_ergodic_convergence_from_near_diagonal_subadditivity_v1,
title = {Mean ergodic convergence from near-diagonal subadditivity},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/mean-ergodic-convergence-from-near-diagonal-subadditivity/v/1}
}Loading discussion…