On Nullhomology and stationary sequences
Abstract
The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measurepreserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same \emph{nullhomology equivalence class}. We also discuss various aspects of nullhomology within the class of Markov random walks, compare nullhomology with a formally stronger notion which we call {\it strictsense nullhomology}. Finally, we also discuss some concrete cases where the notion of nullhomology turns up in a relevant manner.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 DOI:
 10.48550/arXiv.1910.07378
 arXiv:
 arXiv:1910.07378
 Bibcode:
 2019arXiv191007378A
 Keywords:

 Mathematics  Probability;
 Mathematics  Dynamical Systems;
 28D05 (Primary) 60G10 (Secondary)