Liv\v{s}ic regularity for random and sequential dynamics through transfer operators

TL;DR

This paper establishes Livšic-type regularity results for non-autonomous dynamical systems, including random and sequential maps, with implications for variance growth and coboundary characterizations.

Contribution

It extends Livšic regularity results to non-autonomous systems, providing new insights into coboundary representations and variance growth conditions.

Findings

Results apply to random and sequential piecewise expanding maps

Variance growth fails only when functions are coboundaries without regularity restrictions

Provides a more relaxed characterization of coboundaries in non-autonomous systems

Abstract

We prove Liv\v{s}ic-type regularity results of coboundary representations for non-autonomous dynamical systems. Our results have an abstract nature and apply to several important specific situations, such as (higher-dimensional) random or sequential piecewise expanding maps and subshifts of finite type, which have applications to Markov interval maps and to finite state inhomogeneous elliptic Markov shifts, via symbolic representations. We also obtain results for some classes of non-autonomous hyperbolic systems. Our results can be seen as non-autonomous versions of a recent result obtained by Morris. However, we emphasize that our proof differs from the one mentioned previously even in the deterministic case. Finally, we show that our results provide a more relaxed characterization for having variance growth of Birkhoff sums on random and sequential dynamical systems; we show that such growth can fail only when the underlying functions are a coboundary without special restrictions on the regularity of the coboundary. For random systems, we show that this is equivalent to having a coboundary with bounded ``variation", but for sequential systems it turns out that this is no longer true, as demonstrated by examples.