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Livšic regularity for random and sequential dynamics through transfer operators

Lucas Backes, Davor Dragicevic, Yeor Hafouta

TL;DR

This work extends Livšic-type regularity to non-autonomous dynamics by developing a transfer-operator framework for random and sequential systems. It proves that, under suitable random or sequential RPF-type hypotheses, any coboundary F with regularity enforces the same regularity on the transfer function H, and it provides explicit formulas for H via iterates. The authors also derive relaxed criteria for variance growth of Birkhoff sums and furnish coboundary representations and martingale decompositions in both random and sequential contexts. The results further apply to random/sequential SFTs and to small perturbations of hyperbolic maps, yielding uniform Hölder regularity for coboundaries in non-autonomous settings and broadening the Livšic theory to non-autonomous regimes with wide applicability in ergodic theory and dynamical systems.

Abstract

We prove Livšic-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.

Livšic regularity for random and sequential dynamics through transfer operators

TL;DR

This work extends Livšic-type regularity to non-autonomous dynamics by developing a transfer-operator framework for random and sequential systems. It proves that, under suitable random or sequential RPF-type hypotheses, any coboundary F with regularity enforces the same regularity on the transfer function H, and it provides explicit formulas for H via iterates. The authors also derive relaxed criteria for variance growth of Birkhoff sums and furnish coboundary representations and martingale decompositions in both random and sequential contexts. The results further apply to random/sequential SFTs and to small perturbations of hyperbolic maps, yielding uniform Hölder regularity for coboundaries in non-autonomous settings and broadening the Livšic theory to non-autonomous regimes with wide applicability in ergodic theory and dynamical systems.

Abstract

We prove Livšic-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.

Paper Structure

This paper contains 20 sections, 17 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Random dynamics
  4. Setting
  5. Statement of the first main result
  6. Proof of Theorem \ref{['MT']}
  7. Sequential dynamics
  8. Setting
  9. Perturbation theory
  10. Statement of the second main result
  11. Proof of Theorem \ref{['MT2']}
  12. Random and sequential two-sided subshfits of finite type
  13. Random SFT
  14. Sequential SFT
  15. Small perturbations of hyperbolic maps
  16. ...and 5 more sections

Key Result

Theorem 2.4

Suppose that a pair $(F, \mathcal{L})$ admits a random RPF triplet. Furthermore, suppose that there exists a measurable map $H\colon{\mathcal{X}}\to\mathbb R$ such that where $\tau$ is given by tau. Then, $H({\omega}, \cdot)\in {\mathcal{B}}_{\omega}$ for $\mathbb P$-a.e. ${\omega} \in {\Omega}$. Moreover, there exists a constant $C>0$ such that $\text{var}_{\omega}(H(\omega, \cdot))\leq C$ for $

Theorems & Definitions (40)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 30 more