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General real measurable Livšic regularity via transfer operators

Ian D. Morris

TL;DR

The paper addresses the problem of when a measurable coboundary $f = h \circ T - h$ over non-invertible dynamics enjoys enhanced regularity for $h$. It develops a general Livšic regularity theorem using a transfer-operator framework with five hypotheses, and a holomorphic perturbation of the operator $\mathscr{L}_t$ to connect coboundaries to the invariant density $\chi$ and a function $g = h\chi$. Consequently, when $\phi/\chi \in X$, one obtains $f = \hat{g} \circ T - \hat{g}$ with $\hat{g} = g/\chi$, establishing a measurable-to-regularity lift in the space $X$. The authors illustrate the method with concrete results across several settings: $f \in \mathscr{BV}$ for β-transformations, holomorphic cocycles over real-analytic expanding maps, and Hölder regularity for Tsujii's virtually expanding maps, thereby unifying diverse Livšic-type conclusions under an operator-theoretic umbrella. This framework enables transferring regularity from measurability through transfer-operator spectral data, offering a versatile tool for a wide range of non-invertible dynamical systems.

Abstract

We prove a general measurable Livšic regularity theorem for real-valued cocycles over non-invertible dynamical systems using only abstract hypotheses on an associated transfer operator. As illustrative applications we derive measurable Livšic regularity results in the analytic regularity class for cocycles over real-analytic expanding maps, in the bounded-variation regularity class for $β$-transformations, and in $C^α$ regularity over the class of virtually expanding maps recently introduced by M. Tsujii.

General real measurable Livšic regularity via transfer operators

TL;DR

The paper addresses the problem of when a measurable coboundary over non-invertible dynamics enjoys enhanced regularity for . It develops a general Livšic regularity theorem using a transfer-operator framework with five hypotheses, and a holomorphic perturbation of the operator to connect coboundaries to the invariant density and a function . Consequently, when , one obtains with , establishing a measurable-to-regularity lift in the space . The authors illustrate the method with concrete results across several settings: for β-transformations, holomorphic cocycles over real-analytic expanding maps, and Hölder regularity for Tsujii's virtually expanding maps, thereby unifying diverse Livšic-type conclusions under an operator-theoretic umbrella. This framework enables transferring regularity from measurability through transfer-operator spectral data, offering a versatile tool for a wide range of non-invertible dynamical systems.

Abstract

We prove a general measurable Livšic regularity theorem for real-valued cocycles over non-invertible dynamical systems using only abstract hypotheses on an associated transfer operator. As illustrative applications we derive measurable Livšic regularity results in the analytic regularity class for cocycles over real-analytic expanding maps, in the bounded-variation regularity class for -transformations, and in regularity over the class of virtually expanding maps recently introduced by M. Tsujii.

Paper Structure

This paper contains 12 sections, 5 theorems, 23 equations.

Table of Contents

  1. Introduction and statement of results
  2. Proof of Theorem \ref{['th:main']}
  3. Fundamentals: action of $\mathscr{L}$ on $L^1(\mu)$
  4. The perturbed operator and its Riesz projection
  5. The leading eigenvalue of the perturbed operator
  6. The eigenfunction of the perturbed operator
  7. Completion of the proof
  8. Examples
  9. The $\beta$-transformation and bounded-variation cocycles
  10. Real-analytic expanding maps of the circle and bounded holomorphic cocycles
  11. Virtually expanding maps and Hölder cocycles
  12. Acknowledgments

Key Result

Theorem 1.1

Let $T$ be a measurable transformation of a probability space $(X,\mathcal{F},\mu)$, $(\mathfrak{X},\|\cdot\|_\mathfrak{X})$ a Banach space of (equivalence classes of) measurable functions from $X$ to $\mathbb{C}$, and $\mathscr{L}$ a linear operator which acts boundedly on $\mathfrak{X}$. Suppose t Let $f \in \mathfrak{X}$ satisfy the two properties: Then $\chi \in \mathfrak{X}$, and there exist

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['th:app2']}
  • Proposition 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof