The convergence-divergence criterion for the Mañé cohomology lemma and Livšic theorem

Abstract

We establish a new dynamical convergence-divergence criterion for the existence of Lipschitz functions with continuous sub-actions, over general dynamical systems. This unifies and generalises various versions of the Mañé cohomology lemma, in the setting of uniformly expanding maps, and systems with indifferent fixed points. One consequence is that for any compact metric space (X,ρ), and Lipschitz Ruelle-expanding map T:X→ X, a metric transform ω satisfies Dini's criterion ∫₀ ω(s)s⁻¹ ds <∞ if and only if every function that is Lipschitz for ω∘ρ has a continuous sub-action. Another consequence is that if T is an expanding circle map of Manneville-Pomeau type α∈(0,1), then ω satisfies the convergence criterion ∫₀ ω(s)s^−1−α ds <∞ if and only if every function that is Lipschitz for ω∘ρ has a continuous sub-action. For the latter class of maps T, with ω satisfying the convergence criterion, we prove a Livšic theorem for functions that are Lipschitz for ω∘ρ: if ∫ f dμ=0 for all invariant measures μ, then there exists a continuous function φ such that f=φ∘ T−φ, where φ is Lipschitz with respect to D_α(ω)∘ρ, and D_α(ω)(t)= ∫₀^t ω(s)s^−1−α ds.

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BibTeX RIS

BibTeX

@misc{jenkinson2026convergencedivergence,
  title = {{The convergence-divergence criterion for the Mañé cohomology lemma and Livšic theorem}},
  author = {Oliver Jenkinson and Yiwei Zhang and Hang Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32484},
  eprint = {2610.32484},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32484},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The convergence-divergence criterion for the Mañé cohomology lemma and Livšic theorem
AU  - Oliver Jenkinson
AU  - Yiwei Zhang
AU  - Hang Zhao
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32484
UR  - https://arxiv.org/abs/2610.32484
AB  - We establish a new dynamical convergence-divergence criterion for the existence of Lipschitz functions with continuous sub-actions, over general dynamical systems. This unifies and generalises various versions of the Mañé cohomology lemma, in the setting of uniformly expanding maps, and systems with indifferent fixed points. One consequence is that for any compact metric space (X,ρ), and Lipschitz Ruelle-expanding map T:X→ X, a metric transform ω satisfies Dini's criterion ∫_0 ω(s)s^−1 ds <∞ if and only if every function that is Lipschitz for ω∘ρ has a continuous sub-action. Another consequence is that if T is an expanding circle map of Manneville-Pomeau type α∈(0,1), then ω satisfies the convergence criterion ∫_0 ω(s)s^−1−α ds <∞ if and only if every function that is Lipschitz for ω∘ρ has a continuous sub-action. For the latter class of maps T, with ω satisfying the convergence criterion, we prove a Livšic theorem for functions that are Lipschitz for ω∘ρ: if ∫ f dμ=0 for all invariant measures μ, then there exists a continuous function φ such that f=φ∘ T−φ, where φ is Lipschitz with respect to D_α(ω)∘ρ, and D_α(ω)(t)= ∫_0^t ω(s)s^−1−α ds.
ER  -

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