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On the Birkhoff Spectrum for Hyperbolic Dynamics

Sergio Romaña

TL;DR

The paper analyzes Birkhoff spectra for hyperbolic dynamics, introducing dispersed and concentrated observables via the spectrum $\mathcal{B}(f,\varphi,\Lambda)$. It proves a density result: if the spectrum contains both positive and negative sums, it is dense in $\mathbb{R}$, extending Shaobo's work to general basic sets and flows using the density of periodic measures (Sigmund). When the spectrum is not dense, it establishes rigidity: bounded spectra imply $f$ is cohomologous to a constant (Livšic-type), and arithmetically sparse spectra force cohomology to a constant as well. The results extend to flows, including Anosov flows and geodesic flows, yielding density for Birkhoff integrals along closed orbits and geometric rigidity statements for curvature-type observables, with strong implications for inverse problems in hyperbolic geometry.

Abstract

In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. For a Hölder observable $f$ on a basic set $Λ$, we prove that if the Birkhoff sums take both positive and negative values, then the Birkhoff spectrum $\mathcal{B}(f,\varphi,Λ)$ is dense in $\mathbb{R}$. This extends the density result of Gan, Shi, and Xia \cite{Shaobo} from transitive Anosov diffeomorphisms in infranilmanifolds to general basic sets, and yields new density theorems for Axiom~A systems, including Anosov diffeos.\\ Conversely, when the spectrum is not dense, we characterize \emph{concentrated} observables, whose spectrum is confined to one side of zero. For these, we establish two rigidity results: (i) boundedness of the spectrum is equivalent to the function being cohomologous to a zero, which constitutes an extension of the Liv\v sic theorem; (ii) if the spectrum exhibits an arithmetic structure, the function is cohomologous to a constant. Finally, we extend the results to continuous time. For Anosov flows$-$ including geodesic flows on Anosov manifolds$-$we obtain analogous density results for Birkhoff integrals along closed orbits. In particular, we generalize a theorem of Dairbekov--Sharafutdinov (\cite{Dairbekov}) by showing that a bounded or ``arithmetically sparse'' spectrum forces a smooth function to vanish or be constant, respectively.

On the Birkhoff Spectrum for Hyperbolic Dynamics

TL;DR

The paper analyzes Birkhoff spectra for hyperbolic dynamics, introducing dispersed and concentrated observables via the spectrum . It proves a density result: if the spectrum contains both positive and negative sums, it is dense in , extending Shaobo's work to general basic sets and flows using the density of periodic measures (Sigmund). When the spectrum is not dense, it establishes rigidity: bounded spectra imply is cohomologous to a constant (Livšic-type), and arithmetically sparse spectra force cohomology to a constant as well. The results extend to flows, including Anosov flows and geodesic flows, yielding density for Birkhoff integrals along closed orbits and geometric rigidity statements for curvature-type observables, with strong implications for inverse problems in hyperbolic geometry.

Abstract

In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. For a Hölder observable on a basic set , we prove that if the Birkhoff sums take both positive and negative values, then the Birkhoff spectrum is dense in . This extends the density result of Gan, Shi, and Xia \cite{Shaobo} from transitive Anosov diffeomorphisms in infranilmanifolds to general basic sets, and yields new density theorems for Axiom~A systems, including Anosov diffeos.\\ Conversely, when the spectrum is not dense, we characterize \emph{concentrated} observables, whose spectrum is confined to one side of zero. For these, we establish two rigidity results: (i) boundedness of the spectrum is equivalent to the function being cohomologous to a zero, which constitutes an extension of the Liv\v sic theorem; (ii) if the spectrum exhibits an arithmetic structure, the function is cohomologous to a constant. Finally, we extend the results to continuous time. For Anosov flows including geodesic flows on Anosov manifoldswe obtain analogous density results for Birkhoff integrals along closed orbits. In particular, we generalize a theorem of Dairbekov--Sharafutdinov (\cite{Dairbekov}) by showing that a bounded or ``arithmetically sparse'' spectrum forces a smooth function to vanish or be constant, respectively.
Paper Structure (33 sections, 29 theorems, 138 equations)

This paper contains 33 sections, 29 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic sets and basic sets
  4. Some Hyperbolic Properties
  5. Local Product Structure
  6. Shadowing Property
  7. Invariant measures and periodic measures
  8. Livšic Theorem and cohomology
  9. Functions on geodesic flows
  10. Proof of Main Results for Diffeomorphisms
  11. Proof of Theorem \ref{['Shaobo']}
  12. Strategy of the proof of Theorem \ref{['Shaobo']}
  13. Some Consequences of Theorem \ref{['Shaobo']}
  14. Proof of Theorem \ref{['mainT']}
  15. Proof of Theorem \ref{['Rigidity']}
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $\Lambda \subset U$ be a basic set for $\varphi \colon U \to M$ and let $f \colon U \to \mathbb{R}$ be a Hölder continuous function. If $\mathcal{B}(f,\varphi, \Lambda)\cap \mathbb{R}^+ \neq \emptyset$ and $\mathcal{B}(f,\varphi, \Lambda)\cap \mathbb{R}^- \neq \emptyset$, then $\mathcal{B}(f,\va

Theorems & Definitions (62)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2: Bracket Operation
  • ...and 52 more