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Rapid mixing for compact group extensions of hyperbolic flows

Mark Pollicott, Daofei Zhang

TL;DR

The article proves that compact group extensions f_t of hyperbolic flows g_t exhibit rapid mixing with respect to the natural local product measure when a Diophantine condition on the Brin holonomy group holds, with decay faster than any polynomial for Hölder observables. The authors build a symbolic model via Markov partitions and skew products, and obtain a Dolgopyat-type estimate for the associated transfer operators across irreducible representations, enabling analytic continuation of Laplace transforms of correlations. This yields both rapid decay of correlations for the extension flow and superpolynomial equidistribution of holonomies around closed orbits, with frame flows in negative curvature serving as key applications. In particular, for frame flows with suitable Brin-group density (e.g., many negative curvature settings in dimension d ≥ 4), rapid mixing and superpolynomial equidistribution hold, reinforcing the statistical understanding of holonomy dynamics in geometric settings.

Abstract

In this article, we give explicit conditions for compact group extensions of hyperbolic flows (including geodesic flows on negatively curved manifolds) to exhibit quantifiable rates of mixing (or decay of correlations) with respect to the natural probability measures, which are locally the product of a Gibbs measure for a Hölder potential and the Haar measure. More precisely, we show that the mixing rate with respect to Hölder functions will be faster than any given polynomial (i.e., rapid mixing). We also give error estimates on the equidistribution of the holonomy around closed orbits. In particular, these results apply to some frame flows for manifolds with negative sectional curvatures.

Rapid mixing for compact group extensions of hyperbolic flows

TL;DR

The article proves that compact group extensions f_t of hyperbolic flows g_t exhibit rapid mixing with respect to the natural local product measure when a Diophantine condition on the Brin holonomy group holds, with decay faster than any polynomial for Hölder observables. The authors build a symbolic model via Markov partitions and skew products, and obtain a Dolgopyat-type estimate for the associated transfer operators across irreducible representations, enabling analytic continuation of Laplace transforms of correlations. This yields both rapid decay of correlations for the extension flow and superpolynomial equidistribution of holonomies around closed orbits, with frame flows in negative curvature serving as key applications. In particular, for frame flows with suitable Brin-group density (e.g., many negative curvature settings in dimension d ≥ 4), rapid mixing and superpolynomial equidistribution hold, reinforcing the statistical understanding of holonomy dynamics in geometric settings.

Abstract

In this article, we give explicit conditions for compact group extensions of hyperbolic flows (including geodesic flows on negatively curved manifolds) to exhibit quantifiable rates of mixing (or decay of correlations) with respect to the natural probability measures, which are locally the product of a Gibbs measure for a Hölder potential and the Haar measure. More precisely, we show that the mixing rate with respect to Hölder functions will be faster than any given polynomial (i.e., rapid mixing). We also give error estimates on the equidistribution of the holonomy around closed orbits. In particular, these results apply to some frame flows for manifolds with negative sectional curvatures.
Paper Structure (34 sections, 53 theorems, 136 equations, 5 figures)

This paper contains 34 sections, 53 theorems, 136 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory of compact Lie groups
  4. Diophantine subsets
  5. Pressure and equilibrium states
  6. Skew products
  7. Suspension semi-flows
  8. Hyperbolic flows and their properties
  9. Hyperbolic flows, Gibbs measures and u-s paths
  10. Rate of mixing for hyperbolic flows
  11. A Diophantine condition on closed orbits
  12. Jointly non-integrable Anosov flows
  13. Compact group extensions
  14. Compact group extensions of hyperbolic flows
  15. Rapid mixing and superpolynomial equidistribution
  16. ...and 19 more sections

Key Result

Theorem 1

Assume that $G$ is semi-simple and the Brin group for the flow $f_t$ is dense in $G$ (i.e., $\overline H = G$) then $f_{t}$ is rapidly mixing with respect to $\widehat{\mu}_{\Phi}$.

Figures (5)

  • Figure 1: The commutative diagram for the flows $f_t$ and $g_t$
  • Figure 2: A closed chain $W=\{x_{i}\}_{i=0}^{p}$ on $M$
  • Figure 3: The frame flow when $n=3$ where $f_t (x,v_1,v_2, v_3) = (x',\phi_t(v_1), v_2',v_3')$
  • Figure 4: The correlation function $\rho_{E_\pi, F_\pi}$ is replaced by $\chi_{E_\pi, F_\pi}$.
  • Figure 5: The curve $\mathcal{C}_\pi$

Theorems & Definitions (116)

  • Example
  • Definition 1.0.1
  • Definition 1.0.2
  • Theorem 1
  • Theorem 2
  • Example
  • Lemma 2.1.1
  • Lemma 2.1.2
  • Lemma 2.1.3
  • Corollary 2.1.4
  • ...and 106 more