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Dynamics and Rigidity through the Lens of Circles

Hee Oh

TL;DR

The paper studies dynamics and rigidity in infinite-volume homogeneous spaces through the lens of circle packings, notably Apollonian and Sierpiński-type configurations that arise as limit sets of Kleinian groups. It develops circle-counting theorems, orbit-closure classifications, and representation rigidity, connecting these geometric problems to the dynamics of unipotent and diagonal flows in both rank-one and higher-rank settings. A central methodological theme is mixing in infinite volume, including local, exponential, and uniform exponential mixing, which underpins counting, equidistribution, and sieve-type results via self-joinings and Anosov dynamics. The work also constructs higher-rank analogues of classical rigidity (Mostow–Prasad–Sullivan) through boundary maps on limit sets and develops torus-counting results from circle-counting by leveraging quasiconformal deformations and higher-rank Patterson–Sullivan theory, with applications to affine sieve and arithmetic in orbits.

Abstract

We report on recent developments in the dynamics and rigidity of infinite-volume homogeneous spaces, viewed through the lens of circles. By addressing four natural questions about circle packings, we highlight the interplay between dynamics, geometry, and rigidity that defines the emerging frontier of homogeneous dynamics.

Dynamics and Rigidity through the Lens of Circles

TL;DR

The paper studies dynamics and rigidity in infinite-volume homogeneous spaces through the lens of circle packings, notably Apollonian and Sierpiński-type configurations that arise as limit sets of Kleinian groups. It develops circle-counting theorems, orbit-closure classifications, and representation rigidity, connecting these geometric problems to the dynamics of unipotent and diagonal flows in both rank-one and higher-rank settings. A central methodological theme is mixing in infinite volume, including local, exponential, and uniform exponential mixing, which underpins counting, equidistribution, and sieve-type results via self-joinings and Anosov dynamics. The work also constructs higher-rank analogues of classical rigidity (Mostow–Prasad–Sullivan) through boundary maps on limit sets and develops torus-counting results from circle-counting by leveraging quasiconformal deformations and higher-rank Patterson–Sullivan theory, with applications to affine sieve and arithmetic in orbits.

Abstract

We report on recent developments in the dynamics and rigidity of infinite-volume homogeneous spaces, viewed through the lens of circles. By addressing four natural questions about circle packings, we highlight the interplay between dynamics, geometry, and rigidity that defines the emerging frontier of homogeneous dynamics.

Paper Structure

This paper contains 50 sections, 36 theorems, 91 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Circle packings
  3. Four motivating questions
  4. Organization
  5. Circle counting and Kleinian groups
  6. Counting Apollonian circles
  7. Kleinian groups
  8. Circle-counting theorem for geometrically finite Kleinian groups
  9. Decoding the constant and the measure
  10. Bowen-Margulis-Sullivan measure
  11. Skinning measure of a properly immersed geodesic plane
  12. Orbit closures of circles and rigidity of geodesic planes
  13. Orbit closures in round Sierpiński carpets
  14. Geometric structure and limit sets of acylindrical hyperbolic $3$-manifolds
  15. Orbit closures for geometrically finite acylindrical hyperbolic $3$-manifolds
  16. ...and 35 more sections

Key Result

Theorem 2.1

For any bounded Apollonian packing $\mathcal{P}$, there exists a constant $c_{\mathcal{P}}>0$ such that where $\delta_{\mathsf A}$ is the Hausdorff dimension of the closure $\overline{\mathcal{P}}$.

Figures (24)

  • Figure 1.1: First few generations of Apollonian circle packings
  • Figure 1.2: A Sierpiński-type circle packing
  • Figure 1.3: Torus packing
  • Figure 2.1: Dual circles
  • Figure 2.2: Geodesic planes
  • ...and 19 more figures

Theorems & Definitions (45)

  • Theorem 2.1: KO
  • Theorem 2.2: OS_inv
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Sullivan1979density, Sullivan1984entropy
  • Theorem 2.7: OS_inv
  • Theorem 2.8: Patterson1976limit, Sullivan1979density
  • Theorem 3.1: MMO_inventiones, BO_etds
  • Theorem 3.2: Thurston1986hyperbolic, Mc_iter
  • ...and 35 more