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On unions of geodesics and projections of invariant sets

Longhui Li

Abstract

Let $M$ be a $d$-dimensional complete Riemannian manifold and let $π: SM \to M$ denote the canonical projection from the unit tangent bundle. We prove that if $E \subset SM$ is a set that invariant under the geodesic flow with Hausdorff dimension $\dim_{\mathcal{H}} E \ge 2(k-1)+1 +β$ for some integer $1 \le k \le d-1$ and some $β\in [0,1]$, then the projection $π(E)$ satisfies $\dim_{\mathcal{H}} π(E) \ge k + β$. In other words, this yields a lower bound on the Hausdorff dimension of unions of geodesics in $M$. Our theorem extends a result of J. Zahl concerning unions of lines in $\mathbb{R}^d$. The proof relies on the transversal property of geodesics, an appropriate $(k+1)$-linear curved Kakeya estimate, and the Bourgain-Guth argument.

On unions of geodesics and projections of invariant sets

Abstract

Let be a -dimensional complete Riemannian manifold and let denote the canonical projection from the unit tangent bundle. We prove that if is a set that invariant under the geodesic flow with Hausdorff dimension for some integer and some , then the projection satisfies . In other words, this yields a lower bound on the Hausdorff dimension of unions of geodesics in . Our theorem extends a result of J. Zahl concerning unions of lines in . The proof relies on the transversal property of geodesics, an appropriate -linear curved Kakeya estimate, and the Bourgain-Guth argument.
Paper Structure (16 sections, 11 theorems, 161 equations)

This paper contains 16 sections, 11 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Unions of affine $n$-planes
  3. Unions of geodesics and projections of sets invariant under geodesic flows
  4. Structure of the paper
  5. Acknowledgements
  6. Preliminaries
  7. Hausdorff dimension on manifolds
  8. Hausdorff dimension of invariant sets on $S(\mathbb{R}^d)$
  9. Theorem \ref{['Unions of smooth curves']} implies Theorem \ref{['Hausdorff dimension, unions of geodesics']}
  10. Proof of Theorem \ref{['Unions of smooth curves']}
  11. Discretization Argument
  12. A multilinear curved Kakeya estimate
  13. Proof of Theorem \ref{['curved Kakeya, discretized']}
  14. Further Remarks
  15. Sharpness of Theorem \ref{['Hausdorff dimension, unions of geodesics']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Suppose $M$ is a $d$-dimensional complete Riemannian manifold. Let $1\leq k\leq d-1$ be an integer and $\beta\in[0,1]$. If $E\subset SM$ is an invariant set with $\dim_{\mathcal{H}} E\geq 2(k-1)+1+\beta$, then

Theorems & Definitions (19)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • Definition 4.1: $(\delta,s)$-sets
  • Lemma 4.2: Frostman
  • Theorem 4.3
  • Theorem 4.4: The multilinear Kakeya theorem, CV13
  • ...and 9 more