On unions of geodesics and projections of invariant sets

TL;DR

This paper establishes a lower bound on the Hausdorff dimension of the projection of invariant sets under the geodesic flow on a Riemannian manifold, extending results related to unions of lines in Euclidean space.

Contribution

It extends a known Euclidean result to Riemannian manifolds, providing new bounds on the Hausdorff dimension of unions of geodesics using advanced geometric and harmonic analysis techniques.

Findings

Lower bound on Hausdorff dimension of projections of invariant sets

Extension of Zahl's result from Euclidean lines to Riemannian geodesics

Application of curved Kakeya estimates and Bourgain-Guth argument

Abstract

Let $M$ be a $d$-dimensional complete Riemannian manifold and let $\pi: SM \to M$ 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 $\dim_{\mathcal{H}} E \ge 2(k-1)+1 +\beta$ for some integer $1 \le k \le d-1$ and some $\beta \in [0,1]$, then the projection $\pi(E)$ satisfies $\dim_{\mathcal{H}} \pi(E) \ge k + \beta$. 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.