Variations on a theorem of Birman and Series

Lenzhen, Anna; Souto, Juan

Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 171-194.

Soient $\Sigma$ une surface hyperbolique et $f:{ℝ}_{+}\to {ℝ}_{+}$ une fonction monotone. Nous étudions l’adherence dans le fibré projectif tangent $PT\Sigma$ de l’ensemble des géodésiques $\gamma$ telles que $\text{i}(\gamma ,\gamma )\le f\left({\ell }_{\Sigma }\left(\gamma \right)\right)$. En particulier nous montrons que si $f$ est non bornée et sous-linéaire alors la dimension de Hausdorff de cet ensemble est strictement entre $1$ et $3$.

Suppose that $\Sigma$ is a hyperbolic surface and $f:{ℝ}_{+}\to {ℝ}_{+}$ a monotonic function. We study the closure in the projective tangent bundle $PT\Sigma$ of the set of all geodesics $\gamma$ satisfying $I(\gamma ,\gamma )\le f\left({\ell }_{\Sigma }\left(\gamma \right)\right)$. For instance we prove that if $f$ is unbounded and sublinear then this set has Hausdorff dimension strictly bounded between $1$ and $3$.

Classification : 30F10, 30F60
Keywords: geodesics, hyperbolic surface, self-intersection, Hausdorff dimension
Mot clés : géodesiques, surfaces hyperboliques, auto-intersection, dimension de Hausdorff

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