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Hausdorff dimension of divergent diagonal geodesics on product of finite-volume hyperbolic spaces

Published online by Cambridge University Press:  25 September 2017

LEI YANG
LEI YANG*
Affiliation:
Mathematical Sciences Research Institute, Berkeley, CA 94720, USA email [email protected]
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Abstract

In this paper, we consider the product space of several non-compact finite-volume hyperbolic spaces, $V_{1},V_{2},\ldots ,V_{k}$ of dimension $n$. Let $\text{T}^{1}(V_{i})$ denote the unit tangent bundle of $V_{i}$ and $g_{t}$ denote the geodesic flow on $\text{T}^{1}(V_{i})$ for each $i=1,\ldots ,k$. We define

$$\begin{eqnarray}{\mathcal{D}}_{k}:=\{(v_{1},\ldots ,v_{k})\,\in \,\text{T}^{1}(V_{1})\times \cdots \times \text{T}^{1}(V_{k})\,:\,(g_{t}(v_{1}),\ldots ,g_{t}(v_{k}))\text{ diverges as }t\rightarrow \infty \}.\end{eqnarray}$$
We will prove that the Hausdorff dimension of ${\mathcal{D}}_{k}$ is equal to $k(2n-1)-((n-1)/2)$. This extends a result of Cheung.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1401 - 1439
Copyright
© Cambridge University Press, 2017 

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