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Hausdorff dimension of divergent trajectories on homogeneous spaces

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Ergodic theory

Published online by Cambridge University Press:  19 December 2019

Lifan Guan  and
Ronggang Shi
Lifan Guan
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK email [email protected]
Ronggang Shi
Affiliation:
Shanghai Center for Mathematical Sciences, Fudan University, Shanghai200433, PR China email [email protected]
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Abstract

For a one-parameter subgroup action on a finite-volume homogeneous space, we consider the set of points admitting divergent-on-average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Cheung [Hausdorff dimension of the set of singular pairs, Ann. of Math. (2) 173 (2011), 127–167].

Keywords

homogeneous dynamicsdivergent trajectoryHausdorff dimension

MSC classification

Primary: 37A17: Homogeneous flows
Secondary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 2 , February 2020 , pp. 340 - 359
Copyright
© The Authors 2019

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Footnotes

1

Current address: Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3-5, D-37073 Gottingen, Germany

L. G. is supported by EPSRC Programme Grant EP/J018260/1 and R. S. is supported by NSFC 11871158.

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