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The Chebotarev theorem for ergodic toral automorphisms with respect to (G, ρ)–extensions

The Chebotarev theorem for ergodic toral automorphisms with respect to (G, ρ)–extensions We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism à with respect to a (G, ρ)–extension A. In particular we obtain an analogue of the Chebotarev theorem in this situation which is an asymptotic formula for the number of closed orbits of the base transformation according to how they lift onto the extension space. To arrive at this result we introduce a cyclic extension  of A and deduce that  and A is essentially a group extension and homogeneous extension of à respectively. This observation of a group extension is similar to the setting previously studied by Parry & Pollicott and using the prime orbit theorem of Waddington we then derive at an auxiliary result for the group extension analogoues to Parry & Pollicott. Finally we relate this auxiliary result to the homogeneous extension by resorting to the work of Noorani & Parry. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

The Chebotarev theorem for ergodic toral automorphisms with respect to (G, ρ)–extensions

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Publisher
Springer Journals
Copyright
Copyright © 2003 by Sociedade Brasileira de Matemática
Subject
Mathematics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-003-0013-4
Publisher site
See Article on Publisher Site

Abstract

We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism à with respect to a (G, ρ)–extension A. In particular we obtain an analogue of the Chebotarev theorem in this situation which is an asymptotic formula for the number of closed orbits of the base transformation according to how they lift onto the extension space. To arrive at this result we introduce a cyclic extension  of A and deduce that  and A is essentially a group extension and homogeneous extension of à respectively. This observation of a group extension is similar to the setting previously studied by Parry & Pollicott and using the prime orbit theorem of Waddington we then derive at an auxiliary result for the group extension analogoues to Parry & Pollicott. Finally we relate this auxiliary result to the homogeneous extension by resorting to the work of Noorani & Parry.

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Jan 1, 2003

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