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Session 39. Contributed talks
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Expanding maps and Anosov diffeomorphisms on infra-nilmanifolds |
Jonas Deré, KU Leuven Kulak, Belgium
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Infra-nilmanifolds play an important role in dynamical systems,
especially when studying expanding maps or Anosov
diffeomorphisms. Because of the algebraic nature of these manifolds,
questions about self-maps can be translated into questions about
endomorphisms of their fundamental group. In this way, it was shown
by M. Gromov in [3] that every expanding map on a closed
manifold is topologically conjugate to an affine
infra-nilendomorphism. Up till now it is unknown if a similar
statement also holds for Anosov diffeomorphisms, although some
partial results point in that direction, e.g. for infra-nilmanifolds
and for codimension one Anosov diffeomorphisms.
These results motivate the study of infra-nilmanifolds admitting an
Anosov diffeomorphism or an expanding map. In this talk I will show
how these questions are related to the existence of certain
automorphisms of nilpotent Lie algebras. As a consequence, we can
classify all infra-nilmanifolds modeled on a free nilpotent Lie
group admitting an Anosov diffeomorphism, see [2]. Another
consequence is that for nilmanifolds, the existence of an expanding
map or a non-trivial self-covering does only depend on the
commensurability class of the fundamental group of the manifold,
answering a question stated in [1]. This allows us to
construct examples of nilmanifolds admitting an Anosov
diffeomorphism but no expanding map.
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References- I. Belegradek, On co-{H opfian nilpotent
groups, The Bulletin of the London Mathematical Society 6, 2003, 805-811.
- K. Dekimpe, J. Deré, Existence of Anosov
diffeomorphisms on infra-nilmanifolds modeled on free nilpotent
Lie groups , preprint, 2013, arXiv:1304.6529.
- M. Gromov, Groups of polynomial growth and
expanding maps , Institut des Hautes \'Etudes Scientifiques 53,
1981, 53-73.
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