Simultaneous dense and nondense orbits for noncommuting toral endomorphisms

Simultaneous dense and nondense orbits for noncommuting toral endomorphisms Let S and T be hyperbolic endomorphisms of $${\mathbb {T}}^d$$ T d with the property that the span of the subspace contracted by S along with the subspace contracted by T is $${\mathbb {R}}^d$$ R d . We show that the intersection of the set of points with equidistributing orbit under S with the set of points with nondense orbit under T has maximal Hausdorff dimension. In the case that S and T are quasihyperbolic automorphisms, we prove that the Hausdorff dimension of the intersection is again maximal when we assume that $${\mathbb {R}}^d$$ R d is spanned by the subspaces contracted by S and T along with the central eigenspaces of S and T. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

Simultaneous dense and nondense orbits for noncommuting toral endomorphisms

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Publisher
Springer Vienna
Copyright
Copyright © 2018 by Springer-Verlag GmbH Austria, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-018-1154-2
Publisher site
See Article on Publisher Site

Abstract

Let S and T be hyperbolic endomorphisms of $${\mathbb {T}}^d$$ T d with the property that the span of the subspace contracted by S along with the subspace contracted by T is $${\mathbb {R}}^d$$ R d . We show that the intersection of the set of points with equidistributing orbit under S with the set of points with nondense orbit under T has maximal Hausdorff dimension. In the case that S and T are quasihyperbolic automorphisms, we prove that the Hausdorff dimension of the intersection is again maximal when we assume that $${\mathbb {R}}^d$$ R d is spanned by the subspaces contracted by S and T along with the central eigenspaces of S and T.

Journal

Monatshefte f�r MathematikSpringer Journals

Published: Jan 16, 2018

References

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