Open maps: Small and large holes with unusual properties

Kevin G. Hare; Nikita Sidorov

doi:10.3934/dcds.2018255

Article Contents

2018, Volume 38, Issue 11: 5883-5895. Doi: 10.3934/dcds.2018255

This issue Previous Article Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian Next Article Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow

Open maps: Small and large holes with unusual properties

Kevin G. Hare^1, , and
Nikita Sidorov^2,

1.
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
2.
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

* Corresponding author: Kevin G. Hare
* Corresponding author: Kevin G. Hare

Received: February 2018

Revised: June 2018

Published online: November 2018

Research of K. G. Hare was supported by NSERC Grant RGPIN-2014-03154.

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Abstract

Let X be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in X. We show that there exists an arbitrarily small finite overlapping union of shifted cylinders which intersects every orbit under the shift map.

We also show that for any proper subshift Y of X there exists a finite overlapping unions of shifted cylinders such that its survivor set contains Y (in particular, it can have entropy arbitrarily close to the entropy of X). Both results may be seen as somewhat counter-intuitive.

Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.

Keywords:

Mathematics Subject Classification: Primary: 28D05; Secondary: 37B10.

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References

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