# American Institute of Mathematical Sciences

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November  2018, 38(11): 5883-5895. doi: 10.3934/dcds.2018255

## Open maps: Small and large holes with unusual properties

 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

Received  February 2018 Revised  June 2018 Published  August 2018

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

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.

Citation: Kevin G. Hare, Nikita Sidorov. Open maps: Small and large holes with unusual properties. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5883-5895. doi: 10.3934/dcds.2018255
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