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arXiv · 1710.06491

Open maps: small and large holes with unusual properties

Kevin G. Hare·Nikita Sidorov
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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 exist arbitrarily small finite overlapping union of shifted cylinders which intersect 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.

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Kevin G. Hare, Nikita Sidorov. 2017-10-17. Open maps: small and large holes with unusual properties. https://doi.org/10.3934/dcds.2018255

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