Schmidt’s winning sets in $S$-gap shifts
Volume 264 / 2024
Abstract
We discuss Schmidt’s winning sets in $S$-gap shifts $(\Sigma _S, \sigma )$. Roughly speaking, we say that a dynamical system has the \emph {non-dense orbits winning (NDOW) property} if the exceptional sets of non-dense orbits are all winning. We prove that if $S\subset \mathbb {Z}_+$ either is a finite nontrivial subset or is piecewise syndetic, then $(\Sigma _S, \sigma )$ has the NDOW property; on the other hand, if $S$ is the set of all primes, then $(\Sigma _S, \sigma )$ does not have the NDOW property. Additionally, we show that all exceptional sets in any $S$-gap shift have positive Hausdorff dimension, but there exists an invertible symbolic dynamical system with positive Hausdorff dimension and an exceptional set of some point such that the exceptional set is a countable winning set (and hence has zero Hausdorff dimension).
