Cylinder absolute games on solenoids

Abstract

Let $A$ be any affine surjective endomorphism of a solenoid $\Sigma_{\mathcal{P}}$ over the circle $S^1$ which is not an infinite-order translation of $\Sigma_{\mathcal{P}}$. We prove the existence of a cylinder absolute winning (CAW) subset $F \subset \Sigma_{\mathcal{P}}$ with the property that for any $x \in F$, the orbit closure $\overline{\{ A^{\ell} x \mid \ell \in \mathbb{N} \}}$ does not contain any periodic orbits. The class of infinite solenoids considered in this paper provides, to our knowledge, some of the first examples of non-Federer spaces where absolute games can be played and won. Dimension maximality and incompressibility of CAW sets is also discussed for a number of possibilities in addition to their winning nature for the games known from before.