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.

Typo in the statement of main theorem corrected, Abstract expanded