Cylinder absolute games on solenoids

TL;DR

This paper demonstrates the existence of large, robust sets within solenoids where points' orbits under certain affine endomorphisms avoid periodic orbits, expanding the scope of absolute winning sets in non-Federer spaces.

Contribution

It introduces the first examples of non-Federer spaces where cylinder absolute winning sets can be constructed and analyzed, specifically in the context of affine endomorphisms on solenoids.

Findings

Existence of CAW sets in solenoids avoiding periodic orbits

Dimension maximality and incompressibility of CAW sets established

Solenoids as new examples for absolute game theory in non-Federer spaces

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.